Yield to Maturity as Internal Rate of Return

From Bond Price to Yield

A bond’s market price is the value today of its promised future payments, discounted at a rate appropriate to those payments. Yield to maturity, or YTM, reverses that calculation: it is the rate that makes the present value of the bond’s scheduled cash flows equal to its market price. This interpretation makes YTM a bond-specific application of the internal rate of return (IRR).

The equation is useful because it puts coupons and principal on a common basis: their present value. It also clarifies what the yield does and does not say. YTM is the rate implied by the bond’s price and promised cash flows under the stated timing and compounding convention; it is not itself a guarantee of the investor’s eventual realized return.

Definition: For a bond with market price $P_0$, a yield to maturity $y$ is a per-period discount rate that satisfies $-P_0+\sum_{t=1}^{n} CF_t/(1+y)^t=0$, where $CF_t$ is the bond’s payment at time $t$. Thus $y$ is the internal rate of return on the cash flows consisting of the purchase price at time 0 and the promised payments thereafter.

Throughout this tutorial, the bond has a positive face value $F$, pays a fixed coupon $C$ at the end of each year, and returns its face value with the final coupon at the end of year $n$. The purchase occurs at time 0, just after a coupon date, so there are $n$ full annual periods until maturity. Payments are deterministic as promised; no default, taxes, transaction costs, or interim sale are modeled. The yield $y$ is an annual effective rate, applied once per annual period. There is no mortality assumption or probability distribution in these examples.

These assumptions matter. The equation below is for annual coupon payments and annual compounding. If payment dates or the yield-quoting convention differ, the periods and discount factors must be adjusted to match them.

Deriving the Coupon-Bond YTM Equation

At time $t=1,\ldots,n-1$, the bond pays a coupon $C$. At time $n$, it pays both the final coupon and face value, for a total of $C+F$. Discounting each payment at the same annual rate $y$ gives:

P_0=\frac{C}{1+y}+\frac{C}{(1+y)^2}+\cdots+\frac{C}{(1+y)^{n-1}}+\frac{C+F}{(1+y)^n}.

Equivalently, one can write the last payment as a coupon plus the principal, keeping the two cash-flow components separate:

P_0=\sum_{t=1}^{n}\frac{C}{(1+y)^t}+\frac{F}{(1+y)^n} =C\,a_{\overline{n}|\,y}+F(1+y)^{-n}, \qquad a_{\overline{n}|\,y}=\sum_{t=1}^{n}(1+y)^{-t}.

Here $a_{\overline{n}|\,y}$ is the present-value factor for $n$ annual end-of-year payments of 1 at rate $y$. The equation is not a formula that directly gives $y$ from the other quantities in every case. Rather, it defines the yield as the rate that makes the equality hold.

Derivation: Under the stated model, the present value at time 0 of a payment $CF_t$ due at the end of year $t$ is $CF_t/(1+y)^t$. The bond’s full future cash-flow stream therefore has present value $\sum_{t=1}^{n}CF_t/(1+y)^t$. Buying the bond costs $P_0$ at time 0, so the net present value of buying it and receiving its payments is $-P_0+\sum_{t=1}^{n}CF_t/(1+y)^t$. By definition, an internal rate of return makes that net present value zero. Rearranging gives $P_0=\sum_{t=1}^{n}CF_t/(1+y)^t$, the YTM equation.

For a conventional bond with positive price and positive payments, there is at most one solution with $y>-1$. To see why, define the present value of the future payments as $V(y)=\sum_{t=1}^{n}CF_t/(1+y)^t$. For $y>-1$, each denominator is positive, and

V'(y)=-\sum_{t=1}^{n}\frac{t\,CF_t}{(1+y)^{t+1}}<0.

The inequality holds because the payment amounts are positive. Thus $V(y)$ decreases strictly as $y$ increases, so it can equal a given positive price at most once. For a standard coupon bond, the final payment is positive; as $y$ approaches $-1$ from above, its present value grows without bound, while $V(y)$ approaches zero as $y$ increases without bound. Consequently, a unique solution greater than $-1$ exists for a positive price. The yield is positive precisely when the market price is less than the sum of the undiscounted future payments, since $V(0)=\sum CF_t$.

