CH-24 Liability-Driven Strategies – Bond Markets, Analysis and Strategies : Book Guide

CHAPTER 24 LIABILITY-DRIVEN STRATEGIES CHAPTER SUMMARY In this chapter, we begin with the basic principles underlying the management of assets relative to liabilities, popularly referred to as asset/liability management. We then describe several structured portfolio strategies, strategies that seek to match the performance of a predetermined benchmark. We discuss liability-driven strategies that select assets so that cash flows will equal or exceed a client’s liabilities. In the last section of this chapter, we discuss liability-driven strategies for defined benefit pension plans. GENERAL PRINCIPLES OF ASSET/LIABILITY MANAGEMENT The nature of an institutional investor’s liabilities will dictate the investment strategy it will request its portfolio manager to pursue. Classification of Liabilities A liability is a cash outlay that must be made at a specific time to satisfy the contractual terms of an issued obligation. An institutional investor is concerned with both the amount and timing of liabilities, because its assets must produce the cash to meet any payments it has promised to make in a timely way. The descriptions of cash outlays as either known or uncertain are undoubtedly broad. When we refer to a cash outlay as being uncertain, we do not mean that it cannot be predicted. There are some liabilities for which “law of large numbers” makes it easier to predict the timing and/or amount of cash outlays. A type I liability is one for which both the amount and timing of the liabilities are known with certainty. Type I liabilities, however, are not limited to depository institutions. A major product sold by life insurance companies is a guaranteed investment contract (GIC). A type II liability is one for which the amount of cash outlay is known but the timing of the cash outlay is uncertain. The most obvious example of a type II liability is a life insurance policy. A type III liability is one for which the timing of the cash outlay is known but the amount is uncertain. A two-year floating-rate CD in which the interest rate resets quarterly based on a market interest rate is an example. A type IV liability is one in which there is uncertainty as to both the amount and timing of the cash outlay. Probably the most obvious examples are insurance policies issued by property and casualty insurance companies. Liquidity Concerns Because of uncertainty about the timing and/or the amount of the cash outlays, an institution must be prepared to have sufficient cash to satisfy its obligations. Also keep in mind that the entity that holds the obligation against the institution may have the right to change the nature of the obligation, perhaps incurring a penalty. In addition to uncertainty about the timing and amount of the cash outlays and the potential for the depositor or policyholder to withdraw cash early or borrow against a policy, an institution has to be concerned with possible reduction in cash inflows. In the case of a depository institution, this means the inability to obtain deposits. Surplus Management The two goals of a financial institution are to earn an adequate return on funds invested, and to maintain a comfortable surplus of assets beyond liabilities. The task of managing funds of a financial institution to accomplish these goals is referred to as asset/liability management or surplus management. Institutions may calculate three types of surpluses: economic, accounting, and regulatory. The method of valuing assets and liabilities greatly affects the apparent health of a financial institution. Unrealistic valuation, although allowable under accounting procedures and regulations, is not sound investment practice. The economic surplus of any entity is the difference between the market value of all its assets and the market value of its liabilities; that is, The economic surplus can be expressed as economic surplus = market value of assets – present value of liabilities. If interest rates rise, both the assets and liabilities decline, so the economic surplus can increase, decrease, or not change. If the duration of the assets is greater than the duration of the liabilities, the economic surplus will increase if interest rates fall. The net effect on the surplus depends on the duration or interest-rate sensitivity of the assets and liabilities, so it is imperative that portfolio managers be able to measure this sensitivity for all assets and liabilities accurately. Accounting surplus is surplus of assets over liabilities as reported in financial statements based on generally accepted accounting principles (GAAP) accounting. Institutional investors must prepare periodic financial statements that include the reporting of assets and liabilities. With respect to the financial reporting of assets, there are three possible methods for reporting: amortized cost or historical cost, market value, or the lower of cost or market value. FASB 115 specifies which of these three methods must be followed for assets. Specifically, the accounting treatment required for a security depends on how the security is classified. There are three classifications of investment accounts: held to maturity, available for sale, and trading. The held-to-maturity account includes assets that the institution plans to hold until they mature. An asset is classified as in the available-for-sale account if the institution does not have the ability to hold the asset to maturity or intends to sell it. An asset that is acquired for the purpose of earning a short-term trading profit from market movements is classified in the trading account. For all assets in the available-for-sale and trading accounts, market value accounting is used. Institutional investors that are regulated at the state or federal levels must provide financial reports to regulators based on regulatory accounting principles (RAP). The surplus as measured using RAP accounting, called regulatory surplus, may, as in the case of accounting surplus, differ materially from economic surplus. IMMUNIZATION OF A PORTFOLIO TO SATISFY A SINGLE LIABILITY An immunization strategy refers to the investment of the assets in such a way that the existing business is immune to a general change in the rate of interest. Investing in a coupon bond with a yield to maturity equal to the target yield and a maturity equal to the investment horizon does not assure that a portfolio’s target accumulated value will be achieved. This is because an increase in the market yield causes the market value to fall and the portfolio can fail to achieve the target accumulated value. This occurs when the fall in principal is greater than any increase in reinvestment rate. In other words the interest rate (or price) risk has a greater impact that the reinvestment risk. To avoid this loss (and immunize its portfolio from interest rate changes), the portfolio manager should look for a coupon bond so that however the market yield changes, the change in the interest on interest will be offset by the change in the price. Consider, for example, an eight-year 10.125% coupon bond selling at 88.20262 to yield 12.5%. Suppose that $10,000,000 of par value of this bond is purchased for $8,820,262. For this bond, it can be shown that the accumulated value and the total return are never less than the target accumulated value and the target yield. Thus the target accumulated value is assured regardless of what happens to the market yield. For this situation, the coupon issue has the same duration as the liability. The equality of the duration of the asset and the duration of the liability is the key to immunization. When generalizing this observation to portfolios, the key is to immunize a portfolio’s target accumulated value (target yield). To do this, a portfolio manager must construct a bond portfolio such that the duration of the portfolio is equal to the duration of the liability, and the present value of the cash flow from the portfolio equals to the present value of the future liability. Rebalancing an Immunized Portfolio The principles underlying immunization need not just assume a one-time instantaneous change in the market yield. Even in the face of changing market yields, a portfolio can be immunized if it is rebalanced periodically so that its duration is equal to the duration of the liability’s remaining time. On the one hand, the more frequent rebalancing increases transactions costs, thereby reducing the likelihood of achieving the target yield. On the other hand, less frequent rebalancing will result in the duration wandering from the target duration, which will also reduce the likelihood of achieving the target yield. Immunization Risk The sufficient condition for the immunization of a single liability is that the duration of the portfolio be equal to the duration of the liability. However, a portfolio will be immunized against interest-rate changes only if the yield curve is flat and any changes in the yield curve are parallel changes (i.e., interest rates move either up or down by the same number of basis points for all maturities). Immunization risk is the risk of reinvestment. The portfolio that has the least reinvestment risk will have the least immunization risk. When there is a high dispersion of cash flows around the liability due date, the portfolio is exposed to high reinvestment risk. When the cash flows are concentrated around the liability due date, as in the case of the bullet portfolio, the portfolio is subject to low reinvestment risk. Researchers have developed a measure of immunization risk, which demonstrates that if the yield curve shifts in any arbitrary way, the relative change in the portfolio value will depend on the product of two terms. The first term depends solely on the characteristics of the investment portfolio. The second term is a function of interest-rate movement only. The second term characterizes the nature of the change in the shape of the yield curve. Because that change will be impossible to predict a priori, it is not possible to control for it. The first term, however, can be controlled for when constructing the immunized portfolio, because it depends solely on the composition of the portfolio. This first term, then, is a measure of risk for immunized portfolios and is equal to where CFt = cash flow of the portfolio at time period t, H = length (in years) of the investment horizon or liability due date, y = portfolio yield, and n = time to receipt of the last cash flow. The objective in constructing an immunized portfolio is to match the portfolio’s duration to the liability’s duration and select the portfolio that minimizes the immunization risk. The immunization risk measure can be used to construct approximate confidence intervals for the target yield and the target accumulated value. Zero-Coupon Bonds and Immunization An alternative approach to immunizing a portfolio against changes in the market yield is to invest in zero-coupon bonds with a maturity equal to the investment horizon. This is consistent with the basic principle of immunization, because the duration of a zero-coupon bond is equal to the liability’s duration. However, in practice, the yield on zero-coupon bonds is typically lower than the yield on coupon bonds making this strategy more costly. Credit Risk and the Target Yield The target yield may not be achieved if any of the bonds in the portfolio default or decrease in value because of credit quality deterioration. Call Risk When the universe of acceptable issues includes corporate bonds, the target yield may be jeopardized if a callable issue is included that is subsequently called. Call risk can be avoided by restricting the universe of acceptable bonds to noncallable bonds and deep-discount callable bonds. Constructing the Immunized Portfolio When the universe of acceptable issues is established and any constraints