# Yield at Maturity Formula

The yield to maturity is the annual return annual rate (discounted) earned over a bond kept until maturity.  The yield to maturity is the discount rate estimated mathematically that equals the cash flow of payment of interest and principal received with the purchasing price of the bond.

This term is also referred to as internal rate of return or as the expected rate of return of the bond and it is the yield in which most bond investors are interested in.

The yield-to maturity is 8.5%. If you don’t have Microsoft Excel software on your computer, you can use the following approximation formula to determine the yield-to-maturity (YTM for the example:

YTM = Coupon payment + [ (1000 - purchase price) / periods of maturity ]                                (1000 + purchase price) / 2

YTM = 50 + [ 1000 – 770.36) / 10 ]  = 8.24%
( 1000 + 770.36) / 2

Using the approximation formula the 8.24% yield understates the true yield-to-maturity that is calculated using a computer. The reason is that the approximation formula does not use the time value of money for compounding of the coupon payments.

The yield-to-maturity depends on two assumptions:

• The bonds are held to maturity
• The interest payments received are reinvested at the same rate as the yield-to-maturity

If the bond is not kept until maturity, you could estimate the internal return rate of the bond by substituting the sales price of the bond for that of its maturity value and the period kept until the selling date for its maturity date.

The yield-to-maturity rate assumes that bondholders reinvest the received interests from the same yield-to-maturity. If this does not happen the owners return rate will differ from that of the quoted yield-to-maturity rate. For example, if the interest received is spent and not reinvested, the interest does not earn interest; the investor earns much lesser (or greater) rates, the 8% is not achieved. In reality matching the yield-to-maturity rate from the interest received is difficult because interest rates are changing constantly. The interest received is usually reinvested at different rates from the stated yield-to-maturity rate.

However, the yield-to-maturity is useful to compare and evaluate different bonds with variable qualities with different coupon rates and prices. For example when comparing the yield-to-maturity of an A&A-rated bond with a BBB-rated bond you can easily notice in how much would the yield increase when choosing the longer-rated bond. Also you will observe the yield differential between bonds with different periods of maturity.

The relation between the coupon yield, current yield, yield-to-maturity, and the prices of the bonds are summarized next:

 BOND PRICE Discount coupon yield = current yield < YTM Face coupon yield = current yield = YTM Premium coupon yield = current yield > YTM
Yield-to-call: The yield-to-call is the annual rate of return that bondholders receive at a date in which the bond is called. When a bond has a call feature, the bondholder can estimate the yield-to-call by substituting the call price for that of the maturity price in the equation shown above in the yield-to-maturity section. Both, the yield-to-call and the yield-to-maturity should be determined because if the bond is called the yield-to-call will be the annual overall return that the bondholder will receive over his bonds.