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Chapter 6 Interest rate risk measurement: the duration model Answers to end-of-chapter questions Questions and problems 1 What is the difference between book value accounting and market value accounting? How do interest rate changes affect the value of bank assets and liabilities under the two methods? What is marking to market? Book value accounting reports assets and liabilities at the original issue values. Market value accounting reports assets and liabilities at their current market values. Current market values may be different from book values because they reflect current market conditions, such as current interest rates. FIs generally report their balance sheets using book value accounting methods. This is a problem if an asset or liability has to be liquidated immediately. If the asset or liability is held until maturity, then the reporting of book values does not pose a problem. For an FI, a major factor affecting asset and liability values is interest rate changes. If interest rates increase, the value of both loans (assets) and deposits and debt (liabilities) fall. If assets and liabilities are held until maturity, it does not affect the book valuation of the FI. However, if deposits or loans have to be refinanced, then market value accounting presents a better picture of the condition of the FI. The process by which changes in the economic value of assets and liabilities are accounted is called marking to market. The changes can be beneficial as well as detrimental to the total economic health of the FI. 2 What are the two different general interpretations of the concept of duration, and what is the technical definition of this term? How does duration differ from maturity? Duration measures the weighted-average life of an asset or liability in economic terms. As such, duration has economic meaning as the interest sensitivity (or interest elasticity) of an asset’s value to changes in the interest rate. Duration differs from maturity as a measure of interest rate sensitivity because duration takes into account the time of arrival and the rate of reinvestment of all cash flows during the asset’s life. Technically, duration is the weighted-average time to maturity using the relative present values of the cash flows as the weights. 3 A one-year, $100 000 loan carries a coupon rate and a market interest rate of 12 per cent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year. (a) What will be the cash flows at the end of six months and at the end of the year? CF1/2 = ($100 000 × 0.12 × ½) + $50 000 = $56 000 interest and principal CF1 = ($50 000 × 0.12 × ½) + $50 000 = $53 000 interest and principal (b) What is the present value of each cash flow discounted at the market rate? What is the total present value? PV of CF1/2 = $56 000 1.06 = $52 830.19 PV of CF1 = $53 000 (1.06)2 = $47 169.81 PV Total CF = $100 000 (c) What proportion of the total present value of cash flows occurs at the end of six months? What proportion occurs at the end of the year? X½ = $52 830.19 $100 000 = 0.5283 = 52.83% X1 = $47 169.81 $100 000 = 0.4717 = 47.17% (d) What is the duration of this loan? Duration = 0.5283(1/2) + 0.4717(1) = 0.7358 years Or, using the table form: t CF PV of CF PV of CF × t ½ $56 000 $52 830.19 $26 415.09 1 $53 000 $47 169.81 $47 169.81 $100 000.00 $73 584.91 Duration = $73 584.91/$100 000.00 = 0.7358 years 4 Two bonds are available for purchase in the financial markets. The first bond is a two-year, $1000 bond that pays an annual coupon of 10 per cent. The second bond is a two-year, $1000, zero-coupon bond. (a) What is the duration of the coupon bond if the current yield-to-maturity (R) is 8 per cent? 10 per cent? 12 per cent? (Hint: You may wish to create a spread sheet program to assist in the calculations.) Coupon bond: Par value = $1000 Coupon rate = 10% Annual payments R = 8% Maturity = 2 years t CF PV of CF PV of CF × t ½ $100 $92.59 $92.59 1 $1100 $943.07 $1886.15 $1035.67 $1978.74 Duration = $1978.74/$1035.67 = 1.9106 R = 10% Maturity = 2 years t CF PV of CF PV of CF × t ½ $100 $90.91 $90.91 1 $1100 $909.09 $1818.18 $1000.00 $1909.09 Duration = $1909.09/$1000.00 = 1.9091 R = 12% Maturity = 2 years t CF PV of CF PV of CF × t ½ $100 $89.29 $89.23 1 $1100 $876.91 $1753.83 $966.20 $1843.11 Duration = $1843.11/$966.20 = 1.9076 (b) How does the change in the current yield to maturity affect the duration of this coupon bond? Increasing the yield to maturity decreases the duration of the bond. (c) Calculate the duration of the zero-coupon bond with a yield to maturity of 8 per cent, 10 per cent and 12 per cent. Zero-coupon bond: Par value = $1000 Coupon rate = 0% R = 8% Maturity = 2 years t CF PV of CF PV of CF × t 2 $1000 $857.34 $1714.68 $857.34 $1714.68 Duration = $1714.68/$857.34 = 2.0000 R = 10% Maturity = 2 years t CF PV of CF PV of CF × t 2 $1000 $826.45 $1652.89 $826.45 $1652.89 Duration = $1652.89/$826.45 = 2.0000 R = 12% Maturity = 2 years t CF PV of CF PV of CF × t 2 $1000 $797.19 $1594.39 $797.19 $1594.39 Duration = $1594.39/$797.19 = 2.0000 (d) How does the change in the yield to maturity affect the duration of the zero-coupon bond? Changing the yield to maturity does not affect the duration of the zero-coupon bond. (e) Why does the change in the yield to maturity affect the coupon bond differently to the way it affects the zero-coupon bond? Increasing the yield to maturity on the coupon bond allows for a higher reinvestment income that more quickly recovers the initial investment. The zero-coupon bond has no cash flow until maturity. 5 What is the duration of a five-year, $1000 Treasury Bond with a 10 per cent semi-annual coupon selling at par? Selling with a yield to maturity of 12 per cent? 14 per cent? What can you conclude about the relationship between duration and yield to maturity? Plot the relationship. Why does this relationship exist? Five-year Treasury Bond: Par value = $1000 Coupon rate = 10% Semi-annual payments R = 10% Maturity = 5 years t CF PV of CF PV of CF × t 0.5 $50 $47.62 $23.81 1 $50 $45.35 $45.35 1.5 $50 $43.19 $64.79 2 $50 $41.14 $82.27 2.5 $50 $39.18 $97.94 3 $50 $37.31 $111.93 3.5 $50 $35.53 $124.37 4 $50 $33.84 $135.37 4.5 $50 $32.23 $145.04 5 $1050 $644.61 $3223.04 $1000.00 $4053.91 Duration = $4053.91/$1000.00 = 4.0539 R = 12% Maturity = 5 years t CF PV of CF PV of CF × t 0.5 $50 $47.17 $23.58 1 $50 $44.50 $44.50 1.5 $50 $41.98 $62.97 2 $50 $39.60 $79.21 2.5 $50 $37.36 $93.41 3 $50 $35.25 $105.74 3.5 $50 $33.25 $116.38 4 $50 $31.37 $125.48 4.5 $50 $29.59 $133.18 5 $1050 $586.31 $2931.57 $926.40 $3716.03 Duration = $3716.03/$926.40 = 4.0113 R = 14% Maturity = 5 years t CF PV of CF PV of CF × t 0.5 $50 $46.73 $23.36 1 $50 $43.67 $43.67 1.5 $50 $40.81 $61.22 2 $50 $38.14 $76.29 2.5 $50 $35.65 $89.12 3 $50 $33.32 $99.95 3.5 $50 $31.14 $108.98 4 $50 $29.10 $116.40 4.5 $50 $27.20 $122.39 5 $1050 $533.77 $2668.83 $859.53 $3410.22 Duration = $3410.22/$859.53 = 3.9676 6 Consider three Treasury Bonds each of which has a 10 per cent semi-annual coupon and trades at par. Calculate the duration for a bond that has a maturity of four years, three years and two years. Four-year Treasury Bond: Par value = $1000 Coupon rate = 10% Semi-annual payments R = 10% Maturity = 4 years t CF PV of CF PV of CF × t 0.5 $50 $47.62 $23.81 1 $50 $45.35 $45.35 1.5 $50 $43.19 $64.79 2 $50 $41.14 $82.27 2.5 $50 $39.18 $97.94 3 $50 $37.31 $111.93 3.5 $50 $35.53 $124.37 4 $50 $710.53 $2842.73 $1000.00 $3393.19 Duration = $3393.19/$1000.00 = 3.3932 R = 10% Maturity = 3 years t CF PV of CF PV of CF × t 0.5 $50 $47.62 $23.81 1 $50 $45.35 $45.35 1.5 $50 $43.19 $64.79 2 $50 $41.14 $82.27 2.5 $50 $39.18 $97.94 3 $50 $783.53 $2350.58 $1000.00 $3393.19 Duration = $2664.74/$1000.00 = 2.6647 R = 10% Maturity = 2 years t CF PV of CF PV of CF × t 0.5 $50 $47.62 $23.81 1 $50 $45.35 $45.35 1.5 $50 $43.19 $64.79 2 $50 $863.84 $1727.68 $1000.00 $1861.62 Duration = $1861.62/$1000.00 = 1.8616 (b) What conclusions can you reach about the relationship of duration and the time to maturity? Plot the relationship. As maturity decreases, duration decreases at a decreasing rate. Although the graph below does not illustrate with great precision, the change in duration is less than the change in time to maturity. 