are imposed, the portfolio manager has a large number of possible securities from which to construct an initial immunized portfolio and from which to select to rebalance an immunized portfolio. An objective function can be specified, and a portfolio that optimizes the objective function using mathematical programming tools can be determined. Contingent Immunization Contingent immunization is a strategy that consists of identifying both the available immunization target rate and a lower safety net level return with which the investor would be minimally satisfied. To illustrate this strategy, suppose that a client investing $50 million is willing to accept a 10% rate of return over a four-year investment horizon at a time when a possible immunized rate of return is 12%. The 10% return is called the safety net return. The difference between the immunized return and the safety net return is called the safety cushion. In our example, the safety cushion is 200 basis points (12% minus 10%). The three key factors in implementing a contingent immunization strategy are (i) establishing accurate immunized initial and ongoing available target returns, (ii) identifying a suitable and immunizable safety net return, and (iii) designing an effective monitoring procedure to ensure that the safety net return is not violated. STRUCTURING A PORTFOLIO TO SATISFY MULTIPLE LIABILITIES For pension funds, there are multiple liabilities that must be satisfied—payments to the beneficiaries of the pension fund. A stream of liabilities must also be satisfied for a life insurance company that sells an insurance policy requiring multiple payments to policyholders, such as an annuity policy. There are two strategies that can be used to satisfy a liability stream: multiperiod immunization, and cash flow matching. Multiperiod Immunization Multiperiod immunization is a portfolio strategy in which a portfolio is created that will be capable of satisfying more than one predetermined future liability regardless if interest rates change. Even if there is a parallel shift in the yield curve, it has been demonstrated that matching the duration of the portfolio to the duration of the liabilities is not a sufficient condition to immunize a portfolio seeking to satisfy a liability stream. Instead, it is necessary to decompose the portfolio payment stream in such a way that each liability is immunized by one of the component streams. The key to understanding this approach is recognizing that the payment stream on the portfolio, not the portfolio itself, must be decomposed in this manner. There may be no actual bonds that would give the component payment stream. Cash Flow Matching An alternative to multiperiod immunization is cash flow matching. This approach, also referred to as dedicating a portfolio, can be summarized as follows. A bond is selected with a maturity that matches the last liability stream. An amount of principal plus final coupon equal to the amount of the last liability stream is then invested in this bond. The remaining elements of the liability stream are then reduced by the coupon payments on this bond, and another bond is chosen for the new, reduced amount of the next-to-last liability. Going backward in time, this cash flow matching process is continued until all liabilities have been matched by the payment of the securities in the portfolio. A popular variation of multiperiod immunization and cash flow matching to fund liabilities is one that combines the two strategies. This strategy, referred to as combination matching or horizon matching, creates a portfolio that is duration matched with the added constraint that it be cash matched in the first few years, usually five years. EXTENSIONS OF LIABILITY-DRIVEN STRATEGIES Deterministic models assume that the cash flows from assets and liabilities are known with certainty. However, most non-Treasury securities have embedded options that permit the borrower or the investor to alter the cash flows. A number of models have been developed to handle real-world situations in which liability payments and/or asset cash flows are uncertain. Such models are called stochastic models. Such models require that the portfolio manager incorporate an interest-rate model, that is, a model that describes the probability distribution for interest rates. Optimal portfolios then are solved for using a mathematical programming technique known as stochastic programming. There is increasing awareness that stochastic models reduce the likelihood that the liability objective will not be satisfied and that transactions costs can be reduced through less frequent rebalancing of a portfolio derived from these models. COMBINING ACTIVE AND IMMUNIZATION STRATEGIES In contrast to an immunization strategy, an active/immunization combination strategy is a mixture of two strategies that are pursued by the portfolio manager at the same point in time. The immunization component of this strategy could be either a single-liability immunization or a multiple-liability immunization. This component involves an adaptive strategy based on an initial set of liabilities and modified over time to changes in future liabilities. The active portion would continue to be free to maximize expected return, given some acceptable risk level. The following formula can be used to determine the portion of the initial portfolio to be managed actively, with the balance immunized: active component = . In the formula it is assumed that the immunization target return is greater than either the minimum return established by the client or the expected worst-case return from the actively managed portion of the portfolio. LIABILITY-DRIVEN STRATEGIES FOR DEFINED BENEFIT PENSION FUNDS While a defined benefit plan can use an immunization or cash flow matching strategy that has bonds only, the problem is that the liabilities are uncertain due to factors such as changes in the contractual benefits provided by the plan sponsor, the decision by plan beneficiaries to retire early, and the impact of inflation on benefits. In recent years, several liability-driven strategies for pension funds that allow for investing outside of the bond universe have been proposed. These proposals have come about as a result of the poor performance of defined pension plans since the turn of the century due to unfavorable conditions in the capital markets and poor asset allocation decisions. More specifically, a measure of the performance of a pension fund is the funding ratio, which is the ratio of the market value of the assets to the value of the liabilities. There are basically two camps as to what the suitable strategy is to employ in the allocation of assets for defined pension plans. One camp is a bond-only camp. The other camp believes that the liability characteristics of pension funds require the use of equities. Both camps, however, agree that the benchmark should be the liabilities, not general market benchmarks for major asset classes. Ross, Bernstein, Ferguson, and Dalio of Bridgewater Associates propose the following liability-driven strategy for a pension plan, which involves two steps. First, create an immunizing portfolio. The purpose of this portfolio is to hedge the adverse consequences associated with the exposure to the liabilities. Second, create what they refer to as an “excess return portfolio.” The purpose of that portfolio is to generate a return that exceeds the return on the immunizing portfolio. The total return for the pension plan is then total plan return = return on liability-immunizing portfolio + return on excess return portfolio–return on liabilities. The issue associated with liability-driven strategies applied to defined pension plans must deal with risks that may not be handled adequately by investing in the major asset classes. The more obvious of the two risks is the impact of inflation on future pension liabilities. An increase in the inflation rate increases future liabilities as salaries are adjusted upwards. The second of the two risks is longevity risk. This is the risk that beneficiaries may live longer and as a result future liabilities will exceed current actuarial determined liabilities. KEY POINTS • The nature of their liabilities, as well as regulatory considerations, determines the investment strategy pursued by all institutional investors. By nature, liabilities vary with respect to the amount and timing of their payment. • The economic surplus of any entity is the difference between the market value of all its assets and the present value of its liabilities. Institutional investors will pursue a strategy either to maximize economic surplus or to hedge economic surplus against any adverse change in market conditions. • In addition to economic surplus, there are accounting surplus and regulatory surplus. The former is based on GAAP accounting, specifically, FASB 115, and the latter on RAP accounting. To the extent that these two surplus measures may not reflect the true financial condition of an institution, future financial problems may arise. • Liability-driven strategies involve designing a portfolio to produce sufficient funds to satisfy liabilities whether or not interest rates change. • When there is only one future liability to be funded, an immunization strategy can be used. An immunization strategy is designed so that as interest rates change, interest-rate risk and reinvestment risk will offset each other in such a way that the minimum accumulated value (or minimum rate of return) becomes the target accumulated value (or target yield). • An immunization strategy requires that a portfolio manager create a bond portfolio with a duration equal to the duration of the liability. • Because immunization theory is based on parallel shifts in the yield curve, the risk is that a portfolio will not be immunized even if the duration-matching condition is satisfied. Immunization risk can be quantified so that a portfolio that minimizes this risk can be constructed. • When there are multiple liabilities to be satisfied, either multiperiod immunization or cash flow matching can be used. Multiperiod immunization is a duration-matching strategy that exposes the portfolio to immunization risk. The cash flow–matching strategy does not impose any duration requirement. Although the only risk that the liabilities will not be satisfied is that issues will be called or will default, the dollar cost of a cash flow–matched portfolio may be higher than that of a portfolio constructed using a multiperiod immunization strategy. • Liability-driven strategies in which the liability payments and the asset cash flows are known with certainty are deterministic models. In a stochastic model, either the liability payments or the asset cash flows, or both, are uncertain. Stochastic models require specification of a probability distribution for the process that generates interest rates. • A combination of active and immunization strategies can be pursued. Allocation of the portion of the portfolio to be actively managed is based on the immunization target rate, the minimum return acceptable to the client, and the expected worst-case return from the actively managed portfolio. In a contingent immunization strategy, a portfolio manager is either actively managing the portfolio or immunizing it. Because both strategies are not pursued at the same time, contingent immunization is not a combination or mixture strategy. • The liability structure of a defined benefit pension plan is uncertain. There are two liability-driven strategies advocated for defined benefit pension plans. One approach argues that only bonds should be acquired and a dedicated portfolio strategy should be used. The other approach is a liability-driven strategy that uses bonds and equities but uses the liabilities as a benchmark in determining the best asset allocation. ANSWERS TO QUESTIONS FOR CHAPTER 24 (Questions are in bold print followed by answers.) 