By Yuriy Smirnov Ph.D.

The yield to maturity (YTM) of a bond is the internal rate of return (IRR) if the bond is held until the maturity date. In other words, YTM can be defined as the discount rate at which the present value of all coupon payments and face value is equal to the current market price of a bond.

To find the yield to maturity of a bond, the following equation should be solved:

Price = | N | Coupon payment | + | Face Value |

Σ | ||||

(1 + YTM)^{t} |
(1 + YTM)^{N} |
|||

t = 1 |

where Price is the current market price of a bond, and N is the number of periods to maturity.

To solve the equation above, the financial calculator or MS Excel is needed. For an approximate appraisal of yield to maturity, the following formula can be used:

Approx YTM = | Coupon Payment + | Face Value - Price |

N | ||

Face Value + Price | ||

2 |

Please note that coupon payments are usually made semiannually, so the semiannual YTM should be adjusted to the annual YTM as follows:

Annual YTM = (1 + Semiannual YTM)^{2} - 1

A private investor has acquired a 10-year bond at the current market price of $965. The face value is $1,000, and the semiannual coupon rate is 7.5%. Let’s assume that 2 years is left until the maturity date. This means that the investor will receive four semiannual coupon payments of $75 each half-year and $1,000 after 2 years.

Semiannual coupon payment = $1,000 × 7.5% = $75

To calculate the yield to maturity of the bond, we have to use the equation mentioned above.

$965 = | $75 | + | $75 | + | $75 | + | $75 + $1,000 |

(1 + YTM)^{1} |
(1 + YTM)^{2} |
(1 + YTM)^{3} |
(1 + YTM)^{4} |

To solve this equation, you can use the IRR function of MS Excel as in the figure below.

- Select output cell
**B4**. - Click
button, select**fx****All**category, and select**IRR**function from the list. - In field
**Values**, select the data range**B2:F2**, leave empty field**Guess**, and press the**OK**button.

Thus, the yield to maturity of the bond is 8.57%.

Using the IRR function allows you to get a precise appraisal of YTM, but we can also get a rough appraisal by approximating.

Approx YTM = | $75 + | $1,000 - $965 | = 8.52% |

4 | |||

$1,000 + $965 | |||

2 |

As we can see, the approximate appraisal is 0.05% percentage point less, which is unacceptable in financial calculations, but it can be used if a rough appraisal is needed.

The semiannual rate should now be adjusted to an annual basis, so the annual YTM is 17.87%.

Annual YTM = (1 + 0.0857)^{2} - 1 = 17.87%

The relationship between the current market price of a bond and its yield to maturity can be described as follows:

- If YTM is equal to the coupon rate, the bond is currently trading at face value.
- If YTM is higher than the coupon rate, the current market price of a bond will be lower than its face value, which means trading at a discount.
- If YTM is lower than the coupon rate, the current market price of a bond will be higher than its face value, which means trading at a premium.

This relationship can be illustrated on the data of the example above.

Indeed, if the bond is acquired at face value, its yield to maturity is equal to the coupon rate.

The yield to maturity has the same drawback as the internal rate of return and, namely, the assumption that all coupon payments are reinvested at YTM. Taking into account that capital market conditions are constantly changing, this assumption is untenable in the long run. If the actual reinvestment rate would be lower than the expected YTM, the investment will be overpriced, resulting in a loss.

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