Price Risk Versus Reinvestment Risk

Two Effects of a Yield Change

A bond investor faces two different consequences when market yields change. The price of an existing bond moves in the opposite direction from yields: if yields rise, the bond’s fixed payments become less valuable at current market rates, so its price falls. At the same time, a higher rate can benefit the investor when coupons already received are reinvested. If yields fall, the effects reverse: the bond’s price tends to rise, but coupons can be reinvested at lower rates.

These are price risk and reinvestment risk. Neither one alone describes the investor’s total return. For that, account for both the value of the bond at the end of the holding period and the accumulated value of coupons received along the way.

Definition: Price risk is the risk that a bond’s market value will change when yields change. Reinvestment risk is the risk that coupons or other cash flows received before the end of an investment period will earn a different rate when reinvested. A bond’s holding-period return reflects the ending value of the bond and the value at the holding date of its cash flows received during the period.

Unless stated otherwise, the examples use annual coupon payments, an annual effective yield, no default, no taxes or transaction costs, and reinvestment at a constant rate during the stated period. A bond’s price at a sale date is calculated by discounting its remaining contractual cash flows at the yield then required by the market. These assumptions keep the two effects visible; actual yields and reinvestment rates can vary over time.

Calculating Total Wealth and Return

Suppose a bond costs $P_0$, pays an annual coupon $C$, and is held for $H$ years. Let $P_H$ be its market price at the end of year $H$. If coupons are reinvested at a constant annual effective rate $r$, the coupon received at the end of year $k$ accumulates for $H-k$ years. Thus the investor’s ending wealth is the sale price plus the accumulated coupons.

W_H=P_H+\sum_{k=1}^{H} C(1+r)^{H-k}

The holding-period return over all $H$ years is $W_H/P_0-1$. Its equivalent annual effective return is the rate $R_H$ satisfying $P_0(1+R_H)^H=W_H$, or $R_H=(W_H/P_0)^{1/H}-1$. These are different summaries: one is the total return over the entire period, and the other annualizes that return.

Important: The reinvestment rate $r$ applies only to coupons received before the holding date. A coupon received exactly at the end of the period has no time to earn interest within that period. The sale price $P_H$ is based on the yield at the holding date, not on the rate used to reinvest earlier coupons unless the assumptions make those rates the same.

Worked Example: Isolating the Price Effect

A hypothetical bond has a face value of $1,000, pays a $40 annual coupon, and has two years remaining. An investor buys it at par when its annual effective yield is 4%. Immediately afterward, the market yield rises to 6%. Find the bond’s new price and the dollar and percentage price change.

Method. Discount the two remaining payments at the new yield. The cash flows are $40 at the end of year 1 and $1,040 at the end of year 2.

P_{\text{new}}=\frac{40}{1.06}+\frac{1{,}040}{(1.06)^2} =37.7358+925.5963 =963.3321

The initial price was $1,000, so the dollar change is $963.3321-1{,}000=-36.6679$. The percentage change, measured relative to the initial price, is

\frac{-36.6679}{1{,}000}\times 100\%=-3.6668\%

Rounded to the nearest cent and two decimal places for the percentage, the new price is $963.33, the price loss is $36.67, and the percentage price change is -3.67%. This isolates the immediate price effect; it is not a calculation of the investor’s total return over a holding period.

Check. The new yield is above the bond’s 4% coupon rate, so a price below par is consistent with the inverse relationship between price and yield. The common mistake is to call the 3.67% price decline the investor’s full-period return without specifying a period or including any coupons.

Why Price Risk and Reinvestment Risk Offset

A rise in yields lowers the present value of the bond’s remaining fixed payments. But the investor may receive coupons before selling the bond, and those earlier payments can earn more when reinvested at the higher rate. The two effects operate on different cash flows: the yield used to price the remaining bond affects $P_H$, while the reinvestment rate affects the accumulated coupons.

The relationship with yield to maturity makes the offset especially clear. Consider a bond bought at price $P_0$, with yield $y$, annual coupons, and face value $F$. If the bond is sold after $H$ years at the same yield $y$, and coupons are reinvested at $y$, the ending wealth is $P_0(1+y)^H$. This follows from the bond-pricing equation.

P_0=\sum_{k=1}^{n}\frac{C}{(1+y)^k}+\frac{F}{(1+y)^n}

Multiply both sides by $(1+y)^H$. The payments due in years 1 through $H$ become the values of those coupons accumulated to year $H$. The remaining payments, due after year $H$, have the same present value at year $H$ as the bond’s sale price $P_H$, because the yield at sale is still $y$. Therefore, the right-hand side after multiplication is the accumulated coupons plus the sale price: exactly $W_H$.

P_0(1+y)^H = \sum_{k=1}^{H} C(1+y)^{H-k}+P_H = W_H

Consequently, under these conditions the annualized holding-period return is $y$. The result depends on both parts of the assumption: the sale yield remains $y$, and coupons are reinvested at $y$. If either condition fails, yield to maturity alone does not determine the realized holding-period return.

Worked Total-Return Comparisons

Worked Example: Rising Yields and a Two-Year Holding Period

An investor buys a hypothetical three-year bond with a $1,000 face value and a 6% annual coupon at par, when the annual effective yield is 6%. Immediately after purchase, the market yield rises to 8% and remains there. The investor holds the bond for two years, reinvesting coupons at 8%. Calculate the ending wealth and the annualized holding-period return.

Step 1: Find the sale price. At the end of year 2, one year remains. The bond then pays $1,060 at the end of that year, so its price is

P_2=\frac{1{,}060}{1.08}=981.4815

Step 2: Accumulate the coupons. The $60 coupon received at the end of year 1 earns 8% for one year, giving $60(1.08)=64.80$ at the end of year 2. The second $60 coupon arrives at the holding date, so it accumulates no interest. Total accumulated coupons are $64.80+60=124.80$.