1. What are the two dimensions of a liability? The two important dimensions of a liability are the amount and timing of the payment. A liability is a cash outlay that must be made at a specific time to satisfy the contractual terms of an issued obligation. An institutional investor is concerned with both the amount and timing of liabilities, because its assets must produce the cash to meet any payments it has promised to make in a timely way. 2. Why is it not always simple to estimate the liability of an institution? It is not simple to estimate the liability of an institution because a liability can have uncertainty as to both the amount and timing of the cash outlay. When we refer to a cash outlay as being uncertain, we do not mean that it cannot be predicted. There are some liabilities for which “law of large numbers” makes it easier to predict the timing and/or amount of cash outlays. This work is typically done by actuaries, but even actuaries have difficulty predicting natural catastrophes such as floods and earthquakes. 3. Why is asset/liability management best described as surplus management? Asset/liability management is best described as surplus management because it aims at keeping and acquiring assets that are greater than liabilities by a specified amount. A measure of how great assets are to liabilities is called the surplus. The surplus of any entity can be defined as the difference between the value of all its assets and the value of its liabilities. Asset/liability management involves maintaining a surplus which is called a safety net. The asset/liability management task also involves a trade-off between controlling the risk of a decline in the surplus and taking on acceptable risks in order to earn an adequate return on the funds invested. 4. Answer the below questions. (a) What is the economic surplus of an institution? The economic surplus of any entity is the difference between the market value of all its assets and the market value of its liabilities; that is, economic surplus = market value of assets – market value of liabilities. The market value of liabilities is simply the present value of the liabilities, in which the liabilities are discounted at an appropriate interest rate. A rise in interest rates will therefore decrease the present value or market value of the liabilities; a decrease in interest rates will increase the present value or market value of liabilities. Thus the economic surplus can be expressed as economic surplus = market value of assets – present value of liabilities. (b) What is the accounting surplus of an institution? Accounting surplus is surplus of assets over liabilities as reported in financial statements based on generally accepted accounting principles (GAAP) accounting. The accounting treatment for assets is governed by Statement of Financial Accounting Standards No. 115, more popularly referred to as FASB 115. However, it does not deal with the accounting treatment for liabilities. The surplus as measured using GAAP accounting may differ materially from economic surplus. (c) What is the regulatory surplus of an institution? Institutional investors that are regulated at the state or federal levels must provide financial reports to regulators based on regulatory accounting principles (RAP). RAP accounting for a regulated institution need not use the same rules as set forth by FASB 115. Liabilities may or may not be reported at their present value, depending on the type of institution and the type of liability. The surplus as measured using RAP accounting, called regulatory surplus, may, as in the case of accounting surplus, differ materially from economic surplus. (d) Which surplus best reflects the economic well-being of an institution? Economic surplus best reflects the economic well-being of an institution because these values are market values and thus best reflect investor’s beliefs about the true financial condition of a firm. Surplus based on GAAP accounting, specifically, FASB 115, and on RAP accounting may not reflect the true financial condition of an institution. Thus, using these two surplus measures can hide future financial problems. (e) Under what circumstances are all three surplus measures the same? For the three methods to produce the same surplus, the liabilities and assets must all be reported or given as the same. 5. Suppose that the present value of the liabilities of some financial institution is $600 million and the surplus $800 million. The duration of the liabilities is equal to 5. Suppose further that the portfolio of this financial institution includes only bonds and the duration for the portfolio is 6. Answer the below questions. (a) What is the market value of the portfolio of bonds? We rearrange the equation of “economic surplus = market value of assets – market value of liabilities” to get: market value of assets = economic surplus + market value of liabilities. Inserting in our given values, we get: market value of assets = $800 million + $600 million = $1,400 million or $1.4 billion. (b) What does a duration of 6 mean for the portfolio of assets? Duration is a measure of the responsiveness of cash flows to changes in interest rates. Duration can be calculated for liabilities in the same way in which it is calculated for assets. Because the duration of the assets is 6, the market value of the assets will change by about 6% for a 100 basis points change in interest rates. Thus, the increase in the market value of the assets is about 0.06($1.4 billion) = $84 million for a 100 basis points decrease in interest rates. Similarly, the decrease in the market value of the assets is about 0.06($1.4 billion) = $84 million for a 100 basis points increase in interest rates. (c) What does a duration of 5 mean for the liabilities? If the duration of the liabilities is 5, the present value of the liabilities will increase by 5% or 0.05($600 million) = $30 million for a 100 basis point decrease in interest rates. Similarly, it will decrease by $30 million for every 100 basis point increase in interest rates. (d) Suppose that interest rates increase by 50 basis points; what will be the approximate new value for the surplus? Given that we have a 50 basis point change instead of a 100 basis point change, we divided the duration by 2. Doing this gives causes the decrease in assets to be = $42 million. Similarly, the decrease in liabilities will be = $15 million. Thus, the net change in surplus value is: decrease in assets  decrease in liabilities = ($42 million)  ($15 million) = $27 million. Thus, the surplus will experience a negative change where the new value for surplus is: old surplus value + change in surplus value = $800 million + ($27 million) = $773 million. (e) Suppose that interest rates decrease by 50 basis points; what will be the approximate new value for the surplus? Given that we have a 50 basis point change instead of a 100 basis point change, we divided the duration by 2. Doing this gives causes the increase in assets to be = $42 million. Similarly, the increase in liabilities will be = $15 million. Thus, the net change in surplus value is: increase in assets  increase in liabilities = $42 million  $15 million = $27 million. Given the net change in surplus value of $27 million, the surplus will experience a positive change where the new value for surplus is: old surplus value + change in surplus value = $800 million + $27 million = $827 million. 6. Answer the below questions. (a) Why is the interest-rate sensitivity of an institution’s assets and liabilities important? Knowing the interest-rate sensitivity reveals steps that a firm’s manager can perform to avoid financial difficulties caused by being short of assets relative to liabilities. More details are given below. To see why the interest-rate sensitivity of an institution’s assets and liabilities important, consider an institution that has a portfolio consisting only of bonds and liabilities. If interest rates rise, both the bonds and liabilities will decline in value. However, the economic surplus can increase, decrease, or not change. The net effect on economic surplus depends on the relative interest rate sensitivity of the assets compared to the liabilities. Because duration is a measure of the responsiveness of cash flows to changes in interest rates, duration can be calculated for liabilities in the same way in which it is calculated for assets. Thus the duration of liabilities measures their responsiveness to a change in interest rates. If the duration of the assets is greater than the duration of the liabilities, the economic surplus will increase if interest rates fall. (b) In 1986, Martin Leibowitz of Salomon Brothers Inc. wrote a paper titled “Total Portfolio Duration: A New Perspective on Asset Allocation.” What do you think a total portfolio duration means? One would think that total portfolio duration refers to the duration of all assets together. If bonds in the portfolio are all assumed to be option-free bonds, total portfolio duration means modified duration can be used to measure duration. However, when portfolios include securities with embedded options, the effective duration is used. For most institutional portfolios, total portfolio duration will likely mean an effective duration. 7. If an institution has liabilities that are interest-rate sensitive and invests in a portfolio of common stocks, can you determine what will happen to the institution’s economic surplus if interest rates change? The stock market is generally influenced like bonds and other fixed-income securities when interest rates change. That is, stock values fall when interest rates increase and values rise when rates fall. However, there is much more volatility for common stocks than for fixed-income securities. Although it would involve some difficulty to predict any precise relationship between common stock and the institution’s economic surplus if interest rates change, generally one can say that assets would be more volatile relative to liabilities. In conclusion, if interest rates went down then assets would go up more than liabilities giving a greater surplus; the opposite would occur if rates went up. 8. The following quote is taken from Phillip D. Parker (Associate General Counsel of the SEC), “Market Value Accounting—An Idea Whose Time Has Come?” in Elliot P. Williams (ed.), Managing Asset/Liability Portfolios (Charlottesville, VA: Association for Investment Management and Research, 1991), published prior to the passage of FASB 115: “The use of market value accounting would eliminate any incentive to sell or retain investment securities for reasons of accounting treatment rather than business utility.” Explain why this statement is correct. (Note that in historical accounting a loss is recognized only when a security is sold.) Accounting treatment that occurs in annual reports is concerned with making the firm’s book numbers look good. Thus, it is not primarily concerned with providing numbers that reflect market values especially when market value numbers are poor. If book numbers can be manipulated by accounting practices to paint a rosy picture of the company’s earnings and well-being, then the company’s accountants will do that. Thus, the statement is correct when it suggests that using market value accounting can eliminate practices concerned with overvaluing that company’s assets. This statement is especially true to the extent that market values differ from book values and accounting practices do not properly adjust for market numbers unless the asset is sold. 9. Answer the below questions. (a) Indicate why you agree or disagree with the following statement: “Under FASB 115 all assets must be marked to market.” As explained below in more detail, FASB 115 does not require that all assets must be marked to market even though many have to be marked to market. Institutional investors must prepare periodic