Yield to Maturity (YTM)

The internal rate of return that equates the present value of a bond's remaining cash flows (coupons + face) to its current market price. Solved iteratively because no closed-form exists for a coupon bond.

When to use: Infer a nominal annual yield from a positive coupon-date bond price, contractual coupons and face repayment. For callable bonds, also examine every applicable call scenario.

Calculator

Formula

P = \sum_{t=1}^{Nm} \frac{F \cdot CR / m}{(1 + y/m)^{t}} + \frac{F}{(1 + y/m)^{Nm}}, \quad \text{solve for } y

Variables

Symbol	Name	Description	Unit
YTM	Yield to Maturity	IRR of the bond if held to maturity, expressed as a nominal annual rate compounded m times per year	%
P	Bond Price	Market price of the bond per face value unit	$
F	Face Value	Par value paid at maturity	$
CR	Coupon Rate	Annual coupon rate as a decimal (e.g. 0.05 for 5%)	%
N	Years to Maturity	Years remaining until the bond matures	years
m	Coupons per Year	Number of coupons paid per year (e.g. 2 for semi-annual)	integer

Real-Life Examples

Example 1: Discount Bond YTM

10-year, 5% coupon, semi-annual, $1,000 face, trading at $925.61.

Given

P = $925.61F = $1,000.00CR = 5.0000%N = 10.00 yearsm = 2.00

Step-by-Step

1.Set up: 925.61 = 25 × annuity factor(y/2, 20) + 1000 / (1+y/2)²⁰

2.Bisection on y: try 5% → P = 1000 (too high)

3.Try 7% → P ≈ 858 (too low)

4.Converge at y ≈ 6.00% → P = 925.61

5.YTM = 6.00%

Result:6.0000%

The bond's YTM is 6%, exceeding both the 5% coupon and the 5.4% current yield. The extra ~60bps over current yield comes from the $74 pull-to-par capital gain over 10 years.

Example 2: Premium Bond YTM

5-year, 6% coupon, semi-annual, $1,000 face, trading at $1,043.76.

Given

P = $1,043.76F = $1,000.00CR = 6.0000%N = 5.00 yearsm = 2.00

Step-by-Step

1.Bisect: at y = 5%, P = 1043.76

2.YTM = 5.00%

Result:5.0000%

Premium price ($1,044 > $1,000) implies YTM (5%) below coupon (6%). The $44 pull-to-par capital loss over 5 years drags total return below the coupon income.

Frequently Asked Questions

This model settles on a coupon date, with level coupons and a positive whole number of periods remaining. Between coupons, move each discount time to the actual settlement date and account for accrued interest; merely adding accrued interest to an unchanged coupon-date valuation is not exact.

Yield is the rate that equates discounted contractual cash flows to price. Realizing that rate as a compounded terminal-wealth return requires the assumed payments and reinvestment at that rate. Actual reinvestment rates can differ.

Yes, provided the periodic discount factor remains positive. The solver verifies that the returned yield reproduces the entered price.

Related Formulas

Bond Price

Inverse calculation

Yield to Call (YTC)

Yield to Worst (YTW)

Current Yield

Current Yield Yield to Call (YTC)