# Bond Valuation

Bond value equals the present value of the bond cash flows i.e. coupon payments and maturity value at the market discount rate, the rate of return required by investors given the risk of the bond.

There is an inverse relationship between the bond value and market interest rates. If the rates increase, the bond value drops and vice versa. A bond whose coupon rate is lower than the market discount rate is traded at a discount i.e. at a price lower than its par value while a bond with coupon rate that’s higher than the market rate is traded at a premium i.e. at a price higher than its par value. This is because in a high market interest rate environment, bonds with lower coupon rates are not attractive and their prices drop.

## Formula

The value of a conventional bond i.e. a bond with no embedded options (also called straight bond or plain-vanilla bond) can be calculated using the following formula:

$$ Bond\ Value\\=\frac{c}{m}\times F\times\frac{1-\left(1+\frac{i}{m}\right)^{-n\times m}}{\frac{i}{m}}+\frac{F}{\left(1+\frac{i}{m}\right)^{n\times m}} $$

Where **c** is the coupon rate, **F** is the face value, **m** is the number of coupon payments per year, **n** is the number of years till maturity and **i** is the yield to maturity on the bond, i.e. the market interest rates.

The price determined above is the clean price of the bond. To find the full price (i.e. dirty price) of the bond, we must add interest accrued from the last coupon date to the settlement date.

Bond price can also be worked out using Excel PRICE function.

If the market discount rate is not available, a bond can be valued using **matrix pricing**, an approach in which yield to maturity on comparable bonds is used to value a bond.

While the above valuation based on a single market discount rate is valid in most cases, a theoretically better valuation can be arrived at by discounting each coupon payment using an interest rate applicable to that duration. This approach is called the **no-arbitrage valuation approach**. Let’s say you have a 5-year bond paying annual coupons. The first coupon should be discounted using interest rate valid for 1 one year, the second coupon should be discounted using an interest rate applicable for 2 years and so on such that each cash flow is discounted using a different most relevant market rate.

## Example

A $1,000 4% bond paying semiannual coupons is maturing in 3 years. If the applicable market yield is 4.2%, let’s value the bond using the conventional bond pricing approach:

Since the market discount rate is higher than the coupon rate, we can tell that the bond value will be lower than its face value. Let’s find out:

$$ Bond\ Value\\=\frac{4\%}{2}\times$1,000\times\frac{1-\left(1+\frac{4.2\%}{2}\right)^{-3\times2}}{\frac{4.2\%}{2}}+\frac{$1,000}{\left(1+\frac{4.2\%}{2}\right)^{3\times2}}\\=$994.42 $$

If the bond has 1.5 years left till maturity, let’s value it based on the spot rates applicable to each cash flow. Let’s imagine the yield on zero coupon bonds of comparable risk with maturity of 6 months, 1 years and 1.5 years is 4%, 4.1% and 4.5%.

The following equation shows the spot-rate valuation:

$$ Book\ Value\\=\frac{$20}{{(1+\frac{4\%}{2})}^{0.5\times2}}+\frac{$20}{{(1+\frac{4.1\%}{2})}^{1\times2}}+\frac{$20+$1,000}{{(1+\frac{4.5\%}{2})}^{1.5\times2}}\\=$992.95 $$