Problem set 3 (Duration II) with answers PDF

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Problem set 3 (Duration II) Past exam questions: 1. Managing interest rate risk for the firm The following is a simplified balance sheet of EASTPAC Bank. Market Value Market (millions) Value(millions) Assets Duration Liabilities

Duration

Asset 1

$30

5 years

Liability 3

$60

1 year

Asset 2

$90

6 years

Liability 4

$40

9 years

What is the leverage-adjusted duration gap for the bank? What is the change in equity value if the interest rate for all assets and liabilities increases from 10% to 11.1%? Answer: Leverage-adjusted duration gap = DA – DL*k DA = 5*(30/(30+90)) + 6*(90/(30+90)) = 5.75 DL = 1*(60/(60+40)) + 9*(40/(60+40)) = 4.2 k = (60+40)/(30+90) ~= 0.83 Duration gap = 2.26 Change in equity value = -( DA – DL*k)*Asset value*ΔR/(1+R) = -2.26*(30+90)*1.1%/(1+10%) ~= 2.71 2. Convexity An AAA rated bond has a maturity of 2 years, a duration of 1.95 years and a yield to maturity of 5%. The convexity (CX) of this bond is 6. What is the approximate percentage price change for this bond if interest rates increase to 6% by considering the convexity? Answer: ΔP/P = -D*ΔR/(1+R) + 1/2*CX*(ΔR)2 = -1.95*1%/(1+5%)+0.5*6*(1%)^2 ~= -1.83%

End-of-chapter questions: Chapter 9 16. 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 fall by 0.50 percent (50 basis points)? Two-year Bond: Par value = $1,000 Coupon rate = 10% Annual payments R = 14% Maturity = 2 years t CFt DFt CFt x DFt CFt x DFt x t 1 100 0.8772 87.72 87.72 2 1,100 0.7695 846.41 1,692.83 934.13 1,780.55 Duration = $1,780.55/$934.13 = 1.9061 The expected change in price =  D

R 1 R

P  - (1.9061) x (-0.0050/1.14) x $934.13 = $7.81.

This implies a new price of $941.94 ($934.13 + $7.81). The actual price using conventional bond price discounting would be $941.99. The difference of $0.05 is due to convexity, which is not considered in the duration elasticity measure.

22. If an FI uses only duration to immunize its portfolio, what three factors affect changes in the net worth of the FI when interest rates change? The change in net worth for a given change in interest rates is given by the following equation:  E   D A  D L k * A *

R 1 R

where k 

L A

Thus, three factors are important in determining E. 1) [DA - D L k] or the leveraged adjusted duration gap. The larger this gap, the more exposed is the FI to changes in interest rates. 2) A, or the size of the FI. The larger is A, the larger is the exposure to interest rate changes. 3) ΔR/(1 + R), or the interest rate shock. The larger is the shock, the larger is the interest rate risk exposure. 24. The balance sheet for Gotbucks Bank, Inc. (GBI), is presented below ($ millions): Assets Liabilities and Equity Cash $30 Core deposits $20 Federal funds 20 Federal funds 50 Loans (floating) 105 Euro CDs 130 Loans (fixed) 65 Equity 20 Total assets $220 Total liabilities & equity $220 Notes to the balance sheet: The fed funds rate is 8.5 percent, the floating loan rate is LIBOR + 4 percent, and currently LIBOR is 11 percent. Fixed rate loans have five-year maturities, are priced

at par, and pay 12 percent annual interest. The principal is repaid at maturity. Core deposits are fixed rate for two years at 8 percent paid annually. The principal is repaid at maturity. Euro CDs currently yield 9 percent. a. What is the duration of the fixed-rate loan portfolio of Gotbucks Bank? Five-year Loan (values in millions of $s) Par value = $65 Coupon rate = 12% Annual payments R = 12% Maturity = 5 years t CFt DFt CFt x DFt CFt x DFt x t 1 7.8 0.8929 6.964 6.964 2 7.8 0.7972 6.218 12.436 3 7.8 0.7118 5.552 16.656 4 7.8 0.6355 4.957 19.828 5 72.8 05674 41.309 206.543 65.000 262.427 Duration = $262.427/$65.000 = 4.0373 b. If the duration of the floating-rate loans and fed funds is 0.36 year, what is the duration of GBI’s assets? DA = [$30(0) + $20(0.36) + $105(0.36) + $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 millions of $s) Par value = $20 Coupon rate = 8% Annual payments R = 8% Maturity = 2 years t CFt DFt CFt x DFt CFt x DFt x t 1 1.6 0.9259 1.481 1.481 2 21.6 0.8573 18.519 37.037 20.000