financial statements prepared in accordance with GAAP. Thus the assets and liabilities reported are based on GAAP accounting. The accounting treatment for assets is governed by FASB 115. However, it does not deal with the accounting treatment for liabilities. With respect to the financial reporting of assets, there are three possible methods for reporting: amortized cost or historical cost, market value, or the lower of cost or market value. Despite the fact that the real cash flow is the same regardless of the accounting treatment, there can be substantial differences in the financial statements using these three methods. In the amortized cost method, the value reported in the balance sheet reflects an adjustment to the acquisition cost for debt securities purchased at a discount or premium from their maturity value. In the market value accounting method, the balance sheet reported value of an asset is its market value. When an asset is reported in the financial statements of an institution at its market value, it is said to be “marked to market.” Finally, the lower of cost or market method requires comparison of market value to the amortized cost, with the lower of these two values reported in the balance sheet. The value reported cannot exceed the amortized cost. FASB 115 specifies which of these three methods must be followed for assets. Specifically, the accounting treatment required for a security depends on how the security is classified. There are three classifications of investment accounts: (1i) held to maturity, (ii) available for sale, and (iii) trading. The held-to-maturity account includes assets that the institution plans to hold until they mature. The assets classified in this account cannot be common stock because they have no maturity. For all assets in the held-to-maturity account, the amortized cost method must be used. An asset is classified as in the available-for-sale account if the institution does not have the ability to hold the asset to maturity or intends to sell it. An asset that is acquired for the purpose of earning a short-term trading profit from market movements is classified in the trading account. For all assets in the available-for-sale and trading accounts, market value accounting is used. (b) Indicate why you agree or disagree with the following statement: “The greater the price volatility of assets classified in the held-to-maturity account, the greater the volatility of the accounting surplus and reported earnings.” In general one might agree that if there is volatility in assets, there might be volatility in the account numbers that reflect this volatility. However, under FASB 115, the accounting treatment for any unrealized gain or loss depends on the account in which the asset is classified. Specifically, any unrealized gain or loss is ignored for assets in the held-to-maturity account. Thus, it would seem that this account would not generally have much volatility. Thus, for assets in this account there is no affect on reported earnings or the accounting surplus and thus no volatility. For the other two accounts, any unrealized gain or loss affects the accounting surplus. However, there is a difference as to how reported earnings are affected. For assets classified in the available-for-sale account, unrealized gains or losses are not included in reported earnings; in contrast, for assets classified in the trading account, any gains or losses are included in reported earnings. In conclusion, volatility in accounting surplus and reported earnings will reflect the manner in which assets are classified. 10. What is meant by immunizing a bond portfolio? Immunizing a bond portfolio means that the portfolio’s value is protected against a general change in the rate of interest. More details are given below. Investing in a coupon bond with a yield to maturity equal to the target yield and a maturity equal to the investment horizon does not assure that a portfolio’s target accumulated value will be achieved. This is because an increase in the market yield causes the market value to fall and the portfolio can fail to achieve the target accumulated value. This can occur when the fall in principal is greater than any increase in reinvestment rate. In other words the interest rate (or price) risk has a greater impact that the reinvestment risk. To avoid this loss (and immunize its portfolio from interest rate changes), the portfolio manager should look for a coupon bond so that however the market yield changes, the change in the interest on interest will be offset by the change in the price. The equality of the duration of the asset and the duration of the liability is the key to immunization. When generalizing this observation to portfolios, the key is to immunize a portfolio’s target accumulated value (target yield). To do this, a portfolio manager must construct a bond portfolio such that the duration of the portfolio is equal to the duration of the liability, and the present value of the cash flow from the portfolio equals to the present value of the future liability. 11. Answer the below questions. (a) What is the basic underlying principle in an immunization strategy? The basic underlying principle in an immunization strategy is to have the duration of the asset equal the duration of the liability. Generalizing this observation to bond portfolios from individual bonds, the key principle is: To immunize a portfolio’s target accumulated value (target yield), a portfolio manager must construct a bond portfolio such that (i) the duration of the portfolio is equal to the duration of the liability, and (ii) the present value of the cash flow from the portfolio equals to the present value of the future liability. (b) Why may the matching of the maturity of a coupon bond to the remaining time to maturity of a liability fail to immunize a portfolio? Investing in a coupon bond with a yield to maturity equal to the target yield and a maturity equal to the investment horizon does not assure that a portfolio’s target accumulated value will be achieved. This is because an increase in the market yield causes the market value to fall and the portfolio can fail to achieve the target accumulated value. This can occur when the fall in principal is greater than any increase in reinvestment rate. In other words the interest rate (or price) risk has a greater impact that the reinvestment risk. To avoid this loss (and immunize its portfolio from interest rate changes), the portfolio manager should look for a coupon bond so that however the market yield changes, the change in the interest on interest will be offset by the change in the price. 12. Why must an immunized portfolio be rebalanced periodically? The key to immunizing a portfolio of assets is to match the duration of the assets with the liabilities. Because market yields can change periodically affecting the duration of the assets, the portfolio of assets must be rebalanced periodically to insure immunization. More details are given below. Illustrations of the principles underlying immunization often assume a one-time instantaneous change in the market yield. In practice, the market yield will fluctuate over the investment horizon. As a result, the duration of the portfolio will change as the market yield changes. In addition, the duration will change simply because of the passage of time. Even in the face of changing market yields, a portfolio can be immunized if it is rebalanced so that its duration is equal to the duration of the liability’s remaining time. For example, if the liability is initially 5.5 years, the initial portfolio should have a duration of 5.5 years. After six months the liability will be five years, but the duration of the portfolio will probably be different from five years. This is because duration depends on the remaining time to maturity and the new level of yields, and there is no reason why the change in these two values should reduce the duration by exactly six months. Thus the portfolio must be rebalanced so that its duration is five years. Six months later the portfolio must be rebalanced again so that its duration will equal 4.5 years. And so on. There is the question of how often the portfolio should be rebalanced to adjust its duration. On the one hand, the more frequent rebalancing increases transactions costs, thereby reducing the likelihood of achieving the target yield. On the other hand, less frequent rebalancing will result in the duration wandering from the target duration, which will also reduce the likelihood of achieving the target yield. Thus the portfolio manager faces a tradeoff in that some transactions costs must be accepted to prevent the duration from straying from its target, but some adjustment in the duration must be accepted or transactions costs will become prohibitively high. 13. What are the risks associated with a bond immunization strategy? As described below, there are risks that can upset various types of bond immunization strategies. A first risk involves uncertainty as to how the yield curve might shift. For example, if there is a change in interest rates that does not correspond to the shape-preserving shift, matching the portfolio’s duration to the liability’s duration will not assure immunization. The sufficient condition for the immunization of a single liability is that the duration of the portfolio be equal to the duration of the liability. However, a portfolio will be immunized against interest-rate changes only if the yield curve is flat and any changes in the yield curve are parallel changes (i.e., interest rates move either up or down by the same number of basis points for all maturities). Duration is a measure of price volatility for parallel shifts in the yield curve. If there is a change in interest rates that does not correspond to this shape-preserving shift, matching the portfolio’s duration to the liability’s duration will not assure immunization. That is, the target yield will no longer be the minimum total return for the portfolio. A second risk involves the reinvestment rate. For example, consider the example in the text where the accumulated value for a barbell portfolio at the liability due date misses the target accumulated value by more than a bullet portfolio. There are two reasons for this. First, the lower reinvestment rates are experienced on the barbell portfolio for larger interim cash flows over a longer time period than on the bullet portfolio. Second, the portion of the barbell portfolio still outstanding at the end of the liability due date is much longer than the maturity of the bullet portfolio, resulting in a greater capital loss for the barbell than for the bullet. Thus the bullet portfolio has less risk exposure than the barbell portfolio to any changes in the interest-rate structure that might occur. What should be evident from this analysis is that immunization risk is the risk of reinvestment. The portfolio that has the least reinvestment risk will have the least immunization risk. When there is a high dispersion of cash flows around the liability due date, the portfolio is exposed to high reinvestment risk. When the cash flows are concentrated around the liability due date, the portfolio is subject to low reinvestment risk. 14. “I can immunize a portfolio simply by investing in zero-coupon Treasury bonds.” Comment on this statement. If all the cash flows are received at the liability due date, the immunization risk measure is zero. In such a case the portfolio is equivalent to a pure discount security (zero-coupon security) that matures on the liability due date. If a portfolio can be constructed that replicates a pure discount security maturing on the liability due date, that portfolio will be the one with the lowest immunization risk. Typically, however, it is not possible to construct such an ideal portfolio. More details are given below. An alternative approach to immunizing a portfolio against changes in the market yield is to invest in zero-coupon bonds with a maturity equal to the investment horizon. This is consistent with the basic principle of immunization, because the duration of a zero-coupon bond equals the liability’s duration. However, in practice, the yield on zero-coupon bonds is typically lower than the yield on coupon bonds. Thus using zero-coupon bonds to fund a bullet liability requires more funds, because a lower target yield (e.g., yield on the zero-coupon bond) is being locked in. Suppose, for example, that a portfolio manager must invest funds to satisfy a known liability of $20 million five years from now. If a target yield of 10% on a bond-equivalent basis (5% every six months) can be locked in using zero-coupon Treasury bonds, the funds necessary to satisfy the $20 million liability will be $12,278,260, the present value of $20 million using a discount rate of 10% (5% semiannually). Suppose, instead, that by using coupon Treasury securities, a target yield of 10.3% on a bond-equivalent basis (5.15% every six months) is possible. Then the funds needed to satisfy the $20 million liability will be $12,104,240, the present value of $20 million discounted at 10.3% (5.15% semiannually). Thus a target yield higher by 30 basis points would reduce the cost of funding the $20 million by $12,278,260 – $12,104,240 = $174,020. But the reduced cost comes at a price—the risk that the target yield will not be achieved. 