Question

"If the investment horizon is equal to the Macaulay duration of the bond, the investor is hedged against interest rate risk". However, the above statement is only true if interest rates only change before fist coupon payment is received.

Using the following bond to show that if interest rate increases 2% between first and second coupon payment dates, the investor is not hedged against interest rate risk even if his duration gap is zero.:

A four-year 33.7% annual coupon paying bond is priced at 170.4255. The bond's Macaulay duration is 3.

Part of the Solution:

To check whether the investor is hedged against interest rate risk or not, the only thing that you have to do is to calculate his realized rate of return (or horizon yield), given some interest rate changes (which is 2% here). If his realize rate of return is equal to the YTM when he purchased this bond, he is considered as hedged.We need to know the investor realized rate of return. And we also need to know the YTM when he purchased this bond. Because the investor’s duration gap is zero, we can infer that his investment horizon is equal to 3. (Which is the bond’s Macaulay duration. Recall duration gap = Macaulay duration –investment horizon). Now the question becomes the following one:An investor initially purchases a four-year 33.7% annual coupon paying bond which is priced at 170.4255. Assume the interest rates increase 2%between the first and second coupon payment dates, what is his realize rate of return if he sells this bond after three years? Does his realized rate of return equal to the YTM when he purchased this bond?

Answer 1

YTM = Rate (Nper, PMT, PV, FV) = Rate (4, 33.7% x 100, -170.4255, 100) = 11.00%

Price of the bond after three years = - PV (Rate, Nper, PMT, FV) = - PV (11% + 2%, 4 - 3, 33.7% x 100, 100) = 118.32

Hence, his cash flows and return realized:

No 63 Year Cash flows How / Why? 0 (170.43) Purchase cost 65 1 33.70 annual coupon = 33.7% x 100 66 2 33.70 annual coupon = 33.7% x 100 annual coupon + sale price = 152.02 33.7 + 118.32 68 IRR 10.59% "=IRR(064:067)

Does his realized rate of return equal to the YTM when he purchased this bond?

No, the realized rate is 10.59% while YTM at the time of purchase was 11.00%.

Hence, the investor is not hedged against interest rate risk.

Therefore, if interest rate increases 2% between first and second coupon payment dates, the investor is not hedged against interest rate risk even if his duration gap is zero.