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Harbor UCLA Medical Center Measures of Duration Time Duration Worksheet
1.Consider two defaultable 1-year loans with a principal of $1 million
each. The probability of default on each loan is 2.5%. Assume that if one
loan defaults, the other does not. Assume that in the event of default,
the loan leads to a loss that can take any value between $0 and $1
million with equal probability, i.e., the probability that the loss is higher
than $ million is 1 − . If a loan does not default, it yields a profit
equal to $20,000.
a) Compute the 1-year 98% Value at Risk (VaR) and Expected Shortfall
(ES) of a single loan.
b) Compute the 1-year 98% VaR and ES for the portfolio of both loans.
c) Does the VaR and the ES satisfy the subadditivity property in this case?
2.Bond A is a 4-year bond with a 10% coupon rate and Bond B is a 2-year
bond with a 20% coupon rate. Both bonds have a face value of £100,
and all coupons are paid annually, starting in year 1. Consider a portfolio
that consists of one unit of Bond A and one unit of Bond B. The yield
curve is flat at = 5%.
a) What is the Macaulay duration of this portfolio?
b) Compute the approximate percentage change in the value of the
portfolio if the interest rate increases by 200 basis points.
c) Suppose a pension fund holds this portfolio and is worried about
fluctuations in its price. Explain how the fund can hedge its interest
rate risk using 3-year zero-coupon bonds with a face value of £100.
