# Bond Valuation

By Yuriy Smirnov Ph.D.

## Definition

Bond valuation is a process of calculating its fair price. Both investors and issuers use many different techniques, but most of them are based on one fundamental principle—that the fair price of a bond is equal to the present value of all future expected cash flows.

Because of continued economic changes the market price of a bond is usually different from its par value. If its current market price is less than par value, a bond is traded at a discount. Conversely, if its current price is above par value, a bond is traded at a premium.

## Formula

As mentioned above, most techniques used to determine bond valuation use a discounted cash flow approach. In general terms, the formula of a bond’s fair price is calculated as follows:

 Bond Price = C1 + C2 + C3 + … + CN + P (1 + r)1 (1 + r)2 (1 + r)3 (1 + r)N (1 + r)N

or

 Bond Price = N Ct + P Σ (1 + r)t (1 + r)N t = 1

where C is a periodic coupon payment, r is the market interest rate or required rate of return, and P is the par value of a bond.

If a bond has a fixed coupon rate, the formula above can be modified as follows:

 Bond Price = C × 1 – (1 + r)-N + P r (1 + r)N

## Discounted cash flow approach

The traditional discounted cash flow technique is based on the following assumptions:

1. The discount rate used in the model remains fixed, and therefore it applies to each cash flow, including par value paid at maturity.
2. Each cash flow is reinvested at a discount rate.

Thus, bond valuation involves three steps:

1. Determining the interest rate to be used as a discount rate.
2. Estimating expected cash flows.
3. Calculating the present value of each expected cash flow by applying the discount rate.

### How to determine a discount rate

Determining the discount rate is quite complicated and based on several assumptions. Traditionally, the required rate of return or required yield on a corporate bond can be calculated as follows:

Required Yield = Risk-free Interest Rate + Risk Premium

An on-the-run Treasury bond yield is usually used as a risk-free rate provided it compares with the corporate bond maturity date. In turn, the risk premium is based on the issuer’s credit rating. The lower the credit rating, the higher the risk premium and vice versa.

### Bond valuation example

An investor is considering purchasing a new issue of 5-year bonds of \$1,000 par value and an annual fixed coupon rate of 12%, while coupon payments are made semiannually. The minimum semiannual yield that the investor would accept is 6.75%. To find the fair value of a bond, we should calculate the semiannual coupon payment and apply the formula above.

Annual Coupon Payment = \$1,000 × 12% = \$120

Semiannual Coupon Payment = \$120 ÷ 2 = \$60

 Bond Price = \$60 × 1 - (1 + 0.0675)-10 + \$1,000 = \$946.71 0.0675 (1 + 0.0675)10

The cash flow is schematically presented in the figure below.

Thus, the highest price affordable for the investor is \$946.71.

### Zero-coupon bond valuation example

A zero-coupon bond does not have any coupon payments. It is sold at a lower price than the par value, and the par value will be repaid to the investor at maturity. Such a bond has only the cash flow equal to its par value repaid at maturity.

An investor is considering purchasing a 10-year zero-coupon bond of \$1,000 par value. Let’s calculate the fair value of the bond if the current interest rate for equally risky bonds is 12.4%.

 Bond Price = \$1,000 = \$310.70 (1 + 0.124)10

## Floating-rate bond valuation

A floating-rate bond, also known as a floating-rate note (FRN), has a coupon rate tied to a benchmark: Treasury bill, LIBOR (London Interbank Offered Rate), and prime rate or inflation rate. The floating coupon rate usually consists of two parts:

• floating benchmark

Coupon Rate = Benchmark + Premium

For example, if coupon payments are made semiannually, the 6-month LIBOR can be used as the floating benchmark. Let’s assume that the semiannual risk premium equals 3.75%; thus, the formula for calculating the coupon rate will be as follows:

Coupon Rate = 6 Month LIBOR + 3.75%

As the floating coupon rate follows the current level of interest rates, the current market price of a bond is usually close to its par value. In case of a significant increase or decrease in the issuer’s credit rating, the current market price of the floating-rate bond can significantly differ from its par value because the fixed premium depends on the issuer’s credit rating!

We should also highlight the inflation-indexed bond, which can have either a floating coupon rate tied to the inflation rate, e.g., CPI (Consumer Price Index) or the principal indexed to the inflation rate. Because the inflation rate has a greater influence on the current level of interest rates such a bond is also usually traded close to its par value.

## Relative price approach

The relative price approach of bond valuation focuses on determining the required rate of return. Under such an approach, the bond is priced relative to yield to maturity (YTM) of a benchmark (usually a government bond of similar maturity). Then we should add to a benchmark a risk premium depending on the issuer’s credit rating. The resulting interest rate is then used as the discount rate.

## Arbitrage-free pricing approach

Bond valuation under an arbitrage-free pricing approach considers each separate cash flow (coupons and principal) as a zero-coupon bond. The yield to maturity on a zero-coupon bond of similar maturity and equal credit rating is used as the discount rate. Thus, we use several discount rates instead of a single discount rate as in a discount cash flow approach and a relative price approach.