Bonds, returns and business math Updated November 2026

Yield to maturity (YTM)

Yield to maturity is the annual return from buying a bond at its current price and holding it to maturity, with coupons reinvested at that rate. It is the yield y that solves the bond price equation, found by iteration. Approximation: (C + (F - P) / n) / ((F + P) / 2).

Syntax

P = C x (1 - (1 + y)^-n) / y + F / (1 + y)^n, solve for y   approx: (C + (F - P)/n) / ((F + P)/2)

Part	What it is
`P`	Current bond price
`C`	Coupon payment per period
`F`	Face value
`n`	Periods to maturity

Examples

Price 950, 5% annual coupon, F = 1,000, 10 years
Approx: (50 + 50/10) / 975 = 5.64%

Result: Exact YTM = 5.67%

Excel: =RATE(10, 50, -950, 1000)

Price 1,050, 6% coupon paid semiannually, 5 years
C = 30, n = 10

Result: Semiannual rate 2.43%, YTM = 4.86% (bond equivalent)

Excel: =RATE(10, 30, -1050, 1000)*2

Price 900, 4% annual coupon, 3 years
Approx: (40 + 100/3) / 950 = 7.72%

Result: Exact YTM = 7.87%

Common mistakes

Reporting the semiannual rate from RATE as the annual YTM; multiply by 2 for the conventional bond-equivalent yield.
Using the approximation as the final answer: it is usually within 0.2% but not exact, and the error grows for bonds far from par.
Entering price and face value with the same sign in RATE, which returns #NUM!; price is the outflow (negative), coupons and face are inflows.

Practice Yield to maturity (YTM)

These drills check your answer on live data, free, no account needed.

Yield to maturityA $1,000 bond with a 2% coupon, paid semiannually, matures in 15 years and trades at $908.25. What is its yield to maturity, stated as an annual rate compounded semiannually?
Internal rate of returnAn investment costs $25,000 today and returns $17,500, $15,500 and $14,000 at the end of years 1 to 3. What is its internal rate of return?

Questions

What is the difference between YTM and coupon rate?

The coupon rate is fixed at issue and sets the cash paid. YTM is the market's return at today's price. Price below par means YTM above the coupon rate, and vice versa.

Is YTM the return I will actually get?

Only if you hold to maturity, the issuer does not default and you reinvest each coupon at the YTM. If reinvestment rates are lower, the realized return is lower.

Bond priceA bond price is the present value of its coupons plus the present value of its face value at the yield. Annual and semiannual examples, premium and discount.
Current yieldCurrent yield is the bond annual coupon divided by its current price. Examples for discount and premium bonds, how it compares with YTM and mistakes to avoid.
IRR (internal rate of return)The IRR is the discount rate that makes NPV zero. How to find it by trial, what Excel IRR returns, worked examples, the accept rule and where IRR fails.
Effective annual rateThe effective annual rate turns a nominal rate with monthly, quarterly or daily compounding into the true yearly rate: (1 + i/m)^m - 1. Examples and Excel.