38.519

Duration = $38.519/$20.000 = 1.9259 d. If the duration of the Euro CDs and fed funds liabilities 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 x (0.5535) = 0.8942 years Since GBI’s duration gap is positive, an increase in interest rates will lead to a decrease in the market value of equity.

f. What is the impact on the market value of equity if the relative change in all interest rates is an increase of 1 percent (100 basis points)? Note that the relative change in interest rates is R/(1+R) = 0.01. For a 1 percent increase, the change in equity value is: ΔE = -0.8942 x $220,000,000 x (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 percent (-50 basis points)? For a 0.5 percent decrease, the change in equity value is: ΔE = -0.8942 x (-0.005) x $220,000,000 = $983,647 (new net worth will be $20,983,647). h. What variables are available to GBI to immunize the bank? How much would each variable need to change to get DGAP equal to zero? Immunization 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 x 200/220) years by using more fed 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. 34. MLK Bank has an asset portfolio that consists of $100 million of 30-year, 8 percent coupon, $1,000 bonds that sell at par. a. What will be the bonds’ new prices if market yields change immediately by  0.10 percent? What will be the new prices if market yields change immediately by  2.00 percent? At +0.10%: At –0.10%:

Price = $80 x PVAn=30, i=8.1% + $1,000 x PVn=30, i=8.1% = $988.85 Price = $80 x PVAn=30, i=7.9% + $1,000 x PVn=30, i=7.9% = $1,011.36

At +2.0%: At –2.0%:

Price = $80 x PVAn=30, i=10% + $1,000 x PVn=30, i=10% = $811.46 Price = $80 x PVAn=30, i=6.0% + $1,000 x PVn=30, i=6.0% = $1,275.30

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 x [R/(1+R)] x P At +0.10%: At -0.10%:

P = -12.1608 x 0.001/1.08 x $1,000 = -$11.26  P’ = $988.74 P = -12.1608 x (-0.001/1.08) x $1,000 = $11.26  P’ = $1,011.26

At +2.0%: At -2.0%:

P = -12.1608 x 0.02/1.08) x $1,000 = -$225.20  P’ = $774.80 P = -12.1608 x (-0.02/1.08) x $1,000 = $225.20  P’ = $1,225.20

Price - market determined At +0.10%: $988.85 At -0.10%: $1,011.36 At +2.0%: $811.46 At -2.0%: $1,275.30

Price - duration estimation $988.74 $1,011.26 $774.80 $1,225.20

Amount of error $0.11 $0.10 $36.66 $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 x [R/(1+R)] + ½ x CX x (R)2} x P At +0.10%: At -0.10%: At +2.0%: At -2.0%:

At +0.10%: At -0.10%: At +2.0%: At -2.0%:

P = {-12.1608 x 0.001/1.08 + 0.5 x 212.4 x (0.001)2} x $1,000 = -$11.15 P = {-12.1608 x (-0.001/1.08) + 0.5 x 212.4 x (-0.001)2} x $1,000 = $11.366 P = {-12.1608 x 0.02/1.08 + 0.5 x 212.4 x (0.02)2} x $1,000 = -$182.72 P = {-12.1608 x (-0.02/1.08) + 0.5 x 212.4 x (-0.02)2} x $1,000 = $267.68

Price market determined $988.85 $1,011.36 $811.46 $1,275.30

Price duration & convexity estimation -$11.15 $11.37 -$182.72 $267.68

Price duration & convexity estimation $988.85 $1,011.37 $817.28 $1,267.68

Amount of error $0.00 $0.01 $5.82 $7.62

Adding convexity adds more precision, though does not remove the approximation error entirely....