How Investment Horizon Changes Interest-Rate Outcomes

Why the Investment Horizon Matters

A bond’s price can change when market yields change, and coupons received before the investment horizon may earn a different rate when reinvested. The impact of those two effects depends on when the investor plans to use the money. An investor who sells before maturity must account for the bond’s sale price as well as coupons received. An investor who holds to maturity does not sell the bond, but still faces uncertainty about the rates earned on coupons reinvested before maturity.

A bond’s yield to maturity is not automatically the return an investor realizes over every possible holding period. To calculate an outcome for a particular horizon, identify the cash flows received by that date, how coupons are reinvested, and—if the bond is sold—its market price at the sale date.

Definition: The holding-period return over a specified horizon is the investment gain or loss over that period, measured relative to the initial investment. When interim coupons are accumulated to the horizon, terminal wealth is the sum of the sale proceeds (if any) and the accumulated value of coupons. The holding-period return is terminal wealth divided by initial price, minus 1. Its annualized equivalent is the constant annual rate that would produce the same terminal wealth over the stated number of years.

Setting Up a Horizon Calculation

Assume annual coupon payments, annual effective rates, no default, no taxes or transaction costs, and that any stated reinvestment rate applies to coupons from the date they are received. For a bond sold at time $H$, first find its sale price from the remaining promised cash flows, discounted at the market yield at that date. Then accumulate coupons received before $H$ to the horizon using their applicable reinvestment rates.

W_H=P_H+\sum_{t=1}^{H} C_t\prod_{j=t}^{H-1}(1+r_j)

Here $W_H$ is terminal wealth, $P_H$ is the sale price at the horizon, $C_t$ is the coupon received at time $t$, and $r_j$ is the reinvestment rate earned from time $j$ to $j+1$. A product with no periods, such as for a coupon received at the horizon, equals 1. If the bond is held to maturity, the principal repayment replaces the sale price in terminal wealth; there is no sale price to estimate.

The holding-period return over $H$ years is $W_H/P_0-1$, where $P_0$ is the initial price. Its annualized equivalent $R$ satisfies $P_0(1+R)^H=W_H$. This annualized rate summarizes the realized outcome; it does not imply that the investment earned the same rate in each individual year.

Worked Examples

Worked Example: Selling After a Rate Increase

A hypothetical three-year bond has face value $1,000 and pays an annual coupon of $60. Its initial yield is 6%, so its initial price is $1,000. An investor sells immediately after receiving the first coupon, when the market yield for the remaining two years is 8%. Find the sale price, one-year holding-period return, and annualized return. Assume no reinvestment period for the coupon received at the horizon.

Method. Discount the remaining coupons and principal at the yield prevailing at the sale date. Add the coupon just received to sale proceeds to obtain terminal wealth.

Step 1: Sale price. The remaining cash flows are $60 in one year and $1,060 in two years. At an 8% annual effective yield:

P_1=\frac{60}{1.08}+\frac{1{,}060}{1.08^2} =55.555556+908.779835 =964.335391

The sale price is $964.34 to the nearest cent. The price is below the initial $1,000 because the market yield rose above the bond’s coupon rate.

Step 2: Terminal wealth and return. The investor has the $60 coupon and the $964.335391 sale proceeds:

W_1=60+964.335391=1{,}024.335391

The one-year holding-period return is $1{,}024.335391/1{,}000-1=0.024335391$, or 2.43% to two decimal places. Since the horizon is one year, the annualized return is also 2.43%.

Check. Terminal wealth exceeds the initial price by $24.335391, which is 2.4335391% of $1,000. The price decline therefore does not mean the total one-year return is negative: the coupon offsets part of the capital loss.

Common mistake avoided: Discounting the remaining payments at the original 6% yield would give a par sale price, but the bond is being sold at an 8% market yield. The sale price must reflect that current yield.

Worked Example: Holding to Maturity With Different Coupon Reinvestment Rates

A hypothetical three-year bond costs $1,000, has face value $1,000, and pays an annual coupon of $60. The investor holds it to maturity. The first coupon is reinvested for two years at 4% per year, and the second coupon is reinvested for one year at 5%. The third coupon and $1,000 principal are received at maturity. Find terminal wealth, the total holding-period return, and the annualized return.

Method. Accumulate each coupon to the maturity date at its specified reinvestment rate, then add the final coupon and principal. The bond is held to maturity, so a market sale price is not part of the calculation.

Step 1: Accumulate the coupons. The first coupon grows for two years at 4%:

$$ 60(1.04)^2=60(1.0816)=64.9536 $$

The second coupon grows for one year at 5%: $60(1.05)=63$. The third coupon is paid at the horizon, so it remains $60.

Step 2: Add principal and calculate returns.

W_3=64.9536+63+60+1{,}000=1{,}187.9536

The total three-year holding-period return is $1{,}187.9536/1{,}000-1=0.1879536$, or 18.80% to two decimal places. For the annualized equivalent:

R=\left(\frac{1{,}187.9536}{1{,}000}\right)^{1/3}-1 =1.1879536^{1/3}-1 \approx 0.059091

The annualized return is 5.91% to two decimal places. It is not the simple average of the coupon reinvestment rates: those rates apply over different periods, and the annualized return summarizes all terminal wealth relative to the initial price.

Check. The dollar gain is $187.9536, and dividing it by $1,000 gives the stated 18.79536% total return. Compounding 5.9091% annually for three years reproduces the terminal wealth, subject to rounding.