7 A six-year, $10 000 CD pays 6 per cent interest annually and has a 6 per cent yield to maturity. What is the duration of the CD? What would be the duration if interest were paid semi-annually? What is the relationship of duration to the relative frequency of interest payments? Six-year CD: Par value = $10 000 Coupon rate = 6% R = 6% Maturity = 6 years Annual payments t CF PV of CF PV of CF × t 1 $600 $566.04 $566.04 2 $600 $534.00 $1068.00 3 $600 $503.77 $1511.31 4 $600 $475.26 $1901.02 5 $600 $448.35 $2241.77 6 $600 $7472.58 $44 835.49 $10600 $10 00.000 $52 123.64 Duration = $52 123.64/$1000.00 = 5.2124 R = 6% Maturity = 6 years Semi-annual payments t CF PV of CF PV of CF × t 0.5 $300 $291.26 $145.63 1 $300 $282.78 $282.78 1.5 $300 $274.54 $411.81 2 $300 $266.55 $533.09 2.5 $300 $258.78 $646.96 3 $300 $251.25 $753.74 3.5 $300 $243.93 $853.75 4 $300 $236.82 $947.29 4.5 $300 $229.93 $1034.66 5 $300 $223.23 $1116.14 5.5 $300 $216.73 $1192.00 6 $300 $7224.21 $43 345.28 $10 300 $10 000.000 $51 236.12 Duration = $51 263.12/$10 000.00 = 5.1263 Duration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being received more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. 8 What is a consol bond? What is the duration of a consol bond that sells at a yield to maturity of 8 per cent? 10 per cent? 12 per cent? Would a consol trading at a yield to maturity of 10 per cent have a greater duration than a 20-year zero-coupon bond trading at the same yield to maturity? Why? A consol is a bond that pays a fixed coupon each year forever. A consol trading at a yield to maturity of 10 per cent has a duration of 11 years (see table below), while a 20-year zero-coupon bond trading at a YTM of 10 per cent, or any other YTM, has a duration of 20 years because no cash flows occur before the twentieth year Duration of consol bonds at various rates R Duration = 1 + 1/R 8% 1 + 1/0.08 = 13.50 years 10% 11.00 years 12% 9.33 years 9 Maximum Superannuation Fund is attempting to balance one of the bond portfolios under its management. The fund has identified three bonds which have five-year maturities and which trade at a yield to maturity of 9 per cent. The bonds differ only in that the coupons are 7 per cent, 9 per cent and 11 per cent. What is the duration for each bond? Five-year bond: Par value = $1000 Maturity = 5 years Annual payments R = 9% Coupon rate = 7% t CF PV of CF PV of CF × t 1 $70 $64.22 $64.22 2 $70 $58.92 $117.84 3 $70 $54.05 $162.16 4 $70 $49.59 $198.36 5 $1070 $695.43 $347.13 $922.21 $4019.71 Duration = $4019.71/$922.21 = 4.3588 R = 9% Coupon rate = 9% t CF PV of CF PV of CF × t 1 $90 $82.57 $82.57 2 $90 $75.75 $151.50 3 $90 $69.50 $208.49 4 $90 $63.76 $255.03 5 $1090 $708.43 $3542.13 $1000.00 $4239.72 Duration = $4239.72/$1000.00 = 4.2397 R = 9% Coupon rate = 11% t CF PV of CF PV of CF × t 1 $110 $100.92 $100.92 2 $110 $92.58 $185.17 3 $110 $84.94 $254.82 4 $110 $77.93 $311.71 5 $1110 $721.42 $3607.12 $1077.79 $4459.73 Duration = $4459.73/$1077.79 = 4.1378 (b) What is the relationship between duration and the amount of coupon interest that is paid? Plot the relationship. 10 An insurance company is analysing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a yield to maturity of 10 per cent, have $10 000 par values, and have five years to maturity. The bonds differ only in the amount of annual coupon interest that they pay: 8, 10 and 12 per cent. (a) What is the duration for each five-year bond? Five-year bond: Par value = $10 000 R = 10% Maturity = 5 years Annual payments Coupon rate = 8% t CF PV of CF PV of CF × t 1 $800 $727.27 $727.27 2 $800 $661.16 $1 322.31 3 $800 $601.06 $1 803.16 4 $800 $546.41 $2 185.64 5 $10 800 $6705.95 $33 259.75 $9241.84 $39 568.14 Duration = $39 568.14/9241.84 = 4.2814 Coupon rate = 10% t CF PV of CF PV of CF × t 1 $1 000 $909.09 $909.09 2 $1 000 $826.45 $1 652.89 3 $1 000 $751.31 $2 253.94 4 $1 000 $683.01 $2 732.05 5 $11 000 $6 830.13 $34 150.67 $10 000.00 $41 698.65 Duration = $41 698.65/10 000.00 = 4.1699 Coupon rate = 12% t CF PV of CF PV of CF × t 1 $1 200 $1090.91 $1 090.91 2 $1 200 $991.74 $1 983.47 3 $1 200 $901.58 $2 704.73 4 $1 200 $819.62 $3 278.46 5 $11 200 $6 954.32 $34 771.59 $10 758.16 $43 829.17 Duration = $43 829.17/10 758.16 = 4.0740 (b) What is the relationship between duration and the amount of coupon interest that is paid? 11 You can obtain a loan of $100 000 at a rate of 10 per cent for two years. You have a choice of (i) paying the interest (10 per cent) each year and the total principal at the end of the second year or (ii) amortising the loan, that is, paying interest (10 per cent) and principal in equal payments each year. The loan is priced at par. (a) What is the duration of the loan under both methods of payment? Two-year loan: Interest at end of year one; Principal and interest at end of year two Par value = $100 000 Coupon rate = 10% Annual payments R = 10% Maturity = 2 years t CF PV of CF PV of CF × t 1 $10 000 $9 090.91 $9 090.91 2 $110 000 $90 909.09 $181 818.18 $100 000.00 $190 909.09 Duration = $190 909.09/$100 000 = 1.9091 Two-year loan: Amortised over two years Par value = $100 000 Coupon rate = 10% R = 10% Maturity = 2 years Annual amortised payments = $57 619.05 t CF PV of CF PV of CF × t 1 $57 619.05 $52 380.95 $52 380.95 2 $57 619.05 $47 619.05 $95 238.10 $100 000.00 $147 619.05 Duration = $147 619.05/$100 000 = 1.4762 (b) Explain the difference in the two results. Duration decreases dramatically when a portion of the principal is repaid at the end of year one. Duration often is described as the weighted-average maturity of an asset. If more weight is given to early payments, the effective maturity of the asset is reduced. 12 How is duration related to the interest elasticity of a fixed-income security? What is the relationship between duration and the price of the fixed-income security? Taking the first derivative of a bond’s (or any fixed-income security) price (P) with respect to the yield to maturity (R) provides the following: The economic interpretation is that D is a measure of the percentage change in the price of a bond for a given percentage change in yield to maturity (interest elasticity). This equation can be rewritten to provide a practical application: In other words, if duration is known, then the change in the price of a bond due to small changes in interest rates, R, can be estimated using the above formula. 13 You have discovered that the price of a bond rose from $975 to $995 when the yield to maturity fell from 9.75 per cent to 9.25 per cent. What is the duration of the bond? We know: 14 What is dollar duration? How is dollar duration different from duration? Dollar duration is the dollar value change in the price of a security to a 1 per cent change in the return on the security. Duration is a measure of the percentage change in the price of a security for a 1 per cent change in the return on the security. The dollar duration is intuitively appealing in that we multiply the dollar duration by the change in the interest rate to get the actual dollar change in the value of a security to a change in interest rates. 