# Bond Pricing

Bond Pricing/bond valuation is a method of calculating the fair price or value of a bond. The price of a bond is calculated by finding out the present values of future cash flows and discounting them at an appropriate discount factor.

## Understanding Bond Terminology

Before we learn how to determine the price of a bond, let us understand some bond terminology.

A bond is a note issued by governments or corporates that promises its buyer a specified amount of money after a fixed duration (maturity). It also agrees to pay a certain amount as interest to the buyer until maturity. This interest rate is called the coupon rate. Hence, the investor receives a total cash flow of interest paid every year until maturity plus the value of the bond.

### Example

A 5-year bond of \$1000 face value at a 5% coupon rate means a firm has borrowed \$1000 from the buyer of a bond and promises him (the buyer) to repay the \$1000 after five years plus an interest of 5% paid every year. \$1000 is the par value or face value of the bond, 5 years is the maturity period, and 5% is the coupon or interest rate.

So, the total amount the buyer gets is,

1st Year: \$50

2nd Year: \$50

3rd Year: \$50

4th Year: \$50

5th Year: \$50 + \$1000

Total amount: (5*\$50) + \$1000 = \$1250

Hence, the profit made by the buyer of the bond is \$250.

## Need for Bond Pricing

If an investor buys a security, he ensures he gets the best returns out of it. Hence, he always compares among other securities available in the market to choose the best one to buy. So, an investor decides to buy a particular bond if it offers better returns than its peers available in the market, given that the risk associated with them is the same.

However, bonds usually do not trade at par value in the open market. They either trade at a discount or at a premium, depending on the interest rate environment prevailing in the market. Here arises the need to calculate the actual value of the bond (called fair value or intrinsic value) to determine if it is a good buy or not with respect to the current interest rate offered in the market. If the interest rates in the market are always constant, there is no question of valuing a bond.

The main objective of valuing a bond is to compare if the returns offered by the bond (known as bond yield) an investor wishes to buy are greater than or equal to the risk-free interest rate (interest rate offered by no risk securities Example: 3-month treasury bills in the US) present in the market. The

## How to Find the Price of a Bond?

As mentioned above, the right technique to value a bond is to find out the present value of the future cash flows of the bond. Cash flows from the bond are nothing but the coupon payments made every year (or quarter or semi-annually). The final bond price is the sum of all the coupon payments of each year until maturity plus the face value of the bond (as shown in example 1).

Cash flows of each year are calculated by finding an appropriate discount factor and discounting the present value of the coupon payments using this rate. The discount factor is the interest rate, which an investor will get if they hold the bond till maturity. This is called yield to maturity (YTM).

### Formula

The formula for calculating the value of a bond (V) is

I = annual interest payable on the bond

F= Par value of the bond (repayable at maturity)

r = discount factor or required rate of return

n= maturity of the bond

## How to Determine the Discount Rate?

YTM is the return an investor gets if he holds the bond until maturity. The discount rate is the interest rate that the investor wishes to get as a return (in terms of interest %) with respect to the current interest rates prevailing in the market.

After plugging in all the values in the above formula, one can calculate the price of a bond.

The following example helps to understand this concept better.

### Example 2

Calculate the price of a bond whose face value is \$1000. The coupon rate is 10% and will mature after 5 years. The required rate of return is 8%.

Coupon payment every year is \$1000*10% = \$100 every year for a period of 5 years.

Hence,

Therefore, the value of the bond (V) = \$1079.8

The following is the summary of bond pricing:

## Bond Pricing with Interest Rate Changes

The interest rate prevailing in the economy and the bond price or valuation has a definite co-relation. Instead, it is an inverse relationship. With the increase in the interest rates, the bond prices drop because the yield to the investors of bonds reduces, whereas the market interest is higher. To make up for that, the prices fall so that the yield on bonds matches with the market’s prevailing rates.

Similarly, as the interest rate in the economy drops, the bond prices go up so that the yield on the bonds matches with the market yield. In the reducing interest rate scenario, this gives a capital appreciation to the bondholders.

This inverse relationship is much more pronounced where:

(1) The balance maturity period is longer; and

(2) The coupon rate on bonds is quite low. A minor change in interest rate may affect the net yield substantially.

## Bond Valuation with Call and Put Options

Bond pricing and valuation also have an impact on the bond issuance terms and conditions. These can be :

1. Convertible Bonds: The bond issuer may issue a convertible bond. Moreover, that bond may be convertible If the bonds are convertible into equity or preferred stock, fully or partially. Then the standard pricing method discussed above needs tweaking. The standard pricing method is useful only for the non-convertible portion of the bond. And for the convertible portion of the bond value, the application of the stock valuation method happens. Thus, the bond value or price will be the sum of both the values so obtained.
2. Bonds Redeemable at Premium: Sometimes, the issuer of bonds puts a clause that the redemption of the bonds on maturity will happen at a premium of say 5-10% of the bond value. So this is the additional cash flow the bondholder will receive on maturity. Hence, like maturity proceeds, the premium amount receivable should also be added in the cash flow for calculation of the bond prices.
3. Call & Put Options: Long-term bond issuers always face a risk of interest rate uncertainty over such a long period. To overcome this shortcoming or to reduce such interest rate variation uncertainty, the companies often issue bonds with put or call options. Or with both-put and call options.

A put option means the bondholder has the right to submit the bond for redemption and ask for the money of the bond from the issuer. This option is available on certain dates or after certain years of issuance. Similarly, in call options, the company or the issuer has the right to redeem these bonds at certain dates by buying back/ calling back the bonds or extinguishing the bonds by repaying the money.

Of course, the exercise of these rights remains optional with the issuer as well as the bondholders. Moreover, the exercise of these rights depends upon the prevailing interest rate scenario and future interest rate trends in the economy.

Obviously, if the interest rate has already gone up or continues to increase, then the bondholder would like to exercise the call option and invest the funds in other securities to earn higher returns. Similarly, if the funds’ availability is easy and at cheaper rates, then the bond issuer may like to call up the bonds by making the payments to the bondholders. And after that may go for a fresh issue of bonds at a lower interest rate.

Continue reading Clean Price vs. Dirty Price.