You are in charge of the risk management division at XYZ, Inc. XYZ’ assets consist of investments worth $30 million today in 3-year zero-coupon bonds. XYZ’s liabilities consist of a single payment of $43 million due in 10 years. The term structure of interest rates is flat at 5%. Assume that all shifts in the term structure are parallel. a. What is the Macaulay and Modified duration of XYZ’ assets? b. XYZ, Inc. thinks that interest rates may rise to 5.5%. Compute the exact change in the value of XYZ’s assets and compare this with the approximate change using duration.

The step by step solution to your given problem is done below.

Explanation:

a. What is the Macaulay and Modified duration of XYZ’ assets?

The Macaulay duration calculates the weighted average time before a bondholder would receive the bond's cash flows.

D=3 as given in the question.

The modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity.

Modified duration (MD) = D/(1+r)=3/1.05 = 2.857

b. XYZ, Inc. thinks that interest rates may rise to 5.5%. Compute the exact change in the value of XYZ’s assets and compare this with the approximate change using duration.

We have to first calculate the price change in a 3-year zero coupon bond with a face value equal to $100.

If interest rates are 5%: P=100/(1.05)3 = $86.3838

If interest rates are 5.5%: P=100/(1.055)3 = $85.1614

Then the percentage change: (85.1614-86.3838)/86.3838 = -1.415%

Then, the change in the asset portfolio: 30*(-1.415%) = -$0.425m

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