15 Calculate the duration of a two-year, $1000 bond that pays an annual coupon of 10 per cent and trades at a yield of 14 per cent. What is the expected change in the price of the bond if interest rates decline by 0.50 per cent (50 basis points)? Two-year bond: Par value = $1000 Coupon rate = 10% Annual payments R = 14% Maturity = 2 years t CF PV of CF PV of CF × t 1 $100 $87.72 $87.72 2 $1100 $846.41 $1692.83 $934.13 $1780.55 Duration = $1780.55/$934.13 = 1.9061 The expected change in price = – dollar duration × ∆R = – MD × ∆R = –(1.9061/1.14) × (–.005) × $934.13 = $7.81. This implies a new price of $941.94 (that is, $934.13 + $7.81). The actual price using conventional bond price discounting formula would be $941.99. The difference of $0.05 is due to convexity, which is not considered in the duration elasticity measure. 16 The duration of an 11-year, $1000 Treasury Bond paying a 10 per cent semi-annual coupon and selling at par has been estimated at 6.763 years. (a) What is the modified duration of the bond? What is the dollar duration of the bond? Modified duration = D/(1 + R/2) = 6.763/(1 + .10/2) = 6.441 years Dollar duration = MD × P = 6.441 × $1000 = 6441 (b) What will be the estimated price change on the bond if interest rates increase 0.10 per cent (10 basis points)? If rates decrease 0.20 per cent (20 basis points)? For interest rates increase of 0.10 per cent: Estimated change in price = – dollar duration × R = –6441 × 0.001 = –$6.441 => new price = $1000 – $6.441 = $993.559 For interest rates decrease of 0.20 per cent: Estimated change in price = –6441 × –0.002 = $12.882 => new price = $1000 + $12.882 = $1012.882 (c) What would the actual price of the bond be under each rate change situation in part (b) using the traditional present value bond pricing techniques? What is the amount of error in each case? Rate change Price estimated Actual price Error + 0.001 $993.559 $993.535 $0.024 – 0.002 $1012.882 $1013.111 –$0.229 17 Suppose you purchase a six-year, 8 per cent coupon bond (paid annually) that is priced to yield 9 per cent. The face value of the bond is $1000. (a) Show that the duration of this bond is equal to five years. Six-year bond: Par value = $1000 Coupon rate = 8% Annual payments R = 9% Maturity = 6 years t CF PV of CF PV of CF × t 1 $80 $73.39 $73.39 2 $80 $67.33 $134.67 3 $80 $61.77 $185.32 4 $80 $56.67 $226.70 5 $80 $51.99 $259.97 6 $1080 $643.97 $3863.81 $955.14 $4743.87 Duration = $4743.87/955.14 = 4.97 5 years (b) Show that if interest rates rise to 10 per cent within the next year and your investment horizon is five years from today, you will still earn a 9 per cent yield on your investment. Value of bond at end of year five: PV = ($80 + $1000) 1.10 = $981.82. Future value of interest payments at end of year five: $80 × FVIF (n = 4, i = 10%) = $488.41. Future value of all cash flows at n = 5: Coupon interest payments over five years $400.00 Interest on interest at 10 per cent $88.41 Value of bond at end of year five $981.82 Total future value of investment $1470.23 Yield on purchase of asset at $955.14 = $1470.23 × PVIV (n = 5, i=?%) i = 9.00924%. (c) Show that a 9 per cent yield also will be earned if interest rates fall next year to 8 per cent. Value of bond at end of year five: PV = ($80 + $1000) 1.08 = $1000. Future value of interest payments at end of year five: $80 × FVIF (n = 5, i = 8%) = $469.33. Future value of all cash flows at n = 5: Coupon interest payments over five years $400.00 Interest on interest at 8 per cent $69.33 Value of bond at end of year five $1000.00 Total future value of investment $1469.33 Yield on purchase of asset at $955.14 = $1469.33 × PVIV (n = 5, i=?%) i = 8.99596 per cent. 18 Suppose you purchase a five-year, 15 per cent coupon bond (paid annually) that is priced to yield 9 per cent. The face value of the bond is $1000. (a) Show that the duration of this bond is equal to four years. Five-year bond: Par value = $1000 Coupon rate = 15% Annual payments R = 9% Maturity = 5 years t CF PV of CF PV of CF × t 1 $150 $137.62 $137.62 2 $150 $126.25 $252.50 3 $150 $115.83 $347.48 4 $150 $106.26 $425.06 5 $1150 $747.42 $3737.10 $1233.38 $4899.76 Duration = $4899.76/1233.38 = 3.97 4 years (b) Show that if interest rates rise to 10 per cent within the next year and your investment horizon is four years from today, you will still earn a 9 per cent yield on your investment. Value of bond at end of year four: PV = ($150 + $1000) 1.10 = $1045.45. Future value of interest payments at end of year four: $150 × FVIF (n = 4, i = 10%) = $696.15. Future value of all cash flows at n = 4: Coupon interest payments over five years $600.00 Interest on interest at 10 per cent $96.15 Value of bond at end of year five $1045.45 Total future value of investment $1741.60 Yield on purchase of asset at $1233.38 = $1741.60 × PVIV (n = 4, i=?%) i = 9.00%. (c) Show that a 9 per cent yield also will be earned if interest rates fall next year to 8 per cent. Value of bond at end of year four: PV = ($150 + $1000) 1.08 = $1064.81. Future value of interest payments at end of year four: $150 × FVIF (n = 4, i = 8%) = $675.92. Future value of all cash flows at n = 4: Coupon interest payments over five years $600.00 Interest on interest at 8 per cent $75.92 Value of bond at end of year five $1064.81 Total future value of investment $1740.73 Yield on purchase of asset at $1233.38 = $1740.73 × PVIV (n = 4, i=?%) i = 9.00 per cent. 19 Consider the case in which an investor holds a bond for a period of time longer than the duration of the bond, that is, longer than the original investment horizon. (a) If interest rates rise, will the return that is earned exceed or fall short of the original required rate of return? Explain. In this case the actual return earned would exceed the yield expected at the time of purchase. The benefits from a higher reinvestment rate would exceed the price reduction effect if the investor holds the bond for a sufficient length of time. (b) What will happen to the realised return if interest rates decrease? Explain. If interest rates decrease, the realised yield on the bond will be less than the expected yield because the decrease in reinvestment earnings will be greater than the gain in bond value. (c) Recalculate parts (b) and (c) of problem 18 above, assuming that the bond is held for all five years, to verify your answers to parts (a) and (b) of this problem. The case where interest rates rise to 10 per cent, n = five years: Future value of interest payments at end of year five: $150 × FVIF (n = 5, i = 10%) = $915.76. Future value of all cash flows at n = 5: Coupon interest payments over five years $750.00 Interest on interest at 10 per cent $165.76 Value of bond at end of year five $1000.00 Total future value of investment $1915.76 Yield on purchase of asset at $1233.38 = $1915.76 × PVIF (n = 5, i=?%) i = 9.2066 per cent. The case where interest rates fall to 8 per cent, n = five years: Future value of interest payments at end of year five: $150 × FVIF (n = 5, i = 8%) = $879.99. Future value of all cash flows at n = 5: Coupon interest payments over five years $750.00 Interest on interest at 8 per cent $129.99 Value of bond at end of year five $1000.00 Total future value of investment $1879.99 Yield on purchase of asset at $1233.38 = $1879.99 × PVIV (n = 5, i=?