A bank has two, 3-year commercial loans with a present value of $70 million. The

Question

A bank has two, 3-year commercial loans with a present value of $70 million. The first is a $30 million loan that requires a single payment of $37.8 million in 3 years, with no other payments until then.

The second is for $40 million. It requires an annual interest payment of $3.6 million. The principal of

$40 million is due in 3 years.

a. What is the duration of the bank’s commercial loan portfolio?

b. What will happen to the value of its portfolio if the general level of interest rates increased from

8% to 8.5%?

Explanation / Answer

The duration of the first loan is 3 years since it is a zero-coupon loan. The duration
of the second loan is as follows:

Note that I used the 8% YTM referred to in problem (b) to do the calculations
above. But you could also have used 9% as this is the YTM on a 40m loan with
3.6m payments and 40m due at maturity. The answer is very similar due to the
short term to maturity.

(a) The duration of a portfolio is the weighted average duration of its individual
securities. So, the portfolio’s duration = 3/7 * (3) + 4/7 * (2.76) = 2.86

(b) if rates increased, price will change by equal to minus duration multilied by (change in interest rate/(1+previous interest rate) multiplied by value of the loan

=-2.86*(0.005/1.08)*70,000,000 = -926852

Year 1 2 3 sum Payment 3.60 3.60 43.60 PV of Payments 3.33 3.09 34.61 41.03 Time Weighted PV of Payments 3.33 6.18 103.83 Time Weighted PV of Payments
Divided by Price 0.08 0.15 2.53 2.76

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