On this page is a *bond yield to maturity calculator*, which will automatically calculate the internal rate of return earned by an investor who buys a certain bond. This calculator automatically runs, and assumes the investor holds to maturity, reinvests coupons, and all payments and coupons will be paid on time. The page also includes the *approximate* yield to maturity formula, and includes a discussion on how to approach the exact yield to maturity.

## Bond Yield to Maturity Calculator

#### Yield to Maturity Calculator Inputs

**Current Bond Trading Price ($)**– The price of the bond today**Bond Face Value/Par Value ($)**– The face value of the bond, also know as the par value.**Years to Maturity**– The numbers of years until the current bond matures.

#### Yield to Maturity Calculator Outputs

**Yield to Maturity (%):**The converged upon solution for the yield to maturity of the current bond (the internal rate of return)**Yield to Maturity (Estimated) (%):**The estimated yield to maturity using the shortcut equation explained below, so you can compare how the quick estimate would compare with the converged solution.**Current Yield (%):**Simple yield based upon current trading price and face value of the bond.

## Bond Yield to Maturity Formula

For this particular problem, interestingly, *we start with an estimate* before building up to the actual answer. That’s right – the **actual** formula for internal rate of return requires us to converge onto a solution; it doesn’t allow us to isolate a variable and solve.

However, that doesn’t mean we can’t estimate to come close. The formula for the approximate yield to maturity on a bond is:

( (Annual Interest Payment) + ( (Face Value – Current Price) / (Years to Maturity) ) )

/

( ( Face Value + Current Price ) / 2 )

Let’s solve that for the problem we pose by default in the calculator:

- Current Price: $920
- Par Value: $1000
- Years to Maturity: 10
- Annual Coupon Rate: 10%
- Coupon Frequency: 2x a Year

100 + ( ( 1000 – 920 ) / 10)

/

( 1000 + 920 ) / 2

=

100 + 8

/

960

=

11.25%

### And… What’s the *Exact* Yield to Maturity Formula?

If you’ve already tested the calculator, you know the actual yield to maturity on our bond is 11.359%.

So how did we get that?

By calculating the rate an investor would earn if reinvesting every coupon at the current rate, and determining the present value of those cash flows. The summation looks like this:

Price =

Coupon Payment / ( 1 + rate) ^ 1

+

Coupon Payment / ( 1 + rate) ^ 2

…

+

Final Coupon Payment + Face Value / ( 1 + rate) ^ n

As discussing this geometric series is a little heavy for a quick post here, let us note: for further reading, try Karl Sigman’s notes, hosted with Columbia. For most purposes, such as quickly estimating a yield to maturity, the approximation formula should suffice – any advanced valuation should be done procedurally, on a computer, anyway. The calculator internally uses the secant method to converge upon a solution, and uses an adaptation of a method from Github user ndongo.

## Yield to Maturity of Zero Coupon Bonds

A * zero coupon bond* is a bond which doesn’t pay periodic payments, instead having only a face value (value at maturity) and a present value (current value). This makes calculating the yield to maturity of a zero coupon bond straight-forward:

Let’s take as an example the following bond:

- Current Price: $600
- Par Value: $1000
- Years to Maturity: 3
- Annual Coupon Rate: 0%
- Coupon Frequency: 0x a Year

Price =

(Present Value / Face Value) ^ (1/n) – 1 =

(1000 / 600) ^ (1 / 3) – 1=

1.6666… ^ (1/3) – 1 =

18.563%

## Conclusion and Other Financial Basics Calculators

Use the *Yield to Maturity* as you would use other measures of valuation: another factor in your decision whether to buy or avoid (or sell?) a bond. You can compare YTM between various issues to see which ones would perform best… just note the caveat that YTM assumes no missed or delayed payments and reinvesting at the same rate upon coupon payments.

For other calculators in our financial basics series, please see:

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