%) i = 8.7957 per cent. (d) If either calculation in part (c) is greater than the original required rate of return, why would an investor ever try to match the duration of an asset with his or her investment horizon? The answer has to do with the ability to forecast interest rates. Forecasting interest rates is a very difficult task, one that most financial institution money managers are unwilling to do. For most managers, betting that rates would rise to 10 per cent to provide a realised yield of 9.20 per cent over five years is not a sufficient return to offset the possibility that rates could fall to 8 per cent and thus give a yield of only 8.8 per cent over five years. 20 Two banks are being examined by regulators to determine the interest rate sensitivity of their balance sheets. Bank A has assets composed solely of a 10-year $1 million loan with a coupon rate and yield of 12 per cent. The loan is financed with a 10-year $1 million CD with a coupon rate and yield of 10 per cent. Bank B has assets composed solely of a 7-year, 12 per cent zero-coupon bond with a current (market) value of $894 006.20 and a maturity (principal) value of $1 976 362.88. The bond is financed with a 10-year, 8.275 per cent coupon $1 000 000 face value CD with a yield to maturity of 10 per cent. The loan and the CDs pay interest annually, with principal due at maturity. (a) If market interest rates increase 1 per cent (100 basis points), how do the market values of the assets and liabilities of each bank change? That is, what will be the net effect on the market value of the equity for each bank? For Bank A, an increase of 100 basis points in interest rate will cause the market values of assets and liabilities to decrease as follows: Loan: $120 000 × PVIFA (n = 10, i = 13%) + $1 000 000 × PVIF (n = 10, i = 13%) = $945 737.57. CD: $100 000 × PVIFA (n = 10, i = 11%) + $1 000 000 × PVIF (n = 10,i = 11%) = $941 107.68. The loan value decreases $54 262.43 and the CD value falls $58 892.32. Therefore, the decrease in value of the asset is $4629.89 less than the liability. For Bank B: Bond: $1 976 362.88 × PVIF (n = 7, i = 13%) = $840 074.08. CD: $82 750 × PVIFA (n = 10, i = 11%) + $1 000 000 × PVIF (n = 10, i = 11%) = $839 518.43. The bond value decreases $53 932.12 and the CD value falls $54 487.79. Therefore, the decrease in value of the asset is $555.67 less than the liability. (b) What accounts for the differences in the changes in the market value of equity between the two banks? The assets and liabilities of Bank A change in value by different amounts because the durations of the assets and liabilities are not the same, even though the face values and maturities are the same. For Bank B, the maturities of the assets and liabilities are different, but the current market values and durations are the same. Thus, the change in interest rates causes the same (approximate) change in value for both liabilities and assets. (c) Verify your results above by calculating the duration for the assets and liabilities of each bank, and estimate the changes in value for the expected change in interest rates. Summarise your results. Ten-year CD Bank B (values in thousands of $s) Par value = $1000 Coupon rate = 8.275% Annual payments R = 10% Maturity = 10 years t CF PV of CF PV of CF × t 1 $82.75 $75.23 $75.23 2 $82.75 $68.39 $136.78 3 $82.75 $62.17 $186.51 4 $82.75 $56.52 $226.08 5 $82.75 $51.38 $256.91 6 $82.75 $46.71 $280.26 7 $82.75 $42.46 $297.25 8 $82.75 $38.60 $308.83 9 $82.75 $35.09 $315.85 10 $1082.75 $417.45 $4174.47 $894.01 $6528.15 Duration = $6258.15/894.01 = 7.00 The duration of the Bank B CD is calculated above to be 7.00 years. Since the bond is a zero-coupon, the duration is equal to the maturity of 7 years. Using the duration formula to estimate the change in value: Bond: Value = CD: Value = The difference in the change in value of the assets and liabilities for Bank B is $1015.91 using the duration estimation model. The difference in this estimate and the estimate found in part (a) above is due to the convexity of the two financial assets. The duration estimates for the loan and CD for Bank A are presented below: Ten-year loan Bank A (values in thousands of $s) Par value = $1000 Coupon rate = 12% Annual payments R = 12% Maturity = 10 years t CF PV of CF PV of CF × t 1 $120 $107.14 $107.14 2 $120 $95.66 $191.33 3 $120 $85.41 $256.24 4 $120 $76.26 $305.05 5 $120 $68.09 $340.46 6 $120 $60.80 $364.77 7 $120 $54.28 $379.97 8 $120 $48.47 $387.73 9 $120 $43.2701 $389.46 10 $1120 $360.61 $3606.10 $1000.00 $6328.25 Duration = $6328.25/$1000 = 6.3282 Ten-year CD Bank A (values in thousands of $s) Par value = $1000 Coupon rate = 10% Annual payments R = 10% Maturity = 10 years T CF PV of CF PV of CF × t 1 $100 $90.91 $90.91 2 $100 $82.64 $165.29 3 $100 $75.13 $225.39 4 $100 $68.30 $273.21 5 $100 $62.09 $310.46 6 $100 $56.45 $338.68 7 $100 $51.32 $359.21 8 $100 $46.65 $373.21 9 $100 $42.41 $381.69 10 $1000 $424.10 $4240.98 $1000.00 $6759.02 Duration = $6759.02/$1000 = 6.7590 Using the duration formula to estimate the change in value: Loan: Value = CD: Value = The difference in the change in value of the assets and liabilities for Bank A is $4943.66 using the duration estimation model. The difference in this estimate and the estimate found in part (a) above is due to the convexity of the two financial assets. The reason the change in asset values for Bank A is considerably larger than for Bank B is because of the difference in the durations of the loan and CD for Bank A. 21 If you use only duration to immunise your portfolio, what three factors affect changes in the net worth of a financial institution when interest rates change? The change in net worth for a given change in interest rates is given by the following equation: Thus, three factors are important in determining E. (i) [DA – DL k] or the leveraged adjusted duration gap. The larger this gap, the more exposed is the FI to changes in interest rates. (ii) A, or the size of the FI. The larger is A, the larger is the exposure to interest rate changes. (iii) ΔR/(1+R), or the interest rate shock. The larger is the shock, the larger is the interest rate risk exposure. 22 Financial Institution XY has assets of $1 million invested in a 30-year, 10 per cent semi-annual coupon Treasury Bond selling at par. The duration of this bond has been estimated at 9.94 years. The assets are financed with equity and a $900 000, two-year, 7.25 per cent semi-annual coupon capital note selling at par. (a) What is the leverage adjusted duration gap of Financial Institution XY? The duration of the capital note is 1.8975 years. Two-year capital note (values in thousands of $s) Par value = $900 Coupon rate = 7.25% Semi-annual payments R = 7.25% Maturity = 2 years T CF PV of CF PV of CF × t 0.5 $32.625 $31.48 $15.74 1 $32.625 $30.38 $30.38 1.5 $32.625 $29.32 $43.98 2 $932.625 $808.81 $1617.63 $900.00 $1707.73 Duration = $1707.73/$900.00 = 1.8975 The leverage-adjusted duration gap can be found as follows: (b) What is the impact on equity value if the relative change in all market interest rates is a decrease of 20 basis points? Note: The relative change in interest rates is R/(1+R/2) = –0.0020. The change in net worth using leverage adjusted duration gap is given by: (c) Using the information calculated in parts (a) and (b), what can be said about the desired duration gap for a financial institution if interest rates are expected to increase or decrease. If the FI wishes to be immune from the effects of interest rate risk (either positive or negative changes in interest rates), a desirable leverage-adjusted duration gap (DGAP) is zero. If the FI is confident that interest rates will fall, a positive DGAP will provide the greatest benefit. If the FI is confident that rates will increase, then a negative DGAP would be beneficial. (d) Verify your answer to part (c) by calculating the change in the market value of equity assuming that the relative change in all market interest rates is an increase of 30 basis points. (e) What would the duration of the assets need to be to immunise the equity from changes in market interest rates? Immunising the equity from changes in interest rates requires that the DGAP be 0. Thus, (DA – DLk) = 0 DA = DLk, or DA = 1.8975 × 0.9 = 1.70775 years. 