15. Three portfolio managers are discussing a strategy for immunizing a portfolio so as to achieve a target yield. Manager A, whose portfolio consists of Treasury securities and option-free corporates, stated that the duration of the portfolio should be constructed so that the Macaulay duration of the portfolio is equal to the number of years until the liability must be paid. Manager B, with the same types of securities in his portfolio as Manager A, feels that Manager A is wrong because the portfolio should be constructed so that the modified duration of the portfolio is equal to the modified duration of the liabilities. Manager C believes Manager B is correct. However, unlike the portfolios of Managers A and B, Manager C invests in mortgage-backed securities and callable corporate bonds. Discuss the position taken by each manager and explain why they are correct. Manager A wants to use the Macaulay duration for his Treasury securities and option-free corporates. This measure is an acceptable measure if assets and liabilities are option-free. However, Manager A wants the duration of assets to equal the number of years until the liability must be paid. Suppose the duration of the assets is 5 and the duration of the liabilities is not 5 but say 3. If interest rates increase by 100 basis points, then the market value of the assets will increase by approximately 5%. The liabilities will also increase but only by about 3%. Thus, the economic surplus will increase. If the rates decrease by 100 basis points, the opposite would occur, i.e., the surplus would decrease. Manager B wants to overcome the problems faced by Manager A by matching the modified duration of the assets with the modified duration of the assets. Due to the relation between the modified duration and Macaulay duration, the modified duration analysis can be classified in terms of the latter. Manager C should agree with Manager B’s choice but should not agree that Manager B’s strategy applies to its portfolio. This is because Manager C’s portfolio contains securities with embedded option. When portfolios include securities with embedded options, the effective duration is used. 16. Why is there greater risk in a multiperiod immunization strategy than a cash flow-matching strategy? To understand the greater risk in a multiperiod immunization strategy, we need to first understand the differences between the cash flow matching and multiperiod immunization strategies. First, unlike the immunization approach, the cash flow matching approach has no duration requirements. Second, with immunization, rebalancing is required even if interest rates do not change. In contrast, no rebalancing is necessary for cash flow matching except to delete and replace any issue whose quality rating has declined below an acceptable level. Third, there is no risk that the liabilities will not be satisfied (barring any defaults) with a cash flow-matched portfolio. For a portfolio constructed using multiperiod immunization, there is immunization risk due to reinvestment risk. The differences just cited may seem to favor the use of cash flow matching. However, what we have ignored is the relative cost of the two strategies. Cash flow matching is more expensive because, typically, the matching of cash flows to liabilities is not perfect. This means that more funds than necessary must be set aside to match the liabilities. Optimization techniques used to design cash flow-matched portfolios assume that excess funds are reinvested at a conservative reinvestment rate. With multiperiod immunization, all reinvestment returns are assumed to be locked in at a higher target rate of return. In conclusion, portfolio managers face a trade-off in deciding between the two strategies: avoidance of the risk of not satisfying the liability stream under cash flow matching versus the lower cost attainable with multiperiod immunization. 17. Answer the below questions. (a) What is a contingent immunization strategy? A contingent immunization strategy is a strategy that consists of identifying both the available immunization target rate and a lower safety net level return with which the investor would be minimally satisfied. More details are given below. For a contingent immunization strategy, the portfolio manager pursues an active portfolio strategy until an adverse investment experience drives the then-available potential return—the combined active return from actual past experience and immunized return from expected future experience—down to the safety net level. When that point is reached, the portfolio manager is obligated to immunize the portfolio completely and lock in the safety net level return. As long as the safety net is not violated, the portfolio manager can continue to manage the portfolio actively. When the immunization mode is activated because the safety net is violated, the manager can no longer return to the active mode, unless, of course, the contingent immunization plan is abandoned. (b) What is the safety net cushion in a contingent immunization strategy? The safety cushion is the difference between the immunized return and the safety net return. More details including an illustration of the safety cushion is given below. Suppose that a client investing $50 million is willing to accept a 10% rate of return over a four-year investment horizon at a time when a possible immunized rate of return is 12%. The 10% return is called the safety net return. The difference between the immunized return and the safety net return is called the safety cushion. In our example, the safety cushion is 200 basis points (12% minus 10%). Because the initial portfolio value is $50 million, the minimum target value at the end of four years, based on semiannual compounding, is $50,000,000(1.05)8 = $73,872,772. The rate of return at the time is 12%, so the assets required at this time to achieve the minimum target value of $73,872,772 represent the present value of $73,872,772 discounted at 12% on a semiannual basis, which is $73,872,772/(1.06)8 = $43,348,691. Therefore, the safety cushion of 200 basis points translates into an initial dollar safety margin of $50,000,000 – $43,348,691 = $6,651,309. Had the safety net of return been 11% instead of 10%, the safety cushion would have been 100 basis points and the initial dollar safety margin, $1,855,935. In other words, a smaller safety cushion implies a smaller dollar safety margin. (c) Is it proper to classify a contingent immunization as a combination active/immunization strategy? In a contingent strategy, the portfolio manager is permitted to manage the portfolio actively until the safety net is violated. The manager could theoretically employ an active strategy for the whole period. While it is doing this, it is also following the guidelines of an immunization strategy. Given these considerations, one might classify a contingent immunization as a combination active/immunization strategy even though there is no guarantee that an active strategy can be employed for the whole period. However, strictly speaking contingent immunization is not a combination or mixture strategy. The portfolio manager is either in the immunization mode by choice or because the safety net is violated or in the active management mode. In contrast to an immunization strategy, an active/immunization combination strategy is a mixture of two strategies that are pursued by the portfolio manager at the same point in time. The immunization component of this strategy could be either a single-liability immunization or a multiple-liability immunization. In the single-liability immunization case, an assured return would be established so as to serve to stabilize the portfolio’s total return. In the multiple liability immunization case, the component to be immunized would be immunized now, with new requirements, as they become known, taken care of through reimmunization. This would be an adaptive strategy in that the immunization component would be based on an initial set of liabilities and modified over time to changes in future liabilities (e.g., for actuarial changes for the liabilities in the case of a pension fund). The active portion would continue to be free to maximize expected return, given some acceptable risk level. 18. What is a combination matching strategy? A popular variation of multiperiod immunization and cash flow matching to fund liabilities is one that combines the two strategies. This strategy, referred to as combination matching or horizon matching, creates a portfolio that is duration matched with the added constraint that it be cash matched in the first few years, usually five years. The advantage of combination matching over multiperiod immunization is that liquidity needs are provided for in the initial cash flow-matched period. Cash flow matching the initial portion of the liability stream reduces the risk associated with nonparallel shifts of the yield curve. The disadvantage of combination matching over multiperiod immunization is that the cost is slightly greater. 19. In a stochastic liability funding strategy, why is an interest-rate model needed? Changes in interest rates impact the cash flows for a stochastic liability funded strategy. Thus, an interest model is needed to track the cash flows. More details are given below. Since the mid-1980s, a number of models have been developed to handle real-world situations in which liability payments and/or asset cash flows are uncertain. Such models are called stochastic models. Such models require that the portfolio manager incorporate an interest-rate model, that is, a model that describes the probability distribution for interest rates. Optimal portfolios then are solved for using a mathematical programming technique known as stochastic programming. The complexity of stochastic models, however, has limited their application in practice. Nevertheless, they are gaining in popularity as more portfolio managers become comfortable with their sophistication. There is increasing awareness that stochastic models reduce the likelihood that the liability objective will not be satisfied and that transactions costs can be reduced through less frequent rebalancing of a portfolio derived from these models. 