Yield To Maturity From Coupon And Redemption Cash Flows

From Scheduled Cash Flows to Yield

A bond’s coupon rate tells you how its coupon payments are determined; it does not, by itself, tell you the return implied by the price paid. Yield to maturity (YTM) is the rate that makes the present value of the bond’s scheduled coupons and redemption payment equal to its purchase price. Constructing the equation correctly requires matching each payment to its time and discounting in the same periods used to state the yield.

The setup below assumes a conventional fixed-payment bond, purchased on a coupon date, with no default, no missed payments, and no option exercised before maturity. The purchase price is the amount paid on that date, without accrued interest. The scheduled cash flows are received at the ends of their stated periods. These assumptions keep the cash-flow timing explicit; a purchase between coupon dates requires a separate treatment of settlement timing and accrued interest.

Definition: The yield to maturity is the interest rate that equates a bond’s purchase price to the present value of all scheduled coupon payments and its redemption payment, assuming those payments are made as promised and the bond is held to maturity. For a bond with $m$ coupon payments per year, if the quoted yield is a nominal annual rate $y^{(m)}$ convertible $m$-thly, the yield per coupon period is $j=y^{(m)}/m$.

Let $P$ be the purchase price, $C$ the coupon amount paid each period, $R$ the redemption amount paid at maturity, $n$ the number of remaining coupon periods, and $j$ the yield per coupon period. The last coupon and redemption amount are both paid at time $n$. The yield equation is:

P=\sum_{k=1}^{n}\frac{C}{(1+j)^k}+\frac{R}{(1+j)^n} =\ C a_{\overline{n}|j}+R(1+j)^{-n}

Here $a_{\overline{n}|j}$ denotes the present value of an annuity-immediate of $n$ unit payments at the per-period rate $j$. The summation form makes the timing visible: coupon $k$ is discounted for $k$ periods. The final coupon is included in the sum, and redemption is a separate payment at the same final date.

Why the Yield Equation Holds

Under the compound-interest model at rate $j$ per coupon period, one dollar due $k$ periods from the purchase date has present value $(1+j)^{-k}$ dollars. A coupon of $C$ dollars due then has present value $C(1+j)^{-k}$. The total present value is the sum of the present values of the separate payments, because each scheduled payment is a distinct cash flow.

Coupons are paid at periods $1,2,\ldots,n$, so their present values sum to the first term below. The redemption payment is also due at period $n$, so it contributes the second term. By definition, the yield is the rate that makes this total equal the amount paid:

\underbrace{\frac{C}{1+j}+\frac{C}{(1+j)^2}+\cdots+\frac{C}{(1+j)^n}}_{\text{coupon payments}} +\underbrace{\frac{R}{(1+j)^n}}_{\text{redemption payment}} =P

This reasoning depends on using the same period in the payment dates and in the rate. If coupons are semiannual but the quoted yield is nominal annual convertible semiannually, discount each half-year using $j=y^{(2)}/2$, not the full annual quote.

Worked Examples

Worked Example: Annual Coupons and Redemption at Par

A hypothetical bond has a face amount of $1,000, an annual coupon rate of 10%, and two annual coupon periods remaining. It is purchased on a coupon date for $784 and pays $1,000 at maturity. Find its annual yield to maturity. Assume all payments are made as scheduled.

Method. Write the cash flows at the ends of periods 1 and 2, then test the annual rate that equates their present value to the price. Because coupons are annual, the annual yield is also the per-period yield $j$.

The coupon each year is $C=0.10(1{,}000)=\$100$. The cash flows are $100 at the end of year 1 and $100 + $1,000 = $1,100 at the end of year 2. Try $j=25\%$:

\frac{100}{1.25}+\frac{1{,}100}{(1.25)^2} =80+704 =784

The calculated present value equals the $784 purchase price. Therefore the annual yield to maturity is 25%. The check is direct substitution into the price equation.

Common mistake avoided: The maturity cash flow is $1,100, not just the $1,000 redemption payment. The final coupon is still due at maturity and must be included.

Worked Example: Semiannual Coupons and a Nominal Annual Yield

A hypothetical bond with face amount $1,000 pays coupons twice per year at an annual coupon rate of 12%. Two coupon periods remain. The redemption amount is $1,000, and the purchase price on a coupon date is $930.58. Find the nominal annual yield convertible semiannually, to two decimal places. Assume the scheduled payments are made in full.

Method. First determine each half-year cash flow. Then convert the quoted nominal annual yield to a half-year rate before discounting. The coupon is $0.12(1{,}000)/2=\$60$ every six months. The period-1 payment is $60; the period-2 payment is $60 + $1,000 = $1,060.