23 The balance sheet for Got bucks Bank Inc. (GBI), is presented below ($ millions): Assets Liabilities and equity Cash $30 Core deposits $20 Interbank lending $20 Interbank borrowing $50 Loans (floating) $105 Euro CDs $130 Loans (fixed) $65 Equity $20 Total assets $220 Total liabilities and equity $220 Notes to the balance sheet: The interbank cash rate is 8.5 per cent, the floating loan rate is (BBR + 4 per cent), and currently BBR is 11 per cent. Fixed-rate loans have five-year maturities, are priced at par, and pay 12 per cent annual interest. The principal is repaid at maturity. Core deposits are fixed rate for two years at 8 per cent paid annually. The principal is repaid at maturity. Euro CDs currently yield 9 per cent. (a) What is the duration of the fixed-rate loan portfolio of GBI? Five-year loan (values in $ million) Par value = $65 Coupon rate = 12% Annual payments R = 12% Maturity = 5 years T CF PV of CF PV of CF × t 1 $7.80 $6.964 $6.964 2 $7.80 $6.218 $12.436 3 $7.80 $5.552 $16.656 4 $7.80 $4.957 $19.828 5 $72.80 $41.309 $206.543 $65.000 $262.427 Duration = $262.427/$65.000 = 4.0373 The duration is 4.0373 years. (b) If the duration of the floating-rate loans and interbank lending is 0.36 year, what is the duration of GBI’s assets? DA = [30(0) + 20(.36) + 105(.36) + 65(4.0373)]/220 = 1.3974 years (c) What is the duration of the core deposits if they are priced at par? Two-year core deposits (values in $ million) Par value = $20 Coupon rate = 8% Annual payments R = 8% Maturity = 2 years T CF PV of CF PV of CF × t 1 $1.60 $1.481 $1.481 2 $21.60 $18.519 $37.037 $20.000 $38.519 Duration = $38.519/$20.000 = 1.9259 The duration of the core deposits is 1.9259 years. (d) If the duration of the euro CDs and interbank borrowings is 0.401 year, what is the duration of GBI’s liabilities? DL = [20 × (1.9259) + 50 × (0.401) + 130 × (0.401)]/200 = 0.5535 years (e) What is GBI’s duration gap? What is its interest rate risk exposure? GBI’s leveraged adjusted duration gap is: 1.3974 – 200/220 × (0.5535) = 0.8942 years (f) What is the impact on the market value of equity if the relative change in all interest rates is an increase of 1 per cent (100 basis points)? Note that the relative change in interest rates is R/(1 + R) = 0.01. Since GBI’s duration gap is positive, an increase in interest rates will lead to a decrease in the market value of equity. For a 1 per cent increase, the change in equity value is: ΔE = –0.8942 × $220 000 000 × (0.01) = –$1 967 280 (new net worth will be $18 032 720). (g) What is the impact on the market value of equity if the relative change in all interest rates is a decrease of 0.5 per cent (–50 basis points)? Since GBI’s duration gap is positive, a decrease in interest rates will lead to an increase in market value of equity. For a 0.5 per cent decrease, the change in equity value is: ΔE = –0.8942 × (–0.005) × $220 000 000 = $983 647 (new net worth will be $20 983 647). (h) What variables are available to GBI to immunise the bank? How much would each variable need to change to get DGAP equal to zero? Immunisation requires the bank to have a leverage adjusted duration gap of 0. Therefore, GBI could reduce the duration of its assets to 0.5032 (0.5535 × 200/220) years by using more interbank funds and floating-rate loans. Or GBI could use a combination of reducing asset duration and increasing liability duration in such a manner that DGAP is 0. 24 Hands Insurance Company issued a $90 million, one-year, zero-coupon note at 8 per cent add-on annual interest (paying one coupon at the end of the year) or with an 8 per cent yield. The proceeds were used to fund a $100 million, two-year commercial loan with a 10 per cent coupon rate and a 10 per cent yield. Immediately after these transactions were simultaneously closed, all market interest rates increased 1.5 per cent (150 basis points). What is the true market value of the loan investment and the liability after the change in interest rates? The market value of the loan decreased by $2 551 831 to $97 448 169. MVA= $10 000 000 × PVIFA (n = 2, i = 11.5%) + $100 000 000 × PVIF (n = 2, i = 11.5%) = $9 7448 169. The market value of the note decreased $1 232 877 to $88 767 123. MVL = $97 200 000 × PVIF (n = 1, i = 9.5%) = $88 767 123 What impact did these changes in market value have on the market value of the FI’s equity? E = A – L = –$2 551 831 – (–$1 232 877) = –$1 318 954. The increase in interest rates caused the asset to decrease in value more than the liability that caused the market value of equity to decrease by $1 318 954. What was the duration of the loan investment and the liability at the time of issuance? Two-year loan (values in $ million) Par value = $100 Coupon rate = 10% Annual payments R = 10% Maturity = 2 years T CF PV of CF PV of CF × t 1 $10 $9.091 $9.091 2 $110 $90.909 $181.818 $100.000 $190.909 Duration = $190.909/$100.00 = 1.9091 The duration of the loan investment is 1.9091 years. The duration of the liability is one year since it is a one-year note that pays interest and principal at the end of the year. Use these duration values to calculate the expected change in the value of the loan and the liability for the predicted increase of 1.5 per cent in interest rates. The approximate change in the market value of the loan for a 150 basis points change is: The expected market value of the loan using the above formula is $97 396 700. The approximate change in the market value of the note for a 150 basis points change is: The expected market value of the note using the above formula is $88 750 000. What is the duration gap of Hands Insurance Company after the issuance of the asset and note? The leverage-adjusted duration gap was [1.9091 – (0.9)1.0] = 1.0091 years. What is the change in equity value forecasted by this duration gap for the predicted increase in interest rates of 1.5 per cent? MVE = –1.0091 × [0.015/(1.10)] × $100 000 000 = –$1 376 045. Note that this calculation assumes that the change in interest rates is relative to the rate on the loan. Further, this estimated change in equity value compares with the estimates above in part (d) as follows: MVE = MVA – MVL = –$2 603 300 – (–$1250 000) = –$1 353 300. If the interest rate prediction had been available during the time period in which the loan and the liability were being negotiated, what suggestions would you have offered to reduce the possible effect on the equity of the company? What are the difficulties in implementing your ideas? Obviously, the duration of the loan could be shortened relative to the liability, or the liability duration could be lengthened relative to the loan, or some combination of both. Shortening the loan duration would mean the possible use of variable rates, or some earlier payment of principal. The duration of the liability cannot be lengthened without extending the maturity life of the note. In either case, the loan officer may have been up against market or competitive constraints in that the borrower or investor may have had other options. Other methods to reduce the interest rate risk under conditions of this nature include using derivatives such as options, futures and swaps. 