20. Suppose that a client has granted an asset management firm permission to pursue an active/immunized combination strategy. Suppose further that the minimum return expected by the client is 9% and that the asset management firm believes that an achievable immunized target return is 14% and the worst possible return from the actively managed portion of the portfolio is 1%. Approximately how much should be allocated to the active component of the portfolio? The following formula can be used to determine the portion of the initial portfolio to be managed actively, with the balance immunized: active component = . In the formula it is assumed that the immunization target return is greater than either the minimum return established by the client or the expected worst-case return from the actively managed portion of the portfolio. Inserting in our values we get: active component = = = 0.454545 or about 45.45%. 21. A liability-driven strategy for defined benefit pension funds is to create an immunizing portfolio and an excess return portfolio. Explain this strategy. Ross, Bernstein, Ferguson, and Dalio of Bridgewater Associates propose the following liability-driven strategy for a pension plan, which involves two steps. First, create an immunizing portfolio. The purpose of this portfolio is to hedge the adverse consequences associated with the exposure to the liabilities. Second, create what they refer to as an “excess return portfolio.” The excess return is the difference between the expected return of the portfolio minus the risk-free rate, with the risk-free rate being a Treasury rate. The purpose of the excess return portfolio is to generate a return that exceeds the return on the immunizing portfolio. The total return for the pension plan is then total plan return = return on liability-immunizing portfolio + return on excess return portfolio–return on liabilities. The return on liabilities is the change in the present value of the liabilities. If the immunizing portfolio is properly created, its return should be closed to the return on the liabilities. The volatility of the liabilities is neutralized to a great extent. What remains is then the return on the excess return portfolio. 22. The following excerpt is from a January 18, 2008 article (“LDI Strategy that is Liable to Word?”) by Penny Green, Chief Executive of the SAUL Trustee Company (a U.K. that advises on pension management) and deals with liability-driven strategies: “…there is no one asset class that precisely matches a plan’s liabilities. It is the case that bonds provide a cash flow that can be used to meet the cash flows out of a pension plan. But so do equities – it is just that the cash flows from equities (dividends) cannot be guaranteed. However, bonds do not cover longevity risk or salary inflation, so bonds are not a perfect match – but neither do equities. In fact, there is no asset class at present that matches longevity risk or salary inflation. This is the trustee’s dilemma that LDI strategies are supposed to resolve.” Explain why you agree or disagree with this viewpoint. In general one would agree with the views being presented concerning (i) the problems of any asset class rendering cash flows that can perfectly match a plan’s liabilities, and (ii) the shortcomings of any asset class meeting longevity risk. More details are given below. While the general points made are valid, one might disagree a bit with the perspective that equities (like bonds) can provide a cash flow to meet the cash flows needs of a pension fund. First, dividend streams for most equity investments are less than cash flows streams from fixed investments. An exception might be preferred equity but even here their dividend streams might be less than most bonds. Second, cash flow streams produced from selling equity are not guaranteed as equity prices are volatile and can fall sharply in a short period of time and in an untimely fashion to meet pension fund liabilities. Absent the use of derivatives to hedge prices, when stocks perform poorly for a long period of time, they lose their capacity to be an inflation hedge. In the same article from which the previous quote was made, the author states: “From the mid-1950s to the late 1990s, conventional wisdom was that equities were the closest match to a pension plan's liabilities. Thus, whenever an asset liability exercise was performed (the aim of which was to find an investment strategy that matched the liabilities of the plan), the result was a recommendation that the assets of a plan should be weighted to equities. In the late 1990s, this orthodoxy was turned on its head and bonds became the asset class assumed to be most closely matched to the liabilities. Combine this with a fall in the equity markets, and high bond prices (and low yields), and it is easy to see how surpluses in the 1990s turned into deficits in the early part of this century. And in response to the changing theory, the results from the asset liability models changed.” This statement drives home the point that no one asset type is reliable over time, thus pointing out the fact that liability-driven strategies (applied to defined pension plans) must deal with two risks that may not be handled adequately by investing in the major asset classes. First, one risk (that must be dealt with when investing in asset classes) is the impact of inflation on future pension liabilities. An increase in the inflation rate increases future liabilities as salaries are adjusted upwards. At one time the view was that equities were the suitable asset class for dealing with inflation. As pointed out in the above statement, because of the volatility in equity prices, adverse movements in equity values may not mitigate inflation risk. Second, another risk (that must be dealt with when investing in asset classes) is longevity risk. This is the risk that beneficiaries may live longer and as a result future liabilities will exceed current actuarial determined liabilities. 23. In explaining how a pension fund should transition to a liability-driven investment strategy, Duane Rocheleau, managing director of Northern Trust’s global investment solutions team, writes in “Implementing LDI in Pension Plans,” January 2007: “1. Analyze and characterize the liabilities; 2. Quantify the relationship between the assets and liabilities; 3. Develop and implement appropriate investment strategies; 4. Monitor the account, rebalance the assets and liabilities mix and tweak the investment strategy as necessary.” Describe each of the above elements. In terms of analyzing and characterizing the liabilities, the manager wants to know the amounts of cash flows and the timing of the cash flows that are owed pension fund recipients. This is needed to determine what kinds and proportion of funds to allocate to asset types. For example, a diversified mix of asset types with a significant commitment to one type of assets might then be judged to offer the highest probability of building assets, containing costs and meeting the plan’s obligations. In terms of quantifying the relation between the assets and liabilities, the manager wants to be able to match the cash flows from the assets with those of the liabilities so that pension fund recipients will be paid in full and on time. To understand this problem and the need to quantify, consider duration. We can note that for many pension plans, the duration of the liabilities is quite long ranging from twelve years on up to even over twenty years. The duration provides an estimate of how the liability will change as a result of a one percent parallel shift in the yield curve. A plan with a liability duration of ten years could expect to see liabilities grow by approximately ten percent if interest rates dropped by one percent. Over the last decade, interest rates have dropped by far more than one percent for a number of years. In light of this, it becomes important that we match the duration for assets and liabilities. Had the duration of the assets been equally as long, the results would have been very different. In regards to developing and implementing appropriate investment strategies, there are two strategies to choose from: multiperiod immunization and cash flow matching. A multiperiod immunization strategy is one in which a portfolio is created that will be capable of satisfying more than one predetermined future liability regardless if interest rates change. Even if there is a parallel shift in the yield curve, it has been demonstrated that matching the duration of the portfolio to the duration of the liabilities is not a sufficient condition to immunize a portfolio seeking to satisfy a liability stream. Instead, it is necessary to decompose the portfolio payment stream in such a way that each liability is immunized by one of the component streams. The key to understanding this approach is recognizing that the payment stream on the portfolio, not the portfolio itself, must be decomposed in this manner. There may be no actual bonds that would give the component payment stream. A cash flow matching strategy is used to construct a portfolio that will fund a schedule of liabilities from a portfolio’s cash flows, with the portfolio's value diminishing to zero after payment of the last liability. This strategy can be summarized as follows. A bond is selected with a maturity that matches the last liability stream. An amount of principal plus final coupon equal to the amount of the last liability stream is then invested in this bond. The remaining elements of the liability stream are then reduced by the coupon payments on this bond, and another bond is chosen for the new, reduced amount of the next-to-last liability. Going backward in time, this cash flow matching process is continued until all liabilities have been matched by the payment of the securities in the portfolio. In regards to further action (monitoring, rebalancing, and tweaking) on the investment strategy as necessary, we can note that action is required because the market yield will fluctuate over the investment horizon. As a result, the duration of the portfolio will change (and change by more than that caused simply by the passage of time). In the face of changing yields, a portfolio can maintain immunization by rebalancing so that its duration is equal to the duration of the liability’s remaining time. For example, if the liability is initially five years, the initial portfolio assets should have a duration of five years. After a year, the duration for the liabilities and assets will change and it is unlikely they will be matched. This is because duration depends on the remaining time to maturity and the new level of yields, and there is no reason why the change in these two values should change the duration by the exact amount of time. Thus, the portfolio must be rebalanced. A question we can pose is: How often should the portfolio be rebalanced to adjust its duration? On the one hand, the more frequent rebalancing increases transactions costs, thereby reducing the likelihood of achieving the target yield. On the other hand, less frequent rebalancing will result in the duration wandering from the target duration, which will also reduce the likelihood of achieving the target yield. Thus, the portfolio manager faces a trade-off: some transaction costs must be accepted to prevent the duration from wandering too far from its target, but some maladjustment in the duration must be accepted or transaction costs will become prohibitively high. 24. One of your clients, a newcomer to the life insurance business, questioned you about the following excerpt from Peter E. Christensen, Frank J. Fabozzi, and Anthony LoFaso, “Dedicated Bond Portfolios,” Chapter 43 in Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities (Homewood, IL: Richard D. Irwin, 1991): For financial intermediaries such as banks and insurance companies, there is a well-recognized need for a complete funding perspective. This need is best illustrated by the significant interest-rate risk assumed by many insurance carriers in the early years of their Guaranteed Investment Contract (GIC) products. A