Reinvestment Risk and Modified Duration for Exam FM

Reinvestment Risk

Reinvestment risk is the risk that coupon payments or principal repayments must be reinvested at a rate different from the original yield. When rates fall, coupons are reinvested at lower rates, reducing total return. When rates rise, coupons are reinvested at higher rates, increasing total return. Zero-coupon bonds have no reinvestment risk because there are no intermediate cash flows to reinvest.

On Exam FM, reinvestment problems specify that coupons are reinvested at rate j (which may differ from the yield rate i). The accumulated value of the bond investment at time n is: Fr * s-angle-n(j) + C, where the coupons accumulate at j and the redemption is received at time n.

Yield with Reinvestment

The realized yield (or dollar-weighted yield) accounts for reinvestment. The investor pays P at time 0 and receives Fr * s-angle-n(j) + C at time n. The realized annual yield y satisfies P * (1 + y)^n = Fr * s-angle-n(j) + C. When j = i (the yield rate), y = i and there is no reinvestment effect. When j < i, y < i; when j > i, y > i.

Duration and Reinvestment

Duration measures price sensitivity but also relates to reinvestment risk. A bond held for exactly its Macaulay duration achieves approximately the original yield regardless of a one-time parallel shift in rates. This is because the capital loss (gain) from a rate increase (decrease) is approximately offset by higher (lower) reinvestment income over the duration period. This observation underlies immunization theory and explains why duration is the natural holding period for a bond investor seeking to lock in a yield.