Worked Applications

Worked Example: Pricing a Premium Bond at Its YTM

A hypothetical bond has face value $F=\$1{,}000$, three years remaining, and an annual coupon rate of 6%. Coupons are paid annually at year-end. Find its price if its annual effective YTM is 5%, and interpret the result. Assume the bond pays all promised amounts and there are no transaction costs.

Method. First convert the coupon rate to dollars: $C=0.06(\$1{,}000)=\$60$. Discount each promised payment at the specified annual yield, keeping full precision until the final price is rounded to cents.

At year 1, the payment is $\$60$, with present value $\$60/1.05=\$57.142857143$. At year 2, it is also $\$60$, with present value $\$60/1.05^2=\$54.421768707$. At year 3, the bond pays the final coupon and principal, $\$60+\$1{,}000=\$1{,}060$, with present value $\$1{,}060/1.05^3=\$915.667854443$. Therefore,

P_0=57.142857143+54.421768707+915.667854443 =1{,}027.232480293\approx \$1{,}027.23.

The price exceeds the $\$1{,}000$ face value, so the bond sells at a premium. Its coupon rate is 6%, greater than the 5% YTM. As a further check, the current yield is approximately $\$60/\$1{,}027.232480293=5.841\%$, which is not the same as the YTM. The YTM discounts every coupon and the principal, whereas current yield compares only the annual coupon with the price.

Check. Discounting the three promised payments at 5% reproduces the unrounded price used in the equation. A common mistake is to discount only the final $\$1{,}000$ principal at year 3 and omit the final $\$60$ coupon. The correct final payment is $\$1{,}060$.

Worked Example: Reading an IRR From a Discount-Bond Cash-Flow Stream

A hypothetical four-year bond has face value $\$1{,}000$, an annual coupon rate of 4%, and a proposed annual effective yield of 8%. Coupons are paid at each year-end, and the bond is purchased at time 0. Calculate the price implied by the proposed yield, write the corresponding IRR equation, and verify the result. Assume no default or additional costs.

Method. The annual coupon is $0.04(\$1{,}000)=\$40$. The bond pays $\$40$ at years 1, 2, and 3, then $\$1{,}040$ at year 4. Discount each amount at 8%:

\begin{aligned} PV_1&=\frac{40}{1.08}=37.037037037,\\ PV_2&=\frac{40}{1.08^2}=34.293552812,\\ PV_3&=\frac{40}{1.08^3}=31.753289641,\\ PV_4&=\frac{1{,}040}{1.08^4}=764.431046908. \end{aligned}

Adding the unrounded present values gives $P_0=\$867.514926398$, or $\$867.51$ to the nearest cent. The associated investment cash flows are $-\$867.514926398$ at time 0, followed by $\$40,\$40,\$40,\$1{,}040$ at years 1 through 4. Their IRR equation is:

0=-867.514926398+\frac{40}{1+y}+\frac{40}{(1+y)^2} +\frac{40}{(1+y)^3}+\frac{1{,}040}{(1+y)^4}.

Substituting $y=0.08$ makes the future cash flows’ present value $\$867.514926398$, equal to the purchase price, so the net present value is zero. The strict-decrease result above means no other yield greater than $-1$ also satisfies this equation.

Check. The price is below face value, consistent with the 4% coupon being lower than the 8% yield. Avoid summing only the displayed six-decimal present values: those rounded terms add to $\$867.514927$, while the correctly rounded total from unrounded terms is $\$867.514926$. Both give a price of $\$867.51$ to cents, but only the latter preserves the stated precision.

Worked Example: Verifying a Candidate Yield From a Market Price

A hypothetical two-year bond has face value $\$1{,}000$, pays an annual coupon of $\$80$, and has a market price equal to the present value of its promised cash flows at 10%: $\$80/1.10+\$1{,}080/1.10^2$. Confirm that 10% is its YTM, state its IRR cash flows, and compare the coupon rate, current yield, and YTM. Assume annual year-end payments and no default.