Test a half-year yield of $j=10\%$. Its nominal annual equivalent, convertible semiannually, is $y^{(2)}=2j=20\%$. Discount the two payments by one and two half-year periods, respectively:

\frac{60}{1.10}+\frac{1{,}060}{(1.10)^2} =54.54545+876.03306 =930.57851

The calculated price rounds to $930.58, the stated purchase price. Thus the nominal annual yield convertible semiannually is 20.00% to two decimal places. The unrounded calculation checks the rounded price.

Common mistake avoided: Do not discount the first coupon for a full year, or use the nominal annual yield as the six-month rate. There are two payment periods, and the rate per payment period is $20\%/2=10\%$.

Worked Example: Redemption Above Face Amount

A hypothetical bond has face amount $1,000, an annual coupon rate of 2%, and two annual periods remaining. It is purchased for $732.80 on a coupon date and will be redeemed for $1,100 at maturity. Find its annual yield to maturity. Assume no default and no early redemption.

Method. Treat the redemption amount as the amount actually paid at maturity, not as the bond’s face amount. The annual coupon is $0.02(1{,}000)=\$20$. The period-1 payment is $20. At maturity the bond pays both the $20 final coupon and the $1,100 redemption amount, for a total of $1,120.

Test an annual yield $j=25\%$. The present value is:

\frac{20}{1.25}+\frac{1{,}120}{(1.25)^2} =16+716.80 =732.80

This equals the purchase price, so the annual yield to maturity is 25%. The check confirms that both the final coupon and the above-face redemption payment are discounted from period 2.

Common mistake avoided: Replacing the $1,100 redemption amount with the $1,000 face amount understates the final payment by $100 and produces the wrong yield equation.

Keep the Dates, Amounts, and Rate Aligned

A reliable equation starts with a timeline. At each coupon date, record the coupon due; at maturity, record the coupon and redemption together. Then use one yield-period discount for each coupon interval between the purchase date and payment date. If there are $n$ coupon periods remaining, the redemption payment is discounted $n$ periods.

Fix the purchase date and payment schedule.
Identify the date on which the price applies and the number of coupon periods remaining. In the examples here, purchase is on a coupon date, so the first scheduled coupon is one full period away.

Calculate each payment amount.
For a level-coupon bond with annual coupon rate $c$, face amount $F$, and $m$ payments per year, the coupon per payment is $C=Fc/m$. Add the redemption amount $R$ to the final coupon at maturity.

Express the yield per payment period.
If the yield is quoted as nominal annual $y^{(m)}$, use $j=y^{(m)}/m$ for each coupon period. Keep the quote convention and payment frequency consistent.

Discount and equate to price.
Discount each cash flow by its number of periods from purchase. The sum must equal the purchase price; verify a proposed yield by substituting it back into the equation.

The coupon rate and yield answer different questions. The coupon rate determines the scheduled coupon amount from face value. The yield is determined jointly by the purchase price, coupon amounts, payment dates, and redemption amount. In particular, a bond purchased at a price different from its redemption amount has a difference between purchase outlay and maturity proceeds that is reflected in the yield equation.

A useful common-error check is to inspect the final term before solving. It must include the final coupon and redemption, and its discount exponent must match the number of periods until maturity. Another check is to inspect the first coupon: for a purchase immediately after a coupon date, it is discounted for one period, not zero periods.

Key takeaway: Write the yield-to-maturity equation from the actual scheduled cash flows. Coupon payments and redemption are discounted from the purchase date using a rate per payment period; maturity includes both the final coupon and the redemption payment.

Check Your Understanding

For each hypothetical bond, assume purchase on a coupon date, all payments as scheduled, and the stated redemption amount. Use the supplied price to identify the yield to the precision indicated.

A $1,000 bond pays annual coupons at 8%, has two annual periods remaining, and redeems for $1,000. Its price is $965.29. Find the annual yield, to two decimal places. (Answer: approximately 10.00%)
A $1,000 bond pays annual coupons at 5%, has two annual periods remaining, and redeems for $1,100. Its price is $995.87. Find the annual yield, to two decimal places. (Answer: approximately 10.00%)
A $2,000 bond pays coupons semiannually at an annual coupon rate of 8%, has four half-year periods remaining, and redeems for $2,000. Its price is $1,861.40. Find the nominal annual yield convertible semiannually, to two decimal places. (Answer: approximately 12.00%)
A $1,000 bond pays annual coupons at 4%, has three annual periods remaining, and redeems for $1,000. Its price is $972.77. Find the annual yield, to two decimal places. (Answer: approximately 5.00%)

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