Question

6.

An Fl has a $270 million asset portfolio that has an average duration of 7.8 years. The average duration of its $230 million in liabilities is 4.6 years. Assets and liabilities are yielding 15 percent. The Fl uses put options on T-bonds to hedge against unexpected interest ate increases. The average delta (5) of the put options has been estimated at -0.3 and the average duration of the T-bonds is 8.3 years. The current market value of the T-bonds is $97,000. Put options on T-bonds are selling at a premium of $1.30 per face value of $100. a. What is the modified duration of the T-bonds if the current level of interest rates is 15 percent? (Do not round intermediate calculations. Round your answer to 4 decimal places. (e.g., 32.1616)) b. How many put option contracts should the Fl purchase to hedge its exposure against rising interest rates? The face value of the T- bonds is $100,000. (Do not round intermediate calculations. Round your answer to the c. If interest rates increase 50 basis points, what will be the change in value of the equity of the FI? (Enter your answer in dollars not in millions. Negative amount should be indicated by a minus sign. Do not round intermediate calculations. Round your answer to the nearest dollar amount.) d. If interest rates increase 50 basis points, what will be the change in value of the T-bond option hedge position? (Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest dollar amount.) e. What must be the change in interest rates before the change in value of the balance sheet (equity) will offset the cost of placing the hedge? (Do not round intermediate calculations. Negative value should be indicated by a minus sign. Round your answer to 2 decimal places. (e.g., 32.16)) f. How much must interest rates change before the payoff of the hedge will exactly cover the cost of placing the hedge? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16))

Answer 1

Solution:

a. The modified duration, MD of T-bond is calculated as follows:-

MD = D/(1 + i)

MD = 8.3/1.15

MD = 7.2174 years

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b. The number of put options, $N_{p}$ is calculated using the formula

$N_{p} = \frac{(D_{A}-kD_{L})*A}{\delta*-D*B}$

$N_{p} = \frac{(7.8-(230/270)(4.6))*270,000,000}{(-0.3)*-8.3*97,000}$

$N_{p}$ = 4339 contracts

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c. The change in value of equity of the FI is calculated as follows:-

Change in equity value = $-DGAP*A*(\frac{\Delta R}{(1+R)})$

Change in equity value = -3.8815*270,000,000*0.005/1.15

Change in equity value = -$4,556,521.74

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d. The change in value of T-bond option hedge position, $\DeltaP$ $\DeltaP$$\Delta P$

$\Delta P = N_{p}(\delta*-MD*B*\Delta R)$

$\Delta P =$4,339 (-0.3*-7.2174*97,000*0.005)

$\Delta P =$ $4,556,515.96