Price Risk and Reinvestment Risk Over a Holding Period

Why a Bond’s Holding-Period Return Has Two Rate Risks

A bond’s yield to maturity is a rate that equates its purchase price with its promised cash flows under a specified yield assumption. It does not guarantee the return an investor will earn over a shorter holding period. If the investor sells before maturity, the sale price depends on the market yield at that time. Meanwhile, coupons received before sale can earn interest only at the rates available when they are reinvested.

These are two distinct effects. Price risk is the effect of a yield change on the bond’s sale price. Reinvestment risk is the effect of reinvestment rates on the accumulated value of interim cash flows. The fixed-horizon realized return combines both effects; it is not determined by the sale yield or coupon reinvestment rate alone.

Definition: For a bond bought for price $P_0$, held for $H$ years, and producing terminal wealth $W_H$, the annual effective realized return $r$ is the rate satisfying $P_0(1+r)^H=W_H$. Thus $r=(W_H/P_0)^{1/H}-1$, provided $P_0>0$, $W_H>0$, and the holding period is measured in years. Terminal wealth includes the bond’s sale proceeds and all interim payments accumulated to the horizon.

Building Terminal Wealth

Assume annual coupon dates, no default, no taxes or transaction costs, and a sale immediately after any coupon due on the sale date has been paid. If the bond has face value $F$, annual coupon $C$, and $n$ years remaining at sale, its price at a flat annual effective market yield $j$ is the present value of its remaining promised payments:

P_H = C a_{\overline{n}|j} + F(1+j)^{-n}

Here $a_{\overline{n}|j}$ is the present value of $n$ annual payments of 1 at rate $j$. This expression assumes the same yield $j$ discounts each remaining payment. It does not describe a bond priced from maturity-specific spot rates, and the flat-yield calculation should not be used when the problem instead specifies a nonflat spot curve.

For coupons $C_k$ paid at times $k$ before the horizon $H$, let $q_k$ be the effective rate at which coupon $C_k$ can be reinvested from time $k$ to $H$, assuming a constant reinvestment rate over that interval. Then the coupon’s horizon value is $C_k(1+q_k)^{H-k}$. More generally, if rates vary across reinvestment intervals, each interval’s accumulation factor must be applied. The bond’s terminal wealth is the sum of the sale proceeds and the horizon values of all received coupons.

Find the initial outlay.
Use the purchase price $P_0$, based on the bond’s original cash flows and purchase yield.

Price the bond at the holding date.
Discount the cash flows remaining after the sale date using the market yield then assumed. Do not include a coupon already paid at the holding date in the sale price.

Accumulate interim coupons.
Carry each coupon forward to the horizon at its assumed reinvestment rate and for its actual reinvestment period.

Calculate the realized return.
Add the accumulated coupons to the sale price, then solve $P_0(1+r)^H=W_H$.

Why the Two Effects Combine This Way

The method follows directly from valuing every dollar at the same focal date, the holding horizon. At that date, sale proceeds are already available, while earlier coupons must first be accumulated. Adding these horizon values gives total wealth attributable to the initial investment.

For an annual horizon of two years, with a coupon $C$ at the end of year 1, a coupon $C$ at the end of year 2, and sale proceeds $P_2$ just after the second coupon, suppose the first coupon earns reinvestment rate $q$. Then:

$$ W_2=C(1+q)+C+P_2 $$

By the definition of the annual effective realized return, the same terminal wealth must equal the initial price accumulated for two years:

$$ P_0(1+r)^2=C(1+q)+C+P_2 $$

Solving this equality gives $r=\sqrt{W_2/P_0}-1$. This derivation shows why the rate used to price the bond at sale and the rate used to reinvest a coupon enter the calculation differently: one determines the amount received on sale, and the other determines how much an earlier payment grows before the horizon.

