Yield to Maturity and the Investor’s Actual Return
Yield to maturity (YTM) is commonly used to summarize the return implied by a bond’s price and promised cash flows. It is useful, but it is not a guarantee of the return an investor will actually earn. To interpret it correctly, distinguish the bond’s cash flows from what the investor does with each payment after receiving it.
The key issue is reinvestment. A coupon received before maturity is cash in hand; it does not automatically earn the bond’s YTM until maturity. The YTM matches a particular compound return only under specified conditions, including that the investor holds the bond to maturity and reinvests interim coupons at that same yield.
The examples below use annual coupons, annual compounding, and dollars. For an annual-coupon bond, \(y\) is an effective annual yield. A bond with face value \(F\), annual coupon \(C\), and \(n\) years to maturity has promised payments of \(C\) at the end of each year, plus \(F\) with the final coupon. In the standard YTM calculation, the yield is the discount rate that prices those promised payments.
What Must Be Assumed for YTM to Be Realized?
The familiar interpretation of YTM as the investor’s compound annual return relies on a set of conditions. The investor must buy the bond at the price used to calculate its YTM, hold it until maturity, and receive all promised payments as scheduled. Coupons received before maturity must be reinvested at the YTM rate for the remaining time until maturity. The calculation also abstracts from taxes, transaction costs, and other cash flows.
Holding to maturity matters because it avoids having to determine a sale price before the final payment. Reinvestment matters because a coupon received early is not itself compounded at the YTM unless the investor can reinvest it at that rate. If the reinvestment rate differs, terminal wealth changes—even if the bond’s promised payments and purchase price do not.
These are assumptions, not promises about future market rates. A YTM calculation uses one constant yield to discount the bond’s cash flows. Actual reinvestment opportunities may change over time, and the rate available when a coupon is received may be above or below the bond’s YTM.
Why Reinvestment at YTM Produces the YTM Return
The relationship can be shown directly from the bond-pricing equation. Let \(C_t\) be the coupon paid at the end of year \(t\), and let \(F\) be the face value paid at maturity in year \(n\). The final cash flow includes both \(C_n\) and \(F\). If the yield to maturity is \(y\), then the price is
If each coupon received at time \(t\) is reinvested at \(y\) for the remaining \(n-t\) years, it grows to \(C_t(1+y)^{n-t}\) at maturity. The face value is received at maturity, so it does not earn additional interest within this holding period. Adding the grown coupons and the maturity payment gives terminal wealth \(W_n\).
The second line follows by multiplying each term in the present-value equation by \((1+y)^n\); the face-value term becomes \(F\). Since \(W_n=P(1+y)^n\), the annual compound rate from \(P\) to \(W_n\) is exactly \(y\):
This derivation also makes the limitation clear. If coupons are reinvested at a rate other than \(y\), the terms \(C_t(1+y)^{n-t}\) change. The YTM pricing equation still describes the bond’s promised payments and purchase price, but it no longer implies that the investor’s terminal wealth is \(P(1+y)^n\).
Worked Examples
Worked Example: Reinvesting Coupons at the Bond’s YTM
An investor pays $1,000 for a three-year bond with face value $1,000 and annual coupons of $80. The bond’s YTM is 8% per year. Assume the investor holds the bond to maturity, receives every payment, and reinvests each coupon at 8% annually. Find terminal wealth and the realized compound annual return.
Method and assumptions. Grow each coupon from its payment date to year 3 at the stated reinvestment rate. The year-3 coupon and principal are paid at the end of the holding period, so they are not reinvested within the three years.
The year-1 coupon earns interest for two years: \(80(1.08)^2=80(1.1664)=\$93.312\). The year-2 coupon earns interest for one year: \(80(1.08)=\$86.40\). At year 3, the investor receives the final coupon and principal: \(\$80+\$1{,}000=\$1{,}080\).
Thus, \(W_3=93.312+86.40+1{,}080=\$1{,}259.712\). The annualized realized return is \((1{,}259.712/1{,}000)^{1/3}-1=(1.08^3)^{1/3}-1=0.08\), or 8.000%.
Check. The YTM-implied terminal wealth is \(\$1{,}000(1.08)^3=\$1{,}259.712\), matching the coupon-by-coupon calculation. The common mistake this avoids is treating YTM as an automatic realized return without checking the coupon reinvestment assumption.
Worked Example: A Lower Coupon-Reinvestment Rate
A hypothetical three-year bond has face value $1,000, annual coupons of $80, and a purchase price of $1,000. Its YTM is 8% per year. The investor holds it to maturity but can reinvest coupons at only 4% annually. Calculate terminal wealth and the realized compound annual return.
Method and assumptions. Use the actual 4% reinvestment rate to accumulate coupons. The final payment is $1,080, consisting of the last $80 coupon and $1,000 principal. No taxes, costs, or missed payments are assumed.
