You must have heard about Bonds in financial news a lot. When a firm wants to raise money/capital for future operations or payment of previous expenses or any other reason, they often use bond instruments to do so. Even if you are only partly interested in the topic, it is often puzzling to wonder about the reason behind the hue and cry about Bonds and their valuation. This blog attempts to arm you with the tools to demystify the topic and develop a comprehensive understanding about bonds.

### What are Bonds?

A bond is a financial instrument used to raise capital by creating indebtedness of the bond issuer to bond holders. Then, depending on the terms of the bond, the firm is obliged to pay the buyers interest (the coupon) and/or to repay the principal at a later date, termed as maturity. Interest is usually payable at fixed intervals (semiannually, annually, and monthly).

So as an example, suppose there is a \$ 1000 US Treasury bond which pays a 6% semiannual coupon with three years remaining to maturity that is trading at \$ 1100. This would mean that the buyer, in return for his initial payment of \$1100, would receive interest payments of \$60 (=6% of \$1000) every six months till the maturity date is reached (3years from now), at which point the principal (\$ 1000) is paid back to the buyer along with the last interest payment.

### Terminology

First let’s get familiar with some of the terms we will be using in the blog.

 Par Value It is the face value of a bond. It is also the price of a bond when the coupon rate equals to the Yield to measure rate Coupon The interest rate stated on a bond when it is issued. The coupon is typically paid semiannually Maturity Upon maturity of a fixed income investment such as a bond, the borrower has to pay back the full amount of the outstanding principal, plus any applicable interest to the lender Zero coupon bonds A debt security that doesn’t pay interest (a coupon, rendering profit at maturity when the bond is redeemed for its full face value

Valuation of a coupon bond with annual coupon

Consider a security that will pay \$100 per year for ten years and make a single \$1000 payment at maturity. The value of bond is calculated by discounting the cash inflows with a discounting rate (say 8%):

This is derived using the concept of time value of money.

Valuation with semiannual coupon

Considering the same case as above, if the coupon was semi-annual, we would have had a payment of \$ 50 semi-annually rather than \$100 annually. Also, in this case the discount rate will be taken semi annually ie. 8/2=4%.

Valuation of a zero coupon bond

In case of a zero coupon bond, there will be only one cash flow which can be discounted based on the appropriate discounting rate.

Bond Value= Maturity Value/(1+i)n

Where, i= discount rate/Yield to maturity

n=number of periods

Change in price of bond with time

As the bond reaches its maturity its value comes closer to par value since its remaining cash flows are almost equal to the par value.

Change in price of bond with change in yield

It is seen that as the required yield of bonds increases their prices decrease. This is because the yield sets the standard for the level of returns to be provided by a bond. If the yield increases, it would mean that a bond that was trading at par prior to this, would now offer less return than required. Thus its price would decrease and similarly for a decrease in yield would cause increase in price. This can also be seen from the relation:

Yield Measures

Yield measures for a bond tell us about the sort of returns that are provided by a particular bond.

Current Yield = Annual Cash Coupon payment/Bond Price

Yield to Maturity (YTM)

This is the most important of the yield measures. YTM is basically the discount rate at which the present value of the bond payments equals the bond price. ie.

It can be calculated that bonds having a coupon rate greater than YTM are priced at a premium (ie. higher than par) and vice versa. This relationship can be depicted in the following manner:

 Bonds selling at: Relationship Par Coupon rate = YTM Discount Coupon rate < YTM Premium Coupon rate > YTM

Bond Equivalent Yield (BEY)

The BEY allows fixed-income securities whose payments are not annual to be compared with securities with annual yields. In this case, the yields are stated at a sub annual rate, it is then converted to the corresponding annual rate. For eg. a bond with a yield of 4% semi annually, will result in a BEY of (4%*2=)8% annually. This is usually the case with treasury bonds so they are quoted in terms of BEY. The BEY is the yield that is quoted in newspapers.

Cash Flow Yield (CFY)

Is used for mortgage backed securities and other amortizing asset backed securities that have monthly cash flows. It provides a monthly rate of compounding. To convert CFY to BEY we can use the following formula:

BEY = [(1+monthly CFY)6 -1]*2

BEY = [(1+monthly CFY)6 -1]*2

Concepts of duration and convection

The duration/convexity provides an approximate method to calculate the sensitivity of a bond to interest rate change. Its main advantage over the full valuation method is its simplicity. However, it provides only an approximate measure of interest rate susceptibility.

Duration

There are 2 different, equally valid interpretations of the concept of duration:

1. It is a measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. Duration is expressed as a number of years.

2. It is a weighted average of the time until the cash flow will be received. The weights are the proportion of the bond value that each cash flow represents.

3. It is also the approximate change in bond price for a 1% change in yield.

Using the third interpretation, the change in price of a bond caused by a change in yield can be approximated as:

Î”P/P = -D*Î”Y

Where, Î”P is the change in bond price

P is the original bond price

D is the duration of the bond

Î”Y is the change in yield

Types of Duration

Macaulay Duration: Is the weighted average of the time until the cash flow will be received. The weights are the proportion of the bond value that each cash flow represents.

Effective duration: A duration calculation for bonds with embedded options. Effective duration takes into account that expected cash flows will fluctuate as interest rates change.

Convexity

Convexity is a measure of the curvature in the relationship between bond price and bond yields. Higher the value of convexity, greater the deviation of the actual price from the duration approximation. TO incorporate the effects of convexity into the bond price change, the following equation is used to find bond prices:

Î”P/P = -(D*Î”Y) + (Convexity* Î”Y2)