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Chapter Nine

By Frank Grant,2014-02-17 07:25
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Chapter Nine

9. Maximum Pension 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 percent. The bonds differ only in that the coupons

    are 7 percent, 9 percent, and 11 percent.

     a. What is the duration for each bond?

Five-year Bond

     Par value = $1,000 Coupon rate = 7% Annual payments

     R = 9% Maturity = 5 years

    Time Cash Flow PV of CF PV of CF x 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 $1,070 $695.43 $3,477.13

     $922.21 $4,019.71 Duration = $4,019.71/$922.21 = 4.3588

Five-year Bond

     Par value = $1,000 Coupon rate = 9% Annual payments

     R = 9% Maturity = 5 years

    Time Cash Flow PV of CF PV of CF x 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 $1,090 $708.43 $3,542.13

     $1,000.00 $4,239.72 Duration = $4,239.72/$1,000.00 = 4.2397

Five-year Bond

     Par value = $1,000 Coupon rate = 11% Annual payments

     R = 9% Maturity = 5 years

    Time Cash Flow PV of CF PV of CF x 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 $1,110 $721.42 $3,607.12

     $1,077.79 $4,459.73 Duration = $4,459.73/$1,077.79 = 4.1378

     b. What is the relationship between duration and the amount of coupon interest that is paid?

    Plot the relationship.

     9-1

    Duration and Coupon Rates

    Duration decreases as the amount of coupon

    interest increases.

    4.3588 Change in

    4.2397 Duration Coupon Duration

    4.1378Years4.3588 7%

    4.2397 9% -0.1191 4.00 4.1378 11% -0.1019 7%9%11%

    Coupon Rates

13. You have discovered that the price of a bond rose from $975 to $995 when the yield to

    maturity fell from 9.75 percent to 9.25 percent. What is the duration of the bond?

    20?P975P?D????4.5years?D?4.5years We know ?R?.005(1?R)1.0975

14. Calculate the duration of a two-year, $1,000 bond that pays an annual coupon of 10 percent

    and trades at a yield of 14 percent. What is the expected change in the price of the bond if

    interest rates decline by 0.50 percent (50 basis points)?

Two-year Bond

     Par value = $1,000 Coupon rate = 10% Annual payments

     R = 14% Maturity = 2 years

    Time Cash Flow PV of CF PV of CF x t

     1 $100 $87.72 $87.72

     2 $1,100 $846.41 $1,692.83

     $934.13 $1,780.55 Duration = $1,780.55/$934.13 = 1.9061

    ?R?.005The expected change in price = ?DP??1.9061$934.13?$7.81. This implies a 1?R1.14new price of $941.94. The actual price using conventional bond price discounting would be

    $941.99. The difference of $0.05 is due to convexity, which was not considered in the duration

    elasticity measure.

15. The duration of an 11-year, $1,000 Treasury bond paying a 10 percent semiannual coupon

    and selling at par has been estimated at 6.9 years.

     a. What is the modified duration of the bond?

     Modified Duration = D/(1 + R/2) = 6.9/(1 + .10/2) = 6.57 years

     9-2

     b. What will be the estimated price change on the bond if interest rates increase 0.10

    percent (10 basis points)? If rates decrease 0.20 percent (20 basis points)?

     Estimated change in price = -MD x ?R x P = -6.57 x 0.001 x $1,000 = -$6.57.

     Estimated change in price = -MD x ?R x P = -6.57 x -0.002 x $1,000 = $13.14.

     c. What would be the actual price of the bond 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 Price Actual

     Change Estimated Price Error

     + 0.001 $993.43 $993.45 $0.02

     - 0.002 $1,013.14 $1,013.28 -$0.14

18. Two banks are being examined by the regulators to determine the interest rate sensitivity of

    their balance sheets. Bank A has assets composed solely of a 10-year, 12 percent, $1

    million loan. The loan is financed with a 10-year, 10 percent, $1 million CD. Bank B has

    assets composed solely of a 7-year, 12 percent 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 percent coupon, $1,000,000 face value CD with a yield to

    maturity of 10 percent. The loan and the CDs pay interest annually, with principal due at

    maturity.

     a. If market interest rates increase 1 percent (100 basis points), how do the market values

    of the assets and liabilities of each bank change? That is, what will be the net affect 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 + $1,000,000*PVIF = $945,737.57. n=10,i=13%n=10,i=13%

     CD: $100,000*PVIFA + $1,000,000*PVIF = $941,107.68. n=10,i=11%n=10,i=11%

     Therefore, the decrease in value of the asset was $4,629.89 less than the liability.

