# What is Yield to Maturity (YTM)?

## The Mathematics of Bond Valuation

 annual    semi-annual Bond Face Value \$ Bond Price \$ Coupon Rate % Years to Maturity Optional Call Price \$ Years to Call Yield to Maturity = % Yield to Call = %

If you hold bonds that pay annual or semi-annual coupons, you can calculate the Yield to Maturity of each investment. YTM is a metric that puts a bond's yield in terms of an annual (or semi-annual) interest rate. Another way to think about yield to maturity is that it is the composite rate of return on the bond, assuming that you hold the bond until its maturity date and receive all of the coupons. The yield to maturity depends on the bond price (what you paid), the face value (usually \$1,000), the number of years until maturity, and the coupon rate.

Some bonds have a call option, which means you can redeem the bond before its maturity date for an amount that is different from the face value. In this case, you forgo some coupon payments that you would have received had you held the bond to maturity, but you get the redemption value sooner.

The formula that relates the annual YTM (R), the face value (F), the bond price (P), the number of years (N), and the annual coupon amount (C) is

C(1+R)-1 + C(1+R)-2 + ... + C(1+R)-N + F(1+R)-N = P.

The formula for semi-annual YTM (r) is

0.5C(1+r)-1 + 0.5C(1+r)-2 + ... + 0.5C(1+r)-2N + F(1+r)-2N = P.

These equations cannot be solved for exact values of R or r using algebra; they are mathematically too complex. However, you can use the YTM calculator to determine the yield to maturity or yield to call. The calculator uses an iterative algorithm similar to "guess and check."

## Examples

Suppose you have two bonds, Bond A and Bond B. Bond A has a purchase price of \$800, a face value of \$1,000, annual coupon payments of \$50 (i.e. 5%), and it matures in 7 years. Bond B has a purchase price of \$850, a face value of \$1,000, annual coupon payments of \$60, and it matures in 5 years.

Using the calculator, the yield to maturity of Bond A is 8.97% and the yield to maturity of Bond B is 9.95%.

Suppose Bond C has a purchase price of \$700, a face value of \$1,000, annual coupon payments of \$50, and it matures in 15 years. Also, suppose that it has a call date 5 years after the purchase, and the call price is \$900. The yield to maturity is 8.64% and the yield to call is 11.67%.

All of the above results are for bonds that pay annual coupons. If Bond A pays coupons semi-annually (two payments of \$25), then the semi-annual YTM is 4.45%.