large volume of compound interest (zero-coupon) and simple interest (annual pay) GICs were issued in three- through seven-year maturities in the positively sloped yield-curve environment of the mid-1970s. Proceeds from hundreds of the GIC issues were reinvested at higher rates in the longer 10- to 30-year private placement, commercial mortgage, and public bond instruments. At the time, industry expectations were that the GIC product would be very profitable because of the large positive spread between the higher “earned” rate on the longer assets and the lower “credited” rate on the GIC contracts. By pricing GICs on a spread basis and investing the proceeds on a mismatched basis, companies gave little consideration to the rollover risk they were assuming in volatile markets. As rates rose dramatically in the late 1970s and early 1980s, carriers were exposed to disintermediation as GIC liabilities matured and the corresponding assets had 20 years remaining to maturity and were valued at only a fraction of their original cost. Answer the below questions posed to you by your client. (a) “It is not clear to me what risk an issuer of a GIC is facing. A carrier can invest the proceeds in assets offering a higher yield than they are guaranteeing to GIC policyholders, so what’s the problem? Isn’t it just default risk that can be controlled by setting tight credit standards?” An issuer of a GIC is facing reinvestment rate risk. As rates increase, the issuer of a GIC will face paying a higher rate of return on subsequent securities because GICs mature in three to seven years. This liability is matched by a longer term asset that pays an increasingly lower return compared to what GICs yield as interest rates rise. Because the funds are tied up in longer term assets, it is not true that money will be available so that a carrier can invest the proceeds in assets offering a higher yield than they are guaranteeing to GIC policyholders. The problem is that assets and liabilities were not matched in terms of duration. As interest rates increased the assets gave a return too low to meet the return guaranteed on the liabilities. This problem has no immediate relationship to default risk. However, due to the mismatch in assets and liabilities the company has put itself in a precarious net surplus situation. (b) “I understand that disintermediation means that when a policy matures, the funds are withdrawn from the insurance company by the policyholder. But why would a rise in interest rates cause GIC policyholders to withdraw their funds? The insurance company can simply guarantee a higher interest rate.” Under a GIC policy, for a lump-sum payment a life insurance company guarantees that specified dollars will be paid to the policyholder at a specified future date. Or, equivalently, the financial institution (i.e., life insurance company) guarantees a specified rate of return on the payment. However, it cannot simply guarantee the same (or a higher) rate for ensuing periods even if rates increase because its assets used to match the GIC liability may not be earning (or able to earn) that rate. Regardless of the insurance company’s dilemma, there can be a number of reasons why policyholders will withdraw funds. First, the policy matures and client wants to spend the money. Second, the policyholders need for insurance may change. Third, policyholders may feel they can make a rate of return on an investment elsewhere which better suits their needs. (c) “What do the authors mean by ‘pricing GICs on a spread basis and investing the proceeds on a mismatched basis,’ and what is this ‘rollover risk’ they are referring to?” The GICs were priced such that the bank and insurance companies would realize what they believed was a healthy spread compared to other possible liabilities. However, the profit was placed in assets of a different maturity. As interest rates rose, they were faced by investors who wanted rates of return similar to the prevailing rates. However, they had already placed much of their funds in assets with longer maturity or mismatched maturity. The rollover risk stems from the fact that investors want to reinvest in the GICs issued by the institutions but the institutions’ assets cannot pay the prevailing rate. 25. Suppose that a life insurance company sells a five-year guaranteed investment contract that guarantees an interest rate of 7.5% per year on a bond-equivalent yield basis (or equivalently, 3.75% every six months for the next 10 six-month periods). Also suppose that the payment made by the policyholder is $9,642,899. Consider the following three investments that can be made by the portfolio manager: Bond X: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5% yield to maturity that matures in five years. Bond Y: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5% yield to maturity that matures in 12 years. Bond Z: Buy $10,000,000 par value of a six-year 6.75% coupon option-free bond selling at 96.42899 to yield 7.5%. Answer the below questions. (a) Holding aside the spread that the insurance company seeks to make on the invested funds, demonstrate that the target accumulated value to meet the GIC obligation five years from now is $13,934,413. To compute the target accumulated value, we need to take the future value of the annuity resulting from the policyholder’s payment and add it to this payment due at the end of the last period. We have: target accumulated value = (P) + P where the semiannual coupon rate = annual payment  2 = 7.5%  2 = 3.75%; the payment or price of bond (P) = $9,642,899; the yield = y = 7.5%  2 = 3.75%; and, the total number of periods (n) = 5(2) = 10. Inserting these values into our target accumulated formula, we get: target accumulated value = ($9,642,899) + $9,642,899 = $361,608.71[11.86783847] + $9,642,899 = $4,291,513.79 + $9,642,899 = $13,934,412.79. Thus, the target accumulated value is about $13,934,413. (b) Complete Table A assuming that the manager invests in bond X and immediately following the purchase, yields change and stay the same for the five-year investment horizon. For the each row, we have six columns. Below we show how all values are gotten for the first row of 11.00%. The same process can be repeated to get values for the remaining rows. Column One gives the new yield which is 11% for the first row. This means the semiannual yield will be 5.5%. This value changes for each row with each new yield provided for each row. Column Two provides the coupon which is the total interest paid for each of the ten periods. Thus, the interest paid is ten times the semiannual coupon payment. The semiannual coupon payment is the annual coupon rate divided by two (i.e., 0.075  2 = 0.0375) times the par value purchased of $9,642,899. We have: 10[(0.0375)($9,642,899]) = 10[$361,608.71] = $3,616,087.13 or about $3,616,087. This value is the same for each row since the coupon rate of 7.5% does not change. Column Three provides interest on interest. To get interest on interest, we compute (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and (ii) the total interest paid which is the number of periods (10) times the semiannual coupon payment (0.0375 × $9,642,899). For (iii), we take the value in (i) minus the value in (ii). For (i), we have: (P) = ($9,642,899) = $361,608.71[12.87535379] = $4,655,840.11. For (ii), we have: n(semiannual coupon payment) = 10[(0.0375)($9,642,899]) = 10[$361,608.71] = $3,616,087.13. Note that this value is the value also computed in column two. For (iii), we have: $4,655,840.11 – $361,608.71 = $1,039,752.98 or about $1,039,753. This third column value changes for each row because it is a function of the new yield which changes for each row. For example, for the second row we repeat the above process using 10%/2 = 5.0% to compute the future value of semiannual coupon payment annuity. Column Four provides the maturity value of the bond, which is $9,642,899. This value is the same for each row since the investment horizon is the same as the maturity. When the maturity value increases then the value will not equal $9,642,899. Column Five is the total accumulated value. This is computed by adding (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the maturity value given in the fourth column. The value for (i) was given previously as $4,655,840.11 when computing the interest on interest. This value is also given by adding the second and third column values. The value for (ii) is $9,642,899 as given in the fourth column. For the accumulated value, we have: total accumulated value = $4,655,840.11 + $9,642,899 = $14,298,739.11. This value changes for each row as it is a function of the new yield which changes for each row. Column Six, the last column, reports the total return which is given as: total return = 2 where accumulated value is the fifth column value of $14,298,739.11, the policyholder payment is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values, we have: 2 = 2[(1.482825767)0.1 – 1] = 2[1.040181 – 1] = 0.0803625 or about 8.04%. This value changes for each row because it is a function of the new yield which changes for each row. We repeat the above process to get all values for each row. These values are given below with Table A filled in. Table A Accumulated Value and Total Return After Five Years: Five-Year 7.5% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 7.50% Maturity (years): 5 Yield to maturity: 7.50% Price: 100.00000 Par value purchased: $9,642,899 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Yield Coupon Interest on Interest Price of Bond Accumulated Value Total Return 11.00% $3,616,087 $1,039,753 $9,642,899 $14,298,739 8.04% 10.00% $3,616,087 $ 932,188 $9,642,899 $14,191,175 7.88% 9.00% $3,616,087 $ 827,436 $9,642,899 $14,086,423 7.73% 8.00% $3,616,087 $ 725,426 $9,642,899 $13,984,412 7.57% 7.50% $3,616,087 $ 675,427 $9,642,899 $13,934,413 7.50% 7.00% $3,616,087 $ 626,089 $9,642,899 $13,885,073 7.43% 6.00% $3,616,087 $ 529,352 $9,642,899 $13,788,338 7.28% 5.00% $3,616,087 $ 435,153 $9,642,899 $13,694,139 7.14% 4.00% $3,616,087 $ 343,427 $9,642,899 $13,602,414 7.00% (c) Based on Table A, under what circumstances will the investment in bond X fail to satisfy the target accumulated value? Given that the target accumulated value is $13,934,413, we see any new yield below 7.50% will give an accumulated value which is less than the target. This is due to reinvestment rate risk which works to the disadvantage of the life insurance company since the coupon must be invested at a rate below 7.5% which is the guaranteed rate of return. (d) Complete Table B, assuming that the manager invests in bond Y and immediately following the purchase, yields change and stay the same for the five-year investment horizon. For the each row, we have six columns. Below we show how all values are gotten for the first row of 11.00%. The same process can be repeated to get values for the remaining rows in the table. Column One gives the new yield which is 11% for the first row. This means the semiannual yield will be 5.5%. This value changes for each row with each new yield provided for each row. Column Two provides the coupon which is the total interest paid for each of the ten periods. Thus, the interest paid is ten times the semiannual coupon payment. The semiannual coupon payment is the semiannual coupon rate (0.075  2 = 0.0375) times the par value purchased of $9,642,899. We have: 10[(0.0375)($9,642,899]) = 10[$361,609.71] = $3,616,087.13 or about $3,616,087. This value is the same for each row since the coupon rate of 7.5% does not change. The values for this column for Table B are the same values found in Table A. Column Three provides interest on interest. To get interest on interest, we compute (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and (ii) the total interest paid which is the number of periods (10) times the semiannual coupon payment (0.0375 × $9,642,899). For (iii), we