# Bond Valuation

Bond valuation refers to the process of finding the intrinsic value of a bond. A bond's value equals the present value of its cash flows determined at the bond's required rate of return.

There is an inverse relationship between the bond value and required rate of return i.e. the market interest rate. If the interest rate increases, the bond value falls 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. On the other hand, a bond with coupon rate higher than the market interest 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.

There are two ways in which the present value of a bond's cash flows can be determined:

- Discounting all cash flows of a bond at a single interest rate i.e. the yield to maturity.
- Discount each cash flow at the relevant spot interest rate.

## Bond Valuation using Yield to Maturity

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 = c × F × | 1 − (1 + r)^{-n} | + | F |

r | (1 + r)^{n} |

Where **c** is the periodic coupon rate, **F** is the face value, **n** is the total number of coupon payments till maturity and **r** is the periodic yield to maturity on the bond, i.e. the market interest rate.

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.

## Bond Valuation using Spot Interest Rates

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 us 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 us value the bond using the conventional bond pricing approach:

### Yield to Maturity 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 us find out:

BV_{YTM} = 2% × $1,000 × | 1 − (1 + 2.1%)^{-3×2} | + | $1,000 | =$994.42 |

2.1% | (1 + 2.1%)^{3×2} |

### Sport Interest Rate Approach

If the bond has 1.5 years left till maturity, let us value it based on the spot rates applicable to each cash flow. Let us 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%. Since the bond pays semi-annual coupons, we need to include semi-annual spot interest rates.

The following equation shows the spot-rate valuation:

BV_{SR}= | $20 | + | $20 | + | $20 + $1,000 | =$992.95 |

(1 + 2%)^{0.5×2} | (1 + 2.05%)^{1×2} | (1 + 2.25%)^{1.5×2} |

by Obaidullah Jan, ACA, CFA and last modified on