Step 3: Combine the components. Ending wealth is $981.4815+124.80=1{,}106.2815$. The two-year total return is $1{,}106.2815/1{,}000-1=0.1062815$, or 10.63%. Its annualized equivalent is

R_2=(1{,}106.2815/1{,}000)^{1/2}-1 \approx 0.0518

The annualized holding-period return is 5.18% (rounded to two decimal places). For comparison, if the yield had stayed at 6% and the coupons had been reinvested at 6%, the ending wealth would be $1{,}000(1.06)^2=1{,}123.60$. In the 8% scenario, the sale price is $18.52 lower than the $1,000 price in that comparison, while the first coupon earns $1.20 more interest. Together, the effects reduce ending wealth by $17.32.

Check. The investor’s ending wealth exceeds the initial investment, but the annualized return is below the original 6% yield. The price loss outweighs the additional reinvestment income in this example. The common mistake is to treat the 8% reinvestment rate as the return on the entire bond investment.

Worked Example: Falling Yields and Lower Reinvestment Income

A hypothetical three-year bond has a $1,000 face value and an 8% annual coupon. It is bought at par when the annual effective yield is 8%. Immediately after purchase, the yield falls to 6% and stays there. The investor holds the bond for two years and reinvests coupons at 6%. Find the ending wealth and annualized holding-period return.

Step 1: Value the remaining bond. At the end of year 2, the bond has one year remaining and pays $1,080 at maturity. Its sale price is

P_2=\frac{1{,}080}{1.06}=1{,}018.8679

Step 2: Accumulate coupon payments. The first $80 coupon grows for one year to $80(1.06)=84.80$. The second $80 coupon is received at the holding date. The accumulated coupons are $84.80+80=164.80$.

Step 3: Calculate total return. Ending wealth is $1{,}018.8679+164.80=1{,}183.6679$. The two-year total return is $1{,}183.6679/1{,}000-1=18.3668\%$. The annualized return is

R_2=(1{,}183.6679/1{,}000)^{1/2}-1 \approx 0.0880

Rounded to two decimal places, the annualized holding-period return is 8.80%. Relative to a scenario in which the yield and reinvestment rate remain at 8%, the sale price is $18.87 higher, but the first coupon earns $1.60 less interest. The net increase in ending wealth is $17.27.

Check. The price gain is larger than the reinvestment shortfall here, so the annualized return is above the initial 8% yield. A price gain is not automatically the whole return: lower coupon reinvestment income also matters.

Common Mistakes and Interpretation

A price change is a point-in-time change in market value; a holding-period return measures the change in total wealth over a specified period. Confusing them can lead to an incorrect conclusion about how a rate change affected an investor. In the rising-yield example, the bond’s sale price was below its initial price, but the investor also received and reinvested coupons. Conversely, in the falling-yield example, the bond’s price gain did not capture the lower rate earned on an earlier coupon.

For example, a flawed calculation might call the 6% reinvestment rate in the falling-yield example the investor’s annual return. That calculation applies the coupon reinvestment rate to the entire $1,000 investment and ignores the sale price. The corrected calculation uses both components:

W_2=1{,}018.8679+80(1.06)+80=1{,}183.6679, \qquad R_2=(W_2/1{,}000)^{1/2}-1\approx 8.80\%

The investor’s annualized return is not the coupon rate, the reinvestment rate, or the percentage change in the sale price. It is the rate that equates the initial investment with ending wealth, given the specified timing and treatment of cash flows.

A bond held to maturity without default still experiences market-price changes before maturity, even if the investor does not sell at those prices. Not selling avoids realizing a sale-price change at that time, but it does not make reinvestment risk disappear: coupons received before maturity still have to be invested, and the rate they earn affects total wealth.

Key takeaway: A yield increase tends to create a price loss and a reinvestment benefit; a yield decrease tends to create a price gain and a reinvestment shortfall. Calculate total return by combining the bond’s value at the holding date with the accumulated value of coupons received before that date.

Check Your Understanding

For each problem, use annual coupons and annual effective yields, and round dollar amounts to cents and annualized returns to two decimal places.

A two-year bond with a $1,000 face value and a $50 annual coupon is bought at par when its yield is 5%. Immediately, the yield rises to 7%. What is its new price and percentage price change? (Answer: $963.84; price change -3.62%.)
A three-year bond with a $1,000 face value and a $40 annual coupon is bought at par at a 4% yield. Immediately after purchase, the yield rises to 6% and remains there. The investor holds for two years and reinvests the first coupon at 6%. What are the ending wealth and annualized return? (Answer: $1,063.53; 3.13%.)
A three-year bond with a $1,000 face value and a $70 annual coupon is bought at par at a 7% yield. The yield immediately falls to 5% and remains there. The investor holds for two years and reinvests the first coupon at 5%. What are the ending wealth and annualized return? (Answer: $1,162.55; 7.82%.)
A four-year bond with a $1,000 face value and a $60 annual coupon is bought at par when its yield is 6%. The yield stays at 6%, and coupons are reinvested at 6%. If the investor holds for two years, what is the ending wealth and annualized return? (Answer: $1,123.60; 6.00%.)
A two-year bond with a $1,000 face value and a $30 annual coupon is bought at par at a 3% yield. Immediately after purchase, the yield rises to 5% and remains there. The investor holds for one year, receives the first coupon at the holding date, and sells the bond then. What is the sale price and one-year holding-period return? (Answer: $980.95; 1.10%.)

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