The first coupon grows for two years: \(80(1.04)^2=80(1.0816)=\$86.528\). The second coupon grows for one year: \(80(1.04)=\$83.20\). The final payment is \(\$1{,}080\). Therefore, \(W_3=86.528+83.20+1{,}080=\$1{,}249.728\).
Annualize this terminal wealth over the three-year holding period: \[ \left(\frac{1{,}249.728}{1{,}000}\right)^{1/3}-1 \approx 0.07714. \] The realized compound annual return is approximately 7.714%, rounded to three decimal places.
Check and common mistake. This is below the 8% YTM because the earlier coupons earn only 4% after receipt. Using \(1{,}000(1.08)^3\) as the investor’s terminal wealth would incorrectly assume that coupons are reinvested at 8%; it gives $1,259.712, not the actual $1,249.728.
Worked Example: A Higher Coupon-Reinvestment Rate
A two-year bond has face value $1,000, annual coupons of $40, and a purchase price of $1,000. Its YTM is 4% per year. Suppose the investor holds it to maturity and reinvests the first coupon at 8% for one year. Find terminal wealth and the realized compound annual return.
Method and assumptions. The bond’s annual coupon rate equals its 4% YTM, so the price is par under the stated annual-payment terms. The first coupon earns 8% for one year; the year-2 coupon and principal arrive at maturity.
The first coupon accumulates to \(40(1.08)=\$43.20\). The maturity payment is \(\$40+\$1{,}000=\$1{,}040\). Thus, terminal wealth is \(W_2=43.20+1{,}040=\$1{,}083.20\).
The realized annual return \(r\) satisfies \(1{,}000(1+r)^2=1{,}083.20\). Therefore, \[ r=\left(\frac{1{,}083.20}{1{,}000}\right)^{1/2}-1 =\sqrt{1.0832}-1 \approx 0.04077. \] The realized compound annual return is approximately 4.077%, rounded to three decimal places.
Check and common mistake. The result is slightly above the 4% YTM because the first coupon was reinvested at 8%. Simply reporting the bond’s 4% YTM as the realized return would ignore the stated reinvestment rate.
Interpreting the Difference
The reinvestment assumption affects coupons received before maturity, not the promised amount of those coupons. When coupons earn less than the YTM, terminal wealth will be lower than the YTM-compounded purchase price, all else equal. When coupons earn more, terminal wealth will be higher. The realized return is calculated from total terminal wealth, not by averaging the coupon rate and the reinvestment rate.
The size of the difference depends on how much of the bond’s value comes from interim coupons and how long they can earn interest. A coupon received early has more time to accumulate than one received shortly before maturity. A zero-coupon bond has no interim coupons to reinvest, so this particular reinvestment issue does not arise; the promised maturity payment remains subject to the bond’s payment and holding assumptions.
Keep three quantities separate when analyzing a bond:
- YTM: the rate that discounts promised bond payments to the purchase price.
- Reinvestment rate: the rate actually earned on coupons after they are received.
- Realized compound return: the annual rate implied by the initial price and terminal wealth over the actual holding period.
A bondholder who sells before maturity also has a sale price to include in the return calculation. In that case, the maturity-based calculation above does not describe the full holding-period result. For the specific holding-to-maturity interpretation here, the central exam check is whether the coupon-reinvestment rate matches the YTM.
Check Your Understanding
For each problem, use the stated annual coupon-reinvestment rate and assume all promised payments are made, with no taxes or transaction costs.
- A two-year bond costs $1,000, pays annual coupons of $50, and returns $1,000 principal at maturity. Its YTM is 5%. If coupons are reinvested at 5%, find terminal wealth and realized annual return. (Answer: $1,102.50; 5.000%.)
- A three-year bond costs $1,000, has face value $1,000, and pays annual coupons of $70. The YTM is 7%, but coupons earn 0% after receipt. Find terminal wealth and the realized annual return. (Answer: $1,210; approximately 6.560%.)
- A two-year bond costs $1,000, pays annual coupons of $50, and returns $1,000 principal at maturity. Coupons earn 10% after receipt. Find terminal wealth and the realized annual return. (Answer: $1,105; approximately 5.119%.)
- A three-year bond costs $1,000, has face value $1,000, and pays annual coupons of $60. Its YTM is 6%; coupons are reinvested at 3%. Calculate terminal wealth and the realized annual return. (Answer: $1,185.454; approximately 5.835%.)
- An investor buys a bond using its quoted YTM, receives one coupon, and sells the bond before maturity. Is the quoted YTM by itself enough to determine the investor’s realized return? (Answer: No; the sale proceeds and the treatment or reinvestment of the coupon are also needed.)