25 The following balance sheet information is available (amounts in thousands of dollars and duration in years) for a financial institution: Amount Duration T-notes $90 0.50 T-notes $55 0.90 T-notes (5 year) $176 x Loans $2724 7.00 Deposits $2092 1.00 Interbank borrowings $238 0.01 Equity $715 Treasury Bonds are five-year maturities paying 6 per cent semi-annually and selling at par. What is the duration of the T-bond portfolio? Five-year Treasury Bond Par value = $176 Coupon rate = 6% Semi-annual payments R = 6% Maturity = 5 years T CF PV of CF PV of CF × t 0.5 $5.28 $5.13 $2.56 1 $5.28 $4.98 $4.98 1.5 $5.28 $4.83 $7.25 2 $5.28 $4.69 $9.38 2.5 $5.28 $4.55 $11.39 3 $5.28 $4.42 $13.27 3.5 $5.28 $4.29 $15.03 4 $5.28 $4.17 $16.67 4.5 $5.28 $4.05 $18.21 5 $181.28 $134.89 $674.45 $176.00 $773.18 Duration = $773.18/$176.00 = 4.3931 What is the average duration of all the assets? [(0.5)($90) + (0.9)($55) + (4.3931)($176) + (7)($2724)]/$3045 = 6.5470 years What is the average duration of all the liabilities? [(1)($2092) + (0.01)($238)]/$2330 = 0.8989 years What is the leverage adjusted duration gap? What is the interest rate risk exposure? DGAP = DA – kDL = 6.5470 – ($2330/$3045)(0.8989) = 5.8592 years The duration gap is positive, indicating that an increase in interest rates will lead to a decrease in the market value of equity. What is the forecasted impact on the market value of equity caused by a relative upward shift in the entire yield curve of 0.5 per cent [i.e. R/(1 + R) = 0.0050]? The market value of the equity will change by: ΔMVE = –DGAP × (A) × ΔR/(1 + R) = –5.8592($3045)(0.0050) = –$89 207. The loss in equity of $89 207 will reduce the market value of equity to $625 793. If the yield curve shifts downward by 0.25 per cent [i.e. R/(1 + R) = –0.0025], what is the forecasted impact on the market value of equity? The change in the value of equity is ΔMVE = –5.8592($3045)(–0.0025) = $44 603. Thus, the market value of equity will increase by $44 603 to $759 603. What variables are available to the financial institution to immunise the balance sheet? How much would each variable need to change to get DGAP equal to 0? Immunisation requires the bank to have a leverage-adjusted duration gap of 0. Therefore, the FI could reduce the duration of its assets to 0.6878 years by using more T-notes and floating-rate loans. Or the FI could try to increase the duration of its deposits possibly by using fixed-rate CDs with a maturity of three or four years. Finally, the FI could use a combination of reducing asset duration and increasing liability duration in such a manner that DGAP is 0. This duration gap of 5.8592 years is quite large and it is not likely that the FI will be able to reduce it to zero by using only balance sheet adjustments. For example, even if the FI moved all of its loans into T-notes, the duration of the assets still would exceed the duration of the liabilities after adjusting for leverage. This adjustment in asset mix would imply forgoing a large yield advantage from the loan portfolio relative to the T-note yields in most economic environments. 26 Assume that a goal of the regulatory agencies of financial institutions is to immunise the ratio of equity to total assets, that is, (E/A) = 0. Explain how this goal changes the desired duration gap for the institution. Why does this differ from the duration gap necessary to immunise the total equity? How would your answers change to part (h) in problem 23 and part (g) in problem 25 change if immunising equity to total assets was the goal? In this case, the duration of the assets and liabilities should be equal. Thus, if E = A, then by definition the leveraged adjusted duration gap is positive, since E would exceed kA by the amount of (1 – k) and the FI would face the risk of increases in interest rates. In reference to problems 23 and 25, the adjustments on the asset side of the balance sheet would not need to be as strong, although the difference likely would not be large if the FI in question is a depository institution such as a bank or savings institution. 27 Identify and discuss three criticisms of using the duration model to immunise the portfolio of a financial institution. The three criticisms are: Immunisation is a dynamic problem because duration changes over time. Thus, it is necessary to rebalance the portfolio as the duration of the assets and liabilities change over time. Duration matching can be costly because it is not easy to restructure the balance sheet periodically, especially for large FIs. Duration is not an appropriate tool for immunising portfolios when the expected interest rate changes are large because of the existence of convexity. Convexity exists because the relationship between security price changes and interest rate changes is not linear, which is assumed in the estimation of duration. Using convexity to immunise a portfolio will reduce the problem. 28 In general, what changes have occurred in the financial markets that would allow financial institutions to restructure their balance sheets more rapidly and efficiently to meet desired goals? Why is it critical for an investment manager who has a portfolio immunised to match a desired investment horizon to rebalance the portfolio periodically? What is convexity? Why is convexity a desirable feature to capture in a portfolio of assets? The growth of purchased funds markets, asset securitisation and loan sales markets has considerably increased the speed of major balance sheet restructurings. Further, as these markets have developed, the cost of the necessary transactions has also decreased. Finally, the growth and development of the derivative securities markets provides significant alternatives to managing the risk of interest rate movements only with on-balance-sheet adjustments. Assets approach maturity at a different rate of speed than the duration of the same assets approaches zero. Thus, after a period of time, a portfolio or asset that was immunised against interest rate risk will no longer be immunised. In fact, portfolio duration will exceed the remaining time in the investment or target horizon, and changes in interest rates could prove costly to the institution. Convexity is a property of fixed-rate assets that reflects non-linearity in the reflection of price–rate relationships. This characteristic is similar to buying insurance to cover part of the interest rate risk faced by the FI. The more convex is a given asset, the more insurance against interest rate changes is purchased. 