Worked Example: A Higher Sale Yield and Lower Coupon Reinvestment

A bond with face value $1,000 pays annual coupons of $100 and matures in three years. It is purchased for $1,000 at a 10% annual effective yield. The investor holds it for two years and sells immediately after receiving the second coupon. At that time, the market yield is 12%. The first coupon was reinvested for one year at 8%. Find terminal wealth and the annual effective realized return.

Method. Price the remaining one-year cash flow at the sale yield, accumulate the first coupon to year 2, and add the coupon received at year 2. The purchase price is $1,000 because the coupon rate and purchase yield are both 10%.

Sale price. Immediately after the year-2 coupon, only the year-3 coupon and face value remain. Their combined payment is $100 + $1,000 = $1,100, so:

P_2=\frac{1{,}100}{1.12}=\$982.142857\ldots

Accumulated coupons. The year-1 coupon grows for one year to $100(1.08)=\$108$. The year-2 coupon is received at the horizon, so its horizon value is $100.

Terminal wealth.

W_2=982.142857\ldots+108+100=\$1{,}190.142857\ldots

Realized return. Solve $1{,}000(1+r)^2=1{,}190.142857\ldots$:

r=\sqrt{\frac{1{,}190.142857\ldots}{1{,}000}}-1 =0.09094\ldots

The annual effective realized return is 9.094%, rounded to three decimal places. As a check, $1{,}000(1.09094)^2$ is approximately $1,190.15, consistent with the calculated terminal wealth apart from rounding. The mistake this method avoids is using the 12% sale yield as though it also dictated the reinvestment rate on the year-1 coupon.

Separating Price and Reinvestment Effects

A useful comparison holds the initial purchase and holding period fixed, then compares the actual outcome with a stated benchmark. For a clear decomposition, use a benchmark in which the market yield at sale and the coupon reinvestment rate both remain at the initial yield. The difference in sale prices isolates the price effect; the difference in accumulated coupons isolates the reinvestment effect. This dollar decomposition is additive, but the corresponding changes in annualized return are not generally additive because the return is calculated by taking a root.

Worked Example: Isolating Reinvestment Risk When the Sale Yield Is Unchanged

A $1,000 bond pays annual coupons of $80, has three years to maturity, and is bought at a yield of 8%. The investor holds it for two years and sells immediately after receiving the year-2 coupon. The sale yield remains 8%, but the year-1 coupon is reinvested for one year at 4%. Find terminal wealth, the realized return, and the dollar reinvestment effect relative to reinvesting at 8%.

Method. First confirm the initial price. At an 8% yield, the price is $80a_{\overline{3}|0.08}+1{,}000(1.08)^{-3}=\$1{,}000$, since each of the three annual cash flows is consistent with an 8% return on that price.

At the sale date, one year remains. The sale price is:

P_2=\frac{1{,}080}{1.08}=\$1{,}000

The year-1 coupon accumulates to $80(1.04)=\$83.20$. The year-2 coupon is received at the horizon and is worth $80 there. Therefore:

W_2=1{,}000+83.20+80=\$1{,}163.20

The annual effective realized return is:

r=\sqrt{\frac{1{,}163.20}{1{,}000}}-1 =\sqrt{1.1632}-1 =0.07852\ldots

The realized return is 7.852%, rounded to three decimal places. If the year-1 coupon had instead been reinvested at 8%, it would have grown to $80(1.08)=\$86.40$. That benchmark terminal wealth would be $1{,}000+86.40+80=\$1{,}166.40$, which is exactly $1{,}000(1.08)^2$. The lower reinvestment rate reduces terminal wealth by $86.40-83.20=\$3.20$, while the unchanged sale yield causes no sale-price difference. The calculation avoids treating the unchanged bond price as proof that the holding-period return is unchanged.

Worked Example: A Sale-Price Gain Partly Offset by Reinvestment

A $1,000 bond pays annual coupons of $100 and matures in four years. It is purchased at a 10% annual effective yield and held for two years, with sale immediately after the year-2 coupon. At sale, the market yield is 8%. The year-1 coupon is reinvested for one year at 6%. Find the realized return and separate the dollar price and reinvestment effects from a benchmark where both rates remain at 10%.