Method. The year-1 payment is $\$80$; the year-2 payment is $\$80+\$1{,}000=\$1{,}080$. Discounting at the proposed 10% gives:

\begin{aligned} PV_1&=\frac{80}{1.10}=\$72.727272727,\\ PV_2&=\frac{1{,}080}{1.10^2}=\$892.561983471,\\ P_0&=72.727272727+892.561983471 =\$965.289256198. \end{aligned}

The cash-flow stream is $-\$965.289256198$ at time 0, then $\$80$ and $\$1{,}080$. Substituting $y=0.10$ into its IRR equation gives a net present value of zero, so 10% is the YTM. The coupon rate is $\$80/\$1{,}000=8\%$; the current yield is $\$80/\$965.289256198\approx8.288\%$; and the YTM is 10%. In this case, the bond’s discount price makes its YTM greater than both the coupon rate and current yield.

Check. The present values add to the stated market price, and the final payment includes both coupon and principal. A common error is to treat the final payment as $\$1{,}000$, which would omit one $\$80$ coupon and produce an incorrect price and yield equation.

Interpreting the Yield Carefully

The YTM equation links a price to the entire schedule of promised payments, not just to the coupon or to the difference between price and face value. The coupon rate determines the contractual coupon amount relative to face value. Current yield divides the annual coupon by the market price. YTM is the rate that discounts every scheduled payment, including the principal repayment, to that price. These measures coincide only in special cases.

The distinction is especially clear for a bond bought away from face value. A premium price means the investor pays more than the principal scheduled to be repaid at maturity; a discount price means the investor pays less. The YTM equation incorporates this principal gain or loss along with the coupons. It is therefore not correct to infer YTM simply by dividing the coupon by the purchase price.

YTM is an internal rate of return calculated from the bond’s promised cash flows and purchase price. The calculation does not ensure that every payment will be made, that the investor will hold the bond to maturity, or that coupons can be reinvested at the same rate. Those differences matter when comparing a calculated yield with an investor’s eventual experience. For the equation itself, the essential discipline is to specify the payment dates and compounding period, include every coupon and the principal exactly once, and use the same yield consistently for each discount period.

List the dated cash flows.
Record the purchase price as a time-0 outflow, each coupon as a later inflow, and the final coupon plus principal as the maturity inflow.

Match yield and timing.
For annual coupons and annual compounding, use one annual discount period per year; do not mix annual and subannual periods.

Set present value equal to price.
Discount each future cash flow at the candidate yield and require their sum to equal the time-0 purchase price.

Check the interpretation.
Verify the final payment includes principal and coupon, then compare the price with face value as a reasonableness check on the yield.

Check Your Understanding

For each problem, use annual year-end coupon payments, an annual effective yield, and promised payments with no default or transaction costs unless stated otherwise.

A two-year bond has face value $\$1{,}000$, a 5% annual coupon rate, and a 5% YTM. Write its cash-flow stream and calculate its price. (Answer: Cash flows $-\$1{,}000,\$50,\$1{,}050$; price $\$1{,}000.00$.)
A three-year bond has face value $\$1{,}000$, a 4% annual coupon rate, and a 5% YTM. Write the pricing equation and find the price to the nearest cent. (Answer: $P_0=40/1.05+40/1.05^2+1{,}040/1.05^3=\$972.77$.)
A two-year bond pays a $\$60$ annual coupon on $\$1{,}000$ face value and sells at $\$1{,}000$. What yield satisfies its IRR equation, and what is its final cash flow? (Answer: YTM $=6\%$; final cash flow $=\$1{,}060$.)
A three-year bond has face value $\$1{,}000$, annual coupons of $\$70$, and a 8% annual effective yield. Find its price to the nearest cent and identify whether it sells at a premium or discount. (Answer: $P_0=70/1.08+70/1.08^2+1{,}070/1.08^3=\$974.23$; discount.)
A two-year bond has face value $\$1{,}000$, annual coupons of $\$90$, and sells for $\$1{,}050$. Without solving for the exact yield, state whether its YTM is above or below its 9% coupon rate and give the cash-flow equation that defines it. (Answer: YTM is below 9%; $1{,}050=90/(1+y)+1{,}090/(1+y)^2$.)

What You'll Learn