29 A financial institution has an investment horizon of two years 9.33 months (or 2.777 years). The institution has converted all assets into a portfolio of 8 per cent, $1000, three-year bonds that are trading at a yield to maturity of 10 per cent. The bonds pay interest annually. The portfolio manager believes that the assets are immunised against interest rate changes. Is the portfolio immunised at the time of bond purchase? What is the duration of the bonds? Three-year bonds Par value = $1000 Coupon rate = 8% Annual payments R = 10% Maturity = 3 years T CF PV of CF PV of CF × t 1 $80 $72.73 $72.73 2 $80 $66.12 $132.23 3 $1080 $811.42 $2434.26 $950.26 $2639.22 Duration = $2639.22/$950.26 = 2.777 The bonds have a duration of 2.777 years, which is 33.33 months. For practical purposes, the bond investment horizon was immunised at the time of purchase. (b) Will the portfolio be immunised one year later? After one year, the investment horizon will be 1 year, 9.33 months (or 1.777 years). At this time, the bonds will have a duration of 1.9247 years, or 1 year, 11+ months. Thus, the bonds will no longer be immunised. Two-year bonds Par value = $1000 Coupon rate = 8% Annual payments R = 10% Maturity = 2 years T CF PV of CF PV of CF × t 1 $80 $72.73 $72.73 2 $1080 $892.56 $1785.12 $965.29 $1857.85 Duration = $1857.85/$965.29 = 1.9247 (c) Assume that one-year, 8 per cent zero-coupon bonds are available in one year. What proportion of the original portfolio should be placed in these bonds to rebalance the portfolio? The investment horizon is 1 year, 9.33 months, or 21.33 months. Thus, the proportion of bonds that should be replaced with the zero-coupon bonds can be determined by the following analysis: 21.33 months = X × 12 months + (1–X) × 1.9247 × 12 months X = 15.92 per cent Thus, 15.92 per cent of the bond portfolio should be replaced with the zero-coupon bonds after one year. 30 Consider a 12-year, 12 per cent annual coupon bond with a required return of 10 per cent. The bond has a face value of $1000. (a) What is the price of the bond? PV = $120 × PVIFAi = 10%, n = 12 + $1000 × PVIFi = 10%, n = 12 = $1136.27 (b) If interest rates rise to 11 per cent, what is the price of the bond? PV = $120 × PVIFAi = 11%, n = 12 + $1000 × PVIFi = 11%, n = 12 = $1064.92 (c) What has been the percentage change in price? P = ($1064.92 – $1136.27)/$1136.27 = –0.0628 or –6.28 per cent. (d) Repeat parts (a), (b), and (c) for a 16-year bond. PV = $120 × PVIFA i = 10%, n = 16 + $1000 × PVIF i = 10%, n = 16 = $1156.47 PV = $120 × PVIFA i = 11%, n = 16 + $1000 × PVIF i = 11%, n = 16 = $1073.79 P = ($1073.79 – $1156.47)/$1156.47 = –0.0715 or –7.15 per cent. (e) What do the respective changes in bond prices indicate? For the same change in interest rates, longer term fixed-rate assets experience a greater change in price. 31 Consider a five-year, 15 per cent annual coupon bond with a face value of $1000. The bond is trading at a yield to maturity of 12 per cent. (a) What is the price of the bond? PV = $150 × PVIFA i = 12%, n = 5 + $1000 × PVIF i = 12%, n = 5 = $1108.14 (b) If the yield to maturity increases 1 per cent, what will be the bond’s new price? PV = $150 × PVIFA i = 13%, n = 5 + $1000 × PVIF i = 13%, n = 5 = $1070.34 (c) Using your answers to parts (a) and (b), what is the percentage change in the bond’s price as a result of the 1 per cent increase in interest rates? P = ($1070.34 – $1108.14)/$1108.14 = –0.0341 or –3.41 per cent. (d) Repeat parts (b) and (c) assuming a 1 per cent decrease in interest rates. PV = $150 × PVIFA i = 11%, n = 5 + $1000 × PVIF i = 11%, n = 5 = $1147.84 P = ($1147.84 – $1108.14)/$1108.14 = 0.0358 or 3.58 per cent (e) What do the differences in your answers indicate about the rate–price relationships of fixed-rate assets? For a given percentage change in interest rates, the absolute value of the increase in price caused by a decrease in rates is greater than the absolute value of the decrease in price caused by an increase in rates. 32 Consider a $1000 bond with a fixed-rate 10 per cent annual coupon rate and a maturity (N) of 10 years. The bond currently is trading at a yield to maturity (YTM) of 10 per cent. Complete the following table: Change N Coupon rate Yield to maturity Price $ Change in price from par % Change in price from par 8 10% 9% $1055.35 $55.35 5.535% 9 10 9 $1059.95 $59.95 5.995% 10 10 9 $1064.18 $64.18 6.418% 10 10 10 $1000.00 $0.00 0.00% 10 10 11 $941.11 –$58.89 –5.889% 11 10 11 $937.93 –$62.07 –6.207% 12 10 11 $935.07 –$64.93 –6.493% Use this information to verify the principles of interest rate–price relationships for fixed-rate financial assets. Rule 1. Interest rates and prices of fixed-rate financial assets move inversely. See the change in price from $1000 to $941.11 for the change in interest rates from 10 per cent to 11 per cent, or from $1000 to $1064.18 when rates change from 10 per cent to 9 per cent. Rule 2. The longer is the maturity of a fixed-income financial asset, the greater is the change in price for a given change in interest rates. A change in rates from 10 per cent to 11 per cent caused the 10-year bond to decrease in value $58.89, but the 11-year bond decreased in value $62.07, and the 12-year bond decreased $64.93. Rule 3. The change in value of longer term fixed-rate financial assets increases at a decreasing rate. For the increase in rates from 10 per cent to 11 per cent, the difference in the change in price between the 10-year and 11-year assets is $3.18, while the difference in the change in price between the 11-year and 12-year assets is $2.86. Rule 4. Although not mentioned in the Appendix, for a given percentage () change in interest rates, the increase in price for a decrease in rates is greater than the decrease in value for an increase in rates. For rates decreasing from 10 per cent to 9 per cent, the 10-year bond increases $64.18. But for rates increasing from 10 per cent to 11 per cent, the 10-year bond decreases $58.89. The following questions and problems are based on material in Appendix 6B to the chapter. 33 MLK Bank has an asset portfolio that consists of $100 million of 30-year, 8 per cent coupon, $1000 bonds that sell at par. What will be the bonds’ new prices if market yields change immediately by 0.10 per cent? What will be the new prices if market yields change immediately by 2.00 per cent? At + 0.10%: Price = $80 × PVIFA (n = 30, i = 8.1%) + $1000 × PVIF (n = 30, i = 8.1%) = $988.85 At – 0.10%: Price = $80 × PVIFA (n = 30, i = 7.9%) + $1000 × PVIF (n = 30, i = 7.9%) = $1011.36 At + 2.0%: Price = $80 × PVIFA (n = 30, i = 10%) + $1000 × PVIF (n = 30, i = 10%) = $811.46 At – 2.0%: Price = $80 × PVIFA (n = 30, i = 6.0%) + $1000 × PVIF (n = 30, i = 6.0%) = $1275.30 (b) The duration of these bonds is 12.1608 years. What are the predicted bond prices in each of the four cases using the duration rule? What is the amount of error between the duration prediction and the actual market values? P = –D × [R/(1+R)] × P At + 0.10%: P = –12.1608 × 0.001/1.08 × $1000 = –$11.26 P' = $988.74 At – 0.10%: P = –12.1608 × (–0.001/1.08) × $1000 = $11.26 P' = $1011.26 At + 2.0%: P = –12.1608 × 0.02/1.08) × $1000 = –$225.20 P' = $774.80 At – 2.0%: P = –12.1608 × (–0.02/1.08) × $1000 = $225.20 P' = $1225.20 Price—market determined Price—duration estimation Amount of error At + 0.10%: $988.85 $988.74 $0.11 At – 0.10%: $1011.36 $1011.26 $0.10 At + 2.0%: $811.46 $774.80 $36.66 At – 2.0%: $1275.30 $1225.20 $50.10 (c) Given that convexity is 212.4, what are the bond price predictions in each of the four cases using the duration plus convexity relationship? What is the amount of error in these predictions? P = {–D × [R/(1+R)] + ½ × CX × (R)2} × P At + 0.10%: P = {–12.1608 × 0.001/1.08 + 0.5 × 212.4 × (0.001)2} × $1000 = –$11.15 At – 0.10%: P = {–12.1608 × (–0.001/1.08) + 0.5 × 212.4 × (–0.001)2} × $1000 = $11.366 At + 2.0%: P = {–12.1608 × 0.02/1.08 + 0.5 × 212.4 × (0.02)2} × $1000 = –$182.72 At – 2.0%: P = {–12.1608 × (–0.02/1.08) + 0.5 × 212.4 × (–0.02)2} × $1000 = $267.68 Price market determined Price duration and convexity estimation Price duration and convexity estimation Amount of error At + 0.10%: $988.85 –$11.15 $988.85 $0.00 At – 0.10%: $1011.36 $11.37 $1011.37 $0.01 At + 2.0%: $811.46 –$182.72 $817.28 $5.82 At – 2.0%: $1275.30 $267.68 $1267.68 $7.62 (d) Diagram and label clearly the results in parts (a), (b) and (c). The profiles for the estimates based on only 0.10 per cent changes in rates are very close together and do not show clearly in a graph. However, the profile relationship would be similar to that shown above for the 2.0 per cent changes in market rates. 