Method. The initial price is $1,000 because the coupon rate equals the purchase yield. At the sale date, the remaining cash flows are $100 one year later and $1,100 two years later. Price them at the 8% sale yield:

P_2=\frac{100}{1.08}+\frac{1{,}100}{1.08^2} =92.592593+943.072702 =\$1{,}035.665295

The year-1 coupon accumulates to $100(1.06)=\$106$. The year-2 coupon is received at the horizon, so terminal wealth is:

W_2=1{,}035.665295+106+100=\$1{,}241.665295

The realized annual effective return is:

r=\sqrt{\frac{1{,}241.665295}{1{,}000}}-1 =0.11430\ldots

Thus the realized return is 11.430%, rounded to three decimal places. Under the 10% benchmark, the sale price is $1,000: the remaining cash flows have a present value of $100/1.10+1{,}100/1.10^2=\$1{,}000$. The benchmark accumulated coupons are $110 and $100, giving terminal wealth of $1,210. The actual sale-price gain is $1{,}035.665295-1{,}000=\$35.665295$. The reinvestment shortfall is $110-106=\$4$. Together they produce a net terminal-wealth increase of $35.665295-4=\$31.665295$, which agrees with $1{,}241.665295-1{,}210$. The check confirms the dollar decomposition without incorrectly adding the two effects as percentage returns.

Common Traps and Interpretation

A fixed-horizon return describes the investor’s realized outcome under the stated sale and reinvestment assumptions. It does not imply that the bond’s original yield to maturity was wrong: the original yield is a price-and-cash-flow measure, while the realized return reflects what happened during the chosen holding period.

Including a paid coupon in the sale price. If the bond is sold immediately after a coupon is paid, that coupon is a separate receipt, not part of the remaining bond cash flows used to calculate the sale price.
Reinvesting coupons at the sale yield automatically. The problem may specify a different reinvestment rate. Use the rate that applies to each coupon’s actual reinvestment period.
Using the initial yield to price the bond at sale. The sale price must reflect the market yield at the holding date, unless the problem explicitly assumes no yield change.
Comparing returns without a common benchmark. To identify price and reinvestment contributions, state which assumptions are held fixed. A dollar comparison is transparent; annualized percentage-return differences need not add.

This distinction matters whenever an investment is sold before its final promised payment. A yield increase may reduce sale proceeds while reinvestment opportunities improve; a yield decrease may raise the sale price while lowering coupon reinvestment rates. The realized return reflects the net result, not either effect in isolation. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the sale-price and reinvestment assumptions before combining values at the holding horizon.

A $1,000 bond pays annual coupons of $60, matures in three years, and is purchased at a 6% yield. It is held for two years and sold just after the second coupon at a 8% yield. The first coupon was reinvested at 3% for one year. Find terminal wealth and annual effective realized return. (Answer: terminal wealth $1,103.28; realized return approximately 5.04%.)
A $2,000 bond pays annual coupons of $100, matures in three years, and is purchased at a 5% yield. It is held two years and sold just after the second coupon at an unchanged 5% yield. The first coupon is reinvested at 2% for one year. Find the terminal wealth and annual effective realized return. (Answer: terminal wealth $2,202; realized return approximately 4.93%.)
In a two-year holding-period calculation, the benchmark terminal wealth is $1,210. A yield change raises the sale price by $24 relative to that benchmark, while the first coupon’s horizon value is $5 lower. What is the actual terminal wealth, and what is the net dollar effect relative to the benchmark? (Answer: $1,229 terminal wealth; a $19 increase.)
Explain why a bond can have a higher sale price after a yield decrease and still earn less from coupon reinvestment than expected. (Answer: The yield decrease raises the present value of the remaining bond cash flows, but the lower reinvestment rate reduces the accumulated value of coupons already received; both effects enter terminal wealth separately.)

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