# Yield to maturity (expected rate of return)

“It is the rate of return a bond holder is expected to earn if he holds the bond till maturity.”

Or, in other words,

“It is the discount rate which equals the 1) bond current market price to its 2) present value of total remaining future expected cash flows of the bond.”

## Explanation

If you desire to invest in the bonds, you will obviously visit the financial market for purchasing bonds. In financial market following information will be available to you relating to bonds.

1. Bonds face (par) value, Coupon (interest) rate that will be paid on the bond, issue date, maturity date &
2. Current market price of the bond that is required to be paid for purchasing the bond.

The question arises which bond should be purchased. Obviously the answer is to purchase the one which is more beneficial to the investor. How investor will determine which is more beneficial? Here comes the concept of yield to maturity. Yield to maturity gives the expected rate of return on the bond thus enabling the investor to make a better investment decision.

## Yield to maturity: Formula

The bond's current market price, at any point of time, is determined by the following formula.

 V  = C * 1 – (1 + r) -t + F r (1 +r) t

More details about bond valuation / pricing are given here.
Where
r = the discount rate which equals the present value of future expected cash flows to its current market price denoted by “V” in the above formula. Note that this is the same definition we have stated earlier at the start.

Therefore by solving the formula for the value of “r” we can determine the expected rate of return of investor or, in other words, the yield to maturity. The value of “r” is calculated with trial and error method which is best explained by the following example.

## Example

Suppose a bond’s current market price is \$995.14. The face value of bond is \$1000, coupon is 8% and maturity is after 6 years. Calculate the yield to maturity of the bond.

Solution

The bond’s valuation / pricing formula is

 V  = C * 1 – (1 + r) -t + F r (1 +r) t

By putting the values from the question, we get

 995.14  = 80 * 1 – (1 + r) -6 + 1000 r (1 +r) 6

\$80 is the annual coupon interest rate calculated as follows \$1000 * 8% = \$80. Now solving this equation for the value of “r”, we need to use trial and error method.

First of all the current market price of the bond is below par value which means that the yield to maturity rate is more than 8%. How we have concluded this. The answer is by using one of the relationships between bond prices and interest rates. Therefore let’s start with 10% as our trial yield to maturity (expected rate of return) rate.

 995.14  = 80 * 1 – (1 + r) -6 + 1000 0.1 (1 +0.1) 6

The answer we get is \$912.89 which is lower than actual price i.e. \$995.14. This means 10% is too high we need to lower the rate. Let’s use 9% as our rate.

By substituting 9% instead of 10% in the formula we get answer equal to \$995.14 which is equal to our bond price and thus is the yield to maturity of the bond.

Yield to maturity or expected rate of return is then compared with required rate of return of the investor for the purpose of making decision.