34 Estimate the convexity for each of the following three bonds, all of which trade at yield to maturity of 8 per cent and have face values of $1000. A 7-year, zero-coupon bond. A 7-year, 10 per cent annual coupon bond. A 10-year, 10 per cent annual coupon bond that has a duration value of 6.994 years (i.e. approximately 7 years). Market Value at 8.01 per cent Market Value at 7.99 per cent Capital Loss + Capital Gain Divided by Original Price 7-year zero –0.37804819 0.37832833 0.00000048 7-year coupon –0.55606169 0.55643682 0.00000034 10-year coupon –0.73121585 0.73186329 0.00000057 Convexity = 108 × (Capital Loss + Capital Gain) ÷ Original Price at 8.00 per cent 7-year zero CX = 100 000 000 × 0.00000048 = 48 7-year coupon CX = 100 000 000 × 0.00000034 = 34 10-year coupon CX = 100 000 000 × 0.00000057 = 57 An alternative method of calculating convexity for these three bonds using the following equation is illustrated at the end of this problem and onto the following page. Rank the bonds in terms of convexity, and express the convexity relationship between zeros and coupon bonds in terms of maturity and duration equivalencies. Ranking, from least to most convexity: 7-year coupon bond, 7-year zero, 10-year coupon Convexity relationships: Given the same yield to maturity, a zero-coupon bond with the same maturity as a coupon bond will have more convexity. Given the same yield to maturity, a zero-coupon bond with the same duration as a coupon bond will have less convexity. Zero coupon bond Par value = $1000 Coupon = 0% R = 8% Maturity = 7 years t . CF PV of CF PV of CF × t ×(1+t) ×(1+R)2 1 $0.00 $0.00 $0.00 $0.00 2 $0.00 $0.00 $0.00 $0.00 3 $0.00 $0.00 $0.00 $0.00 4 $0.00 $0.00 $0.00 $0.00 5 $0.00 $0.00 $0.00 $0.00 6 $0.00 $0.00 $0.00 $0.00 7 1000.00 $583.49 $4084.43 $32 675.46 $583.49 $4084.43 $32 675.46 680.58 Duration = 7.0000 Convexity = 48.011 7-year coupon bond Par value = $1000 Coupon = 10% R = 8% Maturity = 7 years t . CF PV of CF PV of CF × t ×(1+t) ×(1+R)2 1 $100.00 $92.59 $92.59 $185.19 2 $100.00 $85.73 $171.47 $514.40 3 $100.00 $79.38 $238.15 $952.60 4 $100.00 $73.50 $294.01 $1 470.06 5 $100.00 $68.06 $340.29 $2 041.75 6 $100.00 $63.02 $378.10 $2 646.71 7 $1 100.00 $641.84 $4492.88 $35 943.01 $1104.13 $6007.49 $43 753.72 1287.9 Duration = 5.4409 Convexity = 33.974 10-year coupon bond Par value = $1000 Coupon = 10% R = 8% Maturity = 10 years t . CF PV of CF PV of CF × t ×(1+t) ×(1+R)2 1 $100.00 $92.59 $92.59 $185.19 2 $100.00 $85.73 $171.47 $514.40 3 $100.00 $79.38 $238.15 $952.60 4 $100.00 $73.50 $294.01 $1 470.06 5 $100.00 $68.06 $340.29 $2 041.75 6 $100.00 $63.02 $378.10 $2 646.71 7 $100.00 $58.35 $408.44 $3 267.55 8 $100.00 $54.03 $432.22 $3 889.94 9 $100.00 $50.02 $450.22 $4 502.24 10 $1100.0 $509.51 $5095.13 $56 046.41 $1134.20 $7900.63 $75 516.84 1322.9 Duration = 6.9658 Convexity = 57.083 35 A 10-year, 10 per cent annual coupon, $1000 bond trades at a yield to maturity of 8 per cent. The bond has a duration of 6.994 years. What is the modified duration of this bond? What is the practical value of calculating modified duration? Does modified duration change the result of using the duration relationship to estimate price sensitivity? Modified duration = Duration/(1+ R) = 6.994/1.08 = 6.4759. Some practitioners find this value easier to use because the percentage change in value can be estimated simply by multiplying the existing value times the basis point change in interest rates rather than by the relative change in interest rates. Using modified duration will not change the estimated price sensitivity of the asset. Integrated mini case: calculating and using duration GAP State Bank’s balance sheet is listed below. Market yields and durations (in years) are in parenthesis, and amounts are in millions. Assets Liabilities and equity Cash $20 Demand deposits $250 Interbank lending (5.05%, 0.02) $150 Savings accounts (4.5%, 0.50) $360 T-notes (5.25%, 0.22) $300 CDs (4.3%, 0.48) $715 T-bonds (7.50%, 7.55) $200 CDs (6%, 4.45) $1105 Consumer loans (6%, 2.50) $900 Interbank borrowings (5%, 0.02) $515 Business loans (5.8%, 6.58) $475 Commercial paper (5.05%, 0.45) $400 Fixed-rate mortgages (7.85%, 19.50) $1200 Subordinated debt: fixed-rate (7.25%, 6.65) $200 Variable-rate mortgages, repriced @ quarter (6.3%, 0.25) $580 Premises and equipment $120 Total liabilities $3545 Equity $400 Total assets $3945 Total liabilities and equity $3945 (a) What is State Bank’s duration gap? DA = [20(0) + 150(0.02) + 300(0.22) + 200(7.55) + 900(2.50) + 475(6.85) + 1200(19.50) + 580(0.25) + 120(0)]/3945 = 7.76369 year DL = [250(0) + 360(0.50) + 715(0.48) + 1105(4.45) + 515(0.02) + 400(0.45) + 200(6.65))]/3545 = 1.96354 years DGAP = DA – kDL = 7.76369 – ($3545/$3945)(1.96354) = 5.99924 years (b) Use these duration values to calculate the expected change in the value of the assets and liabilities of State Bank for the predicted increase of 1.5 per cent in interest rates. ΔMVinterbank lending = –0.02 × 0.015/1.0505 × 150m = –$42 837 ΔMVT-notes = –0.22 × 0.015/1.0525 × 300m = –$940 618 ΔMVT-bonds = –7.55 × 0.015/1.0750 × 200m = –$21 069 767 ΔMVconsumer loans = –2.50 × 0.015/1.0600 × 900m = –$31 839 623 ΔMVbusiness loans = –6.58 × 0.015/1.0580 × 475m = –$44 312 382 ΔMVfixed-rate mortgages = –19.50 × 0.015/1.0785 × 1200m = –$325 452 017 ΔMVvariable-rate mortgages = –0.25 × 0.015/1.0630 × 580m = –$2 046 096 =>ΔMVA = –$425 703 339 ΔMVsavings deposits = –0.50 × 0.015/1.045 × 360m = –$2 583 732 ΔMVCDs = –0.48 × 0.015/1.0430 × 715m = –$4 935 762 ΔMVCDs = –4.45 × 0.015/1.0600 × 1105m = –$69 583 726 ΔMVinterbank borrowings = –0.02 × 0.015/1.0500 × 515m = –$147 143 ΔMVcommercial paper = –0.45 × 0.015/1.0505 × 400m = –$2 570 205 ΔMVfixed-rate subordinate debt = –6.65 × 0.015/1.0725 × 200m = –$18 601 399 =>ΔMVL = –$98 421 967 (c) What is the change in equity value forecasted from the duration values for the predicted increase in interest rates of 1.5 per cent? ΔMVE = ΔMVA – ΔMVL = –$425 703 339 – (–$98 421 967) = –$327 281 372 Additional example for Chapter 6 This example is to estimate both the duration and convexity of a 6-year bond paying 5 per cent coupon annually and the annual yield to maturity is 6 per cent. 6-year coupon bond Par value = $1000 Coupon = 5% R = 6% Maturity = 6 years t . CF PV of CF PV of CF × t ×(1+t) ×(1+R)2 1 $50.00 $47.17 $47.17 $94.34 2 $50.00 $44.50 $89.00 $267.00 3 $50.00 $41.98 $125.94 $503.77 4 $50.00 $39.60 $158.42 $792.09 5 $50.00 $37.36 $186.81 $1 120.89 6 $1050.00 $740.21 $4441.25 $31 088.76 $950.83 $5048.60 $33 866.85 1068.3 Duration = 5.3097 Convexity = 31.7 Using the textbook method: CX = 108 [(950.3506 – 950.8268)/950.8268 + (951.3032 – 950.8268)/950.8268] = 108[–0.0005007559 + 0.0005501073] = 31.70 What is the effect of a 2 per cent increase in interest rates, from 6 per cent to 8 per cent? Using present values, the percentage change is: = ($950.8268 – $861.3136)/ $950.8268 = –9.41% Using the duration formula: ΔMVA = –D × ΔR/(1 + R) + 0.5CX(R)2 = –5.3097 × [(0.02)/1.06] + 0.5(31.7)(0.02)2 = –0.1002 + .0063 = –9.38% Adding convexity adds more precision. Duration alone would have given the answer of –0.02%. Solution Manual for Financial Institutions Management Anthony Saunders, Marcia Cornett, Patricia McGraw 9780070979796, 9780071051590