# Yield to Maturity (YTM)

Yield to maturity (YTM) is the rate of return expected on a bond which is held till maturity. It is essentially the internal rate of return on a bond and it equates the present value of bond future cash flows to its current market price.

## Formula

If **m** is the number of coupons in a year and **n** is number of years the following equations can be used to find the yield to maturity.

Bond Price = Par Value × Coupon Rate × | 1 − (1 + r)^{−m×n} | + | Par Value |

r | (1 + r)^{m×n} |

Yield to Maturity = r × m |

The figure is calculated by trial and error in which we plug discount rates into the equation developed above until we find a rate which satisfies the equation (i.e. right hand side of the equation equals left hand side).

## Example

Company D's 10-year bond with par value of $1,000 and semiannual coupon of 8% is currently trading at $950. Find the yield to maturity on the bond.

__Solution__

We defined yield to maturity as the rate which discounts the bond's future cash flows (coupons and par value) such that their present value equals the bond's market price. Company D's bond has a par value of $1,000; semiannual coupon of $40 (=8%/2×$1,000) and price of $950.

We get the following relationship after plugging in the variables:

$950 = $1000 × 4% × | 1 − (1 + r)^{−10×2} | + | $1000 |

r | (1 + r)^{10×2} |

There is an interesting relationship between bond price and yield to maturity:

- If yield to maturity is equal to the coupon rate the bond is trading at par.
- If the yield to maturity is lower than the coupon rate the bond will be trading above par (which means trading at premium).
- If the yield to maturity is higher than the coupon rate the bond will be trading below par (which means trading at discount).

In the example above price is $950 which is lower than the par (which is $1,000). This suggests that the yield to maturity must be higher than the coupon rate (which is 4%). While using trial and error method we will start with discount rates above 4% and narrow down to until the present value is almost equal to the price. In the above example, discount rate of 4% gives price of $1,000; 4.5% gives a price of $935. From this we follow that we need focus on discount rates between 4% and 4.5%. A rate of 4.25% gives a price of $967 and 9.45% gives a price of $941. We keep narrowing down until we get the discount rate which exactly reduces the left hand side of the bond price equation to current bond price. For Company D's bond this rate is 4.38%. Yield to maturity is expressed as an annual rate so it equals 8.76% (=4.38%×2).

## Limitation of Yield to Maturity

Yield to maturity carries the same drawback as an internal rate of return: it assumes that the coupon payments are reinvested at the yield to maturity which is not normally the case. If coupons are to be reinvested at lower rates yield to maturity will be an overstated figure.

There are many other similar measures used such as yield to call, yield to put, cash flows yield, etc.

Written by Obaidullah Jan, ACA, CFA <--- Hire me on Upwork