Reinvestment Risk and Modified Duration for Exam FM

Understanding 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.

Reinvestment risk is distinct from price risk (the risk that the bond's market value changes due to rate movements). These two risks work in opposite directions: a rate increase causes a capital loss but higher reinvestment income, while a rate decrease causes a capital gain but lower reinvestment income. This offsetting relationship is the foundation of immunization theory.

The severity of reinvestment risk depends on the coupon rate and the bond's maturity. Higher coupon bonds have greater reinvestment risk because more cash flow is received before maturity. Longer maturity bonds also have greater reinvestment risk because the reinvested coupons compound over a longer period.

Accumulated Value with Reinvestment

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 the reinvestment rate j and the redemption C is received at time n.

Example: A $1,000 par bond with 6% annual coupons purchased at par (P = $1,000) with 10 years to maturity. Coupons are reinvested at j = 4%. The accumulated value at time 10 is: 60 * s-angle-10 at 4% + 1,000 = 60 * 12.0061 + 1,000 = 720.37 + 1,000 = $1,720.37.

Compare this to the case where coupons are reinvested at the yield rate of 6%: 60 * s-angle-10 at 6% + 1,000 = 60 * 13.1808 + 1,000 = 790.85 + 1,000 = $1,790.85. The lower reinvestment rate reduced the accumulated value by $70.48.

Realized Yield with Reinvestment

The realized yield (or holding period yield) accounts for the actual reinvestment rate. The investor pays P at time 0 and accumulates 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. Solving for y: y = ((Fr * s-angle-n(j) + C) / P)^(1/n) - 1.

Continuing the example: y = (1,720.37 / 1,000)^(1/10) - 1 = (1.72037)^(0.1) - 1 = 0.0558 = 5.58%. The realized yield of 5.58% is less than the 6% yield to maturity because coupons were reinvested at only 4%. When j = i, y = i exactly, confirming that the yield to maturity is the realized yield only when reinvestment occurs at the same rate.

This formula is important because it shows that the yield to maturity is an optimistic measure of return when market rates are expected to fall, and a pessimistic measure when rates are expected to rise.

Duration as the Optimal Holding Period

A remarkable result connects duration 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.

If you hold the bond for less than its duration, price risk dominates and a rate increase hurts you (the bond is worth less when you sell). If you hold for longer than the duration, reinvestment risk dominates and a rate increase helps you (the extra reinvestment income more than compensates for the initial price drop). At exactly the duration, the two effects cancel out.

This observation underlies immunization theory and explains why duration is the natural holding period for a bond investor seeking to lock in a yield. It also explains why immunization requires matching the asset duration to the liability timing.

Modified Duration and Price Sensitivity

Modified duration D* = D / (1+i) directly measures the percentage price sensitivity to a yield change: the approximate percentage change in price for a 1% change in yield is -D* * 0.01. On Exam FM, modified duration problems typically give D* and ask for the estimated price change given a specific yield movement.

Example: A bond has modified duration 6.5 and is priced at $950. If the yield increases by 0.25%, the estimated price change is -6.5 * 0.0025 * 950 = -$15.44. The new estimated price is $934.56. This linear approximation works well for small yield changes but becomes less accurate for larger movements, where convexity adjustments are needed.

Study Tips and Common Pitfalls

The most common mistake in reinvestment problems is using the bond's yield rate instead of the reinvestment rate when computing s-angle-n. Remember: the yield rate determines the purchase price, but the reinvestment rate determines how coupons grow. Another frequent error is confusing Macaulay duration with modified duration. Macaulay duration is measured in time units (years). Modified duration is a rate of change (percent per unit yield change). They differ by a factor of (1+i).

Practice at least 5 reinvestment problems covering different scenarios: reinvestment rate below the yield, above the yield, and equal to the yield. Also practice computing realized yields and comparing them to the yield to maturity. These problems are straightforward once you have the formula memorized, but arithmetic errors are common, so verify your work by checking boundary cases.