     For Bank B:

     Bond: $1,976,362.88*PVIF = $840,074.08. n=7,i=13%

     CD: $82,750*PVIFA + $1,000,000*PVIF = $839,518.43. n=10,i=11%n=10,i=11%

     The bond value decreased $53,932.12, and the CD value fell $54,487.79. Therefore,

    the decrease in value of the asset was $555.67 less than the liability.

     b. What accounts for the differences in the changes of the market value of equity between

    the two banks?

     9-3

     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.

    Summarize your results.

    Ten-year CD Bank B (values in thousands of $s)

     Par value = $1,000 Coupon rate = 8.275% Annual payments

     R = 10% Maturity = 10 years

    Time Cash Flow PV of CF PV of CF x 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 $1,082.75 $417.45 $4,174.47

     $894.01 $6,258.15 Duration = $6,258.15/894.01 = 7.0001

     The duration for the CD of Bank B is calculated above to be 7.0001 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:

    ?R.01 Bond: ?Value = ?DP??7.0$894,006.20??$55,875.391?R1.12

    ?R.01?DP??7.0001$894,006.20??$56,892.12 CD: ?Value = 1?R1.10

     The difference in the change in value of the assets and liabilities for Bank B is $1,016.73

    using the duration estimation model. The small 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 = $1,000 Coupon rate = 12% Annual payments

     R = 12% Maturity = 10 years

     9-4

    Time Cash Flow PV of CF PV of CF x 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 $43.27 $389.46

     10 $1,120 $360.61 $3,606.10

     $1,000.00 $6,328.25 Duration = $6,328.25/$1,000 = 6.3282

Ten-year CD Bank A (values in thousands of $s)

     Par value = $1,000 Coupon rate = 12% Annual payments

     R = 12% Maturity = 10 years

    Time Cash Flow PV of CF PV of CF x 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 $1,100 $424.10 $4,240.98

     $1,000.00 $6,759.02 Duration = $6,759.02/$1,000 = 6.7590

     Using the duration formula to estimate the change in value:

    ?R.01?DP??6.3282$1,000,000??$56,501.79 Loan: ?Value = 1?R1.12

    ?R.01 CD: ?Value = ?DP??6.7590$1,000,000??$61,445.45 1?R1.10

     The difference in the change in value of the assets and liabilities for Bank A is $4,943.66

    using the duration estimation model. The small 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.

20. Financial Institution XY has assets of $1 million invested in a 30-year, 10 percent

    semiannual coupon Treasury bond selling at par. The duration of this bond has been

     9-5

    estimated at 9.94 years. The assets are financed with equity and a $900,000, 2-year, 7.25

    percent semiannual 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% Semiannual payments

     R = 7.25% Maturity = 2 years

    Time Cash Flow PV of CF PV of CF x 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 $1,617.63

     $900.00 $1,707.73 Duration = $1,707.73/$900.00 = 1.8975

     The leverage-adjusted duration gap can be found as follows:

    $900,000??Leverage?adjusteddurationgap?D?Dk?9.94?1.8975?8.23years AL$1,000,000

     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:

    ?R9 ?? ???E??D?Dk*A*??9.94?(1.8975)(1,000,000)(?.002)?$16,464AL10R1?2

     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 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.

     9-6

    ?R ?????E??D?Dk*A*??8.23225(1,000,000)(.003)??$24,697ALR1?2

     e. What would the duration of the assets need to be to immunize the equity from changes

    in market interest rates?

     Immunizing the equity from changes in interest rates requires that the DGAP be 0. Thus,

    (D-Dk) = 0 ? D = Dk, or D = 1.8975x0.9 = 1.70775 years. ALALA

     9-7

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