Worked Example: A Short Horizon With Both Sale-Price and Reinvestment Effects

A hypothetical four-year bond has face value $1,000, an annual coupon of $50, and an initial yield of 5%, giving an initial price of $1,000. An investor holds it for two years and sells just after receiving the second coupon. In the falling-yield scenario, the sale yield for the remaining two years is 4%, and the first coupon earns 3% for one year. In the rising-yield scenario, the sale yield is 6%, and the first coupon earns 6% for one year. Calculate terminal wealth and annualized return in each scenario. No coupon received at the sale date is reinvested.

Method. In each scenario, discount the remaining two annual payments at the sale yield. Add the resulting sale price to the second coupon and the first coupon accumulated to year two at its stated reinvestment rate.

Falling-yield scenario. At a 4% sale yield, the remaining payments are $50 in one year and $1,050 in two years:

P_2=\frac{50}{1.04}+\frac{1{,}050}{1.04^2} =48.076923+970.414201 =1{,}018.491124

The first coupon accumulates to $50(1.03)=51.50$; the second coupon is $50. Therefore $W_2=51.50+50+1{,}018.491124=1{,}119.991124$. The two-year total return is $1{,}119.991124/1{,}000-1=11.9991124\%$. The annualized return is $\sqrt{1.119991124}-1\approx 5.83\%$, rounded to two decimal places.

Rising-yield scenario. At a 6% sale yield:

P_2=\frac{50}{1.06}+\frac{1{,}050}{1.06^2} =47.169811+934.496266 =981.666077

The first coupon accumulates to $50(1.06)=53$, and the second coupon is $50. Thus $W_2=53+50+981.666077=1{,}084.666077$. The two-year total return is 8.4666077%, and the annualized return is $\sqrt{1.084666077}-1\approx 4.15\%$, rounded to two decimal places.

Check. The falling-yield scenario produces a higher sale price and higher terminal wealth, even though its first coupon earns a lower reinvestment rate. Over this short horizon, the difference in sale prices is an important part of the result.

Common mistake avoided: Applying the sale yield as the reinvestment rate for every coupon would incorrectly combine two distinct assumptions. The yield used to price the bond at sale and the rate earned by a previously received coupon need not be the same.

Why the Horizon Changes the Rate Exposure

The horizon calculation follows from the timing of the investor’s actual cash flows. A coupon received before the horizon is not yet part of terminal wealth at its face amount: it must be accumulated from its payment date to the horizon. If the bond is sold early, its remaining promised payments have value at the sale date. Adding those two components gives total wealth available at that date.

For example, for a one-year horizon with one annual coupon and a sale at the end of the year, $W_1=C_1+P_1$. The return is then $W_1/P_0-1$. For a longer horizon, coupons received earlier may earn interest, while the coupon received at the horizon has no time to grow. If the investor holds to maturity, the principal repayment is received at maturity instead of receiving a sale price.

This explains why an interest-rate change can affect investors with different horizons differently. A near-term seller realizes a market price based on the yield at the sale date. A holder who keeps the bond to maturity receives the promised principal, assuming no default, but the final accumulated value of interim coupons still depends on reinvestment rates. Neither outcome can be determined from the initial yield alone unless the relevant future rates and cash flows are specified.

Key takeaway: For a specified investment horizon, calculate the value of every component at that horizon. Use the market yield at the sale date to value remaining bond cash flows, and use reinvestment rates to accumulate coupons already received. A yield to maturity is not a substitute for this cash-flow-by-cash-flow calculation.

Common Errors to Avoid

Using the original yield to price an early sale. The sale price depends on the market yield at the sale date, not necessarily the yield when the bond was purchased.
Ignoring reinvestment timing. A coupon received one year before the horizon can earn interest; a coupon received at the horizon cannot.
Counting principal too early. Principal is included at maturity, or within the sale price if the bond is sold before maturity, but not both.
Calling total return an annualized return. Over multiple years, $W_H/P_0-1$ is the total holding-period return. Annualize by solving $P_0(1+R)^H=W_H$.

Check Your Understanding

For each problem, assume annual coupon payments, annual effective rates, no default, and no taxes or transaction costs. Round dollar values to cents and annualized returns to two decimal places where requested.

A $1,000 three-year bond pays $50 annually and is bought at par when its yield is 5%. An investor sells just after the first coupon when the yield on the remaining two years is 10%. The coupon is not reinvested. Find the sale price and one-year holding-period return. (Answer: Sale price $913.22; holding-period return $-3.68\%$.)
A $1,000 two-year bond pays a $40 annual coupon and is held to maturity. The first coupon is reinvested for one year at 5%; the second coupon and principal are received at maturity. Find terminal wealth and the annualized return. (Answer: Terminal wealth $1,082.00; annualized return approximately $4.02\%$.)
A $1,000 four-year bond pays $60 annually and is bought at par at a 6% yield. It is sold just after the second coupon when the remaining two-year yield is 5%. The first coupon earns 3% for one year; the second is not reinvested. Find terminal wealth and the annualized two-year return. (Answer: Sale price $1,018.59; terminal wealth $1,140.39; annualized return approximately $6.79\%$.)
A $1,000 three-year bond pays $50 annually and is held to maturity. The first coupon earns 2% for two years, the second earns 2% for one year, and the final coupon is received at maturity. Find terminal wealth and the total return. (Answer: Terminal wealth $1,153.02; total three-year return $15.30\%$.)
Two investors buy the same coupon bond. One sells before maturity after a yield increase; the other holds to maturity. Name one component of the first investor’s terminal wealth that the second investor does not use, and one rate-related uncertainty that remains for the second investor. (Answer: The early seller includes the bond’s sale price; the maturity holder does not. The maturity holder still faces uncertainty about the reinvestment rates earned on coupons received before maturity.)

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