take the value in (i) minus the value in (ii). For (i), we have: (P) = ($9,642,899) = $361,608.71[12.87535379] = $4,655,840.11. For (ii), we have: n(semiannual coupon payment) = 10[(0.0375)($9,642,899]) = 10[$361,608.71] = $3,616,087.13. Note that this value is the value also computed in column two. For (iii), we have: $4,655,840.11 – $361,608.71 = $1,039,752.98. This third column value changes for each row because it is a function of the new yield which changes for each row. For example, for the second row we repeat the above process using 10%/2 = 5% to compute the future value of semiannual coupon payment annuity. The values for this column for Table B are the same values found in Table A. Column Four provides the maturity value of the bond, which is $8,024,638.89. This value is computed by taking the present value of the bond value at the end of five years. Because the maturity has seven remaining years after five years, this means we compute (i) the value at the end of five years for an annuity composed of 7(2) = 14 semiannual coupon payments of $361,608.71 discounted at a yield of 5.5% and (ii) the value at the end of five years for the bond value of $9,642,899 received in 14 periods and discounted at a yield of 5.5%. For (i), we get: (P) = ($9,642,899) = $361,608.71[9.5896479] = $3,467,700.23. For (ii), we get: = = $9,642,899(0.4725693866) = $4,556,938.66. Adding (i) and (ii), we get the value of the bond price at the beginning of period 11 (or at the end of period 10) as $3,467,700.23 + $4,556,938.66 = $8,024,638.89 or about $8,024,639. This value changes for each row because it is a function of the new yield which changes for each row. Column Five is the total accumulated value. This is computed by adding (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the maturity value given in the fourth column. The value for (i) was given previously as $4,655,840.11 when computing the interest on interest. This value is also given by adding the second and third column values. The value for (ii) is $8,024,638.89 as given in the fourth column. For the accumulated value, we have: total accumulated value = $4,655,840.11 + $8,024,638.89 = $12,680,479.00. This value changes for each row as it is a function of the new yield which changes for each row. Column Six, the last column, reports the total return which is given by total return = 2 where accumulated value is the fifth column value of $12,680,479.00, the policyholder payment is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values, we have: 2 = 2[(1.315006929)0.1 – 1] = 2[1.0277626 – 1] = 0.0555252 or about 5.55%. This value changes for each row because it is a function of the new yield which changes for each row. We repeat the above process for each row to get all values. These values are given below with Table B filled in. Table B Accumulated Value and Total Return After Five Years: Twelve-Year 7.5% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 7.50% Maturity (years): 12 Yield to maturity: 7.50% Price: 100.00000 Par value purchased: $9,642,899 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Yield Coupon Interest on Interest Price of Bond Accumulated Value Total Return 11.00% $3,616,087 $1,039,753 $8,024,639 $12,680,479 5.55% 10.00% $3,616,087 $ 932,188 $8,449,753 $12,998,030 6.06% 9.00% $3,616,087 $ 827,436 $8,903,566 $13,347,090 6.61% 8.00% $3,616,087 $ 725,426 $9,388,251 $13,729,764 7.19% 7.50% $3,616,087 $ 675,427 $9,642,899 $13,934,413 7.50% 7.00% $3,616,087 $ 626,089 $9,906,163 $14,148,337 7.82% 6.00% $3,616,087 $ 529,352 $10,459,851 $14,605,289 8.48% 5.00% $3,616,087 $ 435,153 $11,052,078 $15,103,318 9.18% 4.00% $3,616,087 $ 343,427 $11,685,837 $15,645,352 9.92% (e) Based on Table B, under what circumstances will the investment in bond Y fail to satisfy the target accumulated value? Given that the target accumulated value is $13,934,413, we see any new yield above 7.50% will give accumulated value which is less than the target. This is due to fact that the increase in interest rates lowers the value of the coupon and principal payments which are discounted at a higher yield in the future. This works to the disadvantage of the life insurance company since they have chosen to match their liability with a longer term twelve-year bond instead of a five-year bond. One should note that when the market yield increases interest on interest will be greater; however, the market price of the bond decreases. The net effect is that the accumulated value is less than the target accumulated value. The reverse is true when the market yield decreases. The change in the price of the bond will more than offset the decline in the interest on interest, resulting in an accumulated value that exceeds the target accumulated value. (f) Complete Table C, assuming that the manager invests in bond Z and immediately following the purchase, yields change and stay the same for the five-year investment horizon. For each row, we have six columns. Below we show how all values are gotten for the first row of 11.00%. The same process can be repeated to get values for the remaining rows in the table. Column One gives the new yield which is 11% for the first row. This means the semiannual yield will be 5.5%. This value changes for each row with each new yield provided for each row. Column Two provides the coupon which is the total interest paid for each of the ten periods. Thus, the interest paid is ten times the semiannual coupon payment. The semiannual coupon payment is the semiannual coupon rate (0.0675  2 = 0.03375) times the par value purchased of $10,000,000. We have: 10[(0.03375)($10,000,000]) = 10[$337,500] = $3,375,000. This value is the same for each row since the coupon rate of 6.75% does not change. Column Three provides interest on interest. To get interest on interest, we compute (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and (ii) the total interest paid which is the number of periods (10) times the semiannual coupon payment (0.03375 × $10,000,000). For (iii), we take the value in (i) minus the value in (ii). For (i), we have: (P) = ($10,000,000) = $337,500[12.87535379] = $4,345,431.90. For (ii), we have: n(semiannual coupon payment) = 10[(0.03375)($ 10,000,000]) = 10[$337,500] = $3,375,000. Note that this value is the value also computed in column two. For (iii), we have: $4,345,431.90 – $3,375,000 = $970,431.90. This third column value changes for each row because it is a function of the new yield which changes for each row. For example, for the second row we repeat the above process using 10%  2 = 5% to compute the future value of semiannual coupon payment annuity. Column Four provides the maturity value of the bond, which is $9,607,657.06. This value is computed by taking the present value of the bond value at the end of five years. Because the maturity has one remaining year after five years, this means we compute (i) the value at the end of five years for an annuity composed of two semiannual coupon payments of $337,500 discounted at a yield of 5.5% and (ii) the value at the end of five years of the bond value of $10,000,000 received in two periods and discounted at a yield of 5.5%. For (i), we get: (P) = ($10,000,000) = $337,500[1.8463197] = $623,132.90. For (ii), we get: = = $10,000,000(0.898452416) = $8,984,524.16. Adding (i) and (ii), we get the value of the bond price at the beginning of period 11 (or at the end of period 10) as $623,132.90 + $8,984,524.16 = $9,607,657.06 or about $9,607,657. This value changes for each row because it is a function of the new yield which changes for each row. Column Five is the total accumulated value. This is computed by adding (i) the future value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the maturity value given in the fourth column. The value for (i) was given previously as $4,345,431.90 when computing the interest on interest. This value is also given by adding the second and third column values. The value for (ii) is $9,607,657.06 as given in the fourth column. For the accumulated value, we have: total accumulated value = $4,345,431.90 + $9,607,657.06 = $13,953,088.96. This value changes for each row as it is a function of the new yield which changes for each row. Column Six, the last column, reports the total return which is given by total return = 2 where accumulated value is the fifth column value of $13,953,088.96, the policyholder payment is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values, we have: 2 = 2[(1.446980723)0.1 – 1] = 2[1.037638971 – 1] = 0.075278 or about 7.53%. This value changes for each row because it is a function of the new yield which changes for each row. We repeat the above process for each row to get all values. These values are given below with Table C filled in. Table C Accumulated Value and Total Return After Five Years: Six-Year 6.75% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 6.75% Maturity (years): 6 Yield to maturity: 7.5% Price: 96.42899 Par value purchased: $10,000,000 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Yield Coupon Interest on Interest Price of Bond Accumulated Value Total Return 11.00% $3,375,000 $970,432 $ 9,607,657 $13,953,089 7.53% 10.00% $3,375,000 $870,039 $ 9,789,325 $13,942,885 7.51% 9.00% $3,375,000 $772,271 $ 9,882,119 $13,936,596 7.50% 8.00% $3,375,000 $677,061 $ 9,929,017 $13,934,180 7.50% 7.50% $3,375,000 $630,395 $ 9,976,254 $13,934,413 7.50% 7.00% $3,375,000 $584,345 $10,071,755 $13,935,599 7.50% 6.00% $3,375,000 $494,059 $10,168,650 $13,940,814 7.51% 5.00% $3,375,000 $406,141 $10,266,965 $13,949,791 7.52% 4.00% $3,375,000 $320,531 $10,266,965 $13,962,495 7.54% (g) Based on Table C, under what circumstances will the investment in bond Z fail to satisfy the target accumulated value? There is only one yield in the table that gives a value below the target of $13,934,413 and that is for 8.00% yield where the target value is $13,934,180 which is $133 off. (h) What is the modified duration of the liability? Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected cash flow does not change with interest rates. One modified duration expression we can use is: modified duration = where C is the semiannual coupon payment, y is the semiannual yield, n is the number of semiannual periods, and P is the bond quote in 100’s. For our bond (expressing numbers in terms of a $100 bond quote), we have: C = $3.75, y = 0.0375, n = 10, and P = $96.43. Inserting these values in our modified duration formula, we can solve as follows: = = = = 8.52. Converting to annual number by dividing by two gives a modified duration for the liabilities of 4.26. The Macaulay duration is 4.13 while the modified duration using the Macaulay relationship is 3.98. (i) Complete the following table for the three bonds assuming that each bond is trading to yield 7.5%: Bond Modified Duration 5-year, 7.5% coupon, selling at par 12-year, 7.5% coupon, selling at par 6-year, 6.75% coupon, selling for 96.42899 Using the formula in part (h) for modified duration, we get the below values as given in the completed table: Bond Modified Duration 5-year, 7.5% coupon, selling at par 4.11 12-year, 7.5% coupon, selling at par 7.82 6-year, 6.75% coupon, selling for 96.42899 4.33 (j) For which bond is the modified duration equal to the duration of the liability? The 6-year, 6.7%% coupon, selling for $96.43 per $100 at the end of 5 years gives the closest duration to the liability. Rounding off to the nearest tenth both are equal at 4.3. (k) Why in this example can one focus on modified duration rather than effective duration? Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected cash flows do not change with changes in interest rates. Modified duration is not an appropriate measure for securities where the projected cash flows change as interest rates change. The effective duration computation allows for changing cash flow when interest rates change. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773