Bond Valuation

Readings: Chapter 7

At the end of this unit students should be able to:

1. Identify the key characteristics common to all bonds.

2. Calculate the value of a bond with annual or semiannual interest payments.

3. Plot and interpret the Time Path to Maturity of a bond

4. Explain why the market value of an outstanding fixed-rate bond will fall when interest rates rise on new bonds of equal risk, or vice versa.

5. Calculate the yield to maturity, and/or the yield to call on a bond.

6. Differentiate between interest rate risk, reinvestment rate risk, and default risk.

7. Explain the importance of bond ratings and list some of the criteria used to rate bonds.

Bonds are normally long-term debt instruments issued by a company to raise funds. The bond contract (indenture) states a rate of interest called the coupon rate, a par value ( face value) and the maturity date

Par Value  this is the nominal value of the bond when it is issued and most bonds are issued at a $1,000 par value. This is the sum that the issuer takes from the original investor and makes a promise to pay at the maturity date while paying interest during the intervening period. (a discount bond is sold below par and a premium bond is sold above par)

Maturity date - this is that future point in time when the issuer repays the par value of the bond

The coupon rate  is the bond's rate of interest that is paid to the bondholder. On specific dates a fixed amount is paid to the bondholder. A coupon interest rate does not change once the bond is issued

Callable Bond - Some bond contracts have a clause termed a call provision which gives the issuer the right to redeem the bond prior to the maturity date. This is so if interest rates fall subsequent to the issue then the issuer would issue a new set of bonds at a lower interest rate and use the proceeds to pay off the old bonds. So callable bonds have attached to them a call premium which attempts to compensate the bond holder for the dis-utility that is caused from a bond that is called. This dis-utility includes:

 1.      Reinvestment rate risk - the loss of having to reinvest money in an uncertain low interest economy

2.      Loss of income

There are usually two classes of Bonds

  1.      Pure discount bond ( zero coupon)

-         This does not pay any interest during the life of the bond but is sold at a discount and redeems at par, given by:  

where:  M = par value / face value/ maturity value of bond k = investor's required rate of return n = remaining time to maturity of the bond

2.      Coupon Bond- given by:  

where:          I  = interest paid per period = ( M x CIR%) CIR% = coupon rate/coupon interest rate

If interest is paid more than once per year, the formula is adjusted as follows:    

Example 1

How much will you be willing to pay for a bond with a $1,000 par paying 15% per annum interest if your required rate of return is 20% and the bond matures in 10 years?  

This is the maximum that you will be willing to pay now for this bond given your 20% required rate of return. This is so because this is the amount of money that you could deposit now in an account that pays 20% per annum for the next 10 years to enable you to make 10 yearly withdrawals of $150 plus a $1,000 withdrawals at the end of the 10^th year.

The bond sells at a discount because current interest rates are higher than that reflected in the coupon interest rate attached to the bond. It means therefore that an investor with money to invest currently can earn more than 15% per annum and in turn must be compensated for this.

A discount bond is one where the investors required rate of return k is greater then the coupon interest rate, i.e..  

Example 2

What would you pay for the bond if your required rate of return were 10% per annum?

The bond sells above par. In this case the investor's k is only 10% while he is now getting 15%, so he must pay an extra cost for this.

 A premium bond is one where the investor's required rate of return is less than the coupon interest rate, i.e.  

Example 3

What would you pay for the bond if your required rate of return were 15% p.a.

The price is equal to the par value. Hence when the CIR = Investor's required rate of return(k), the price of the bond is always  = to the par value. These are called Par Bonds. When bonds are originally issued they are so done at Par to attract investors.

Let's now look at the price of the bond after one year:

1.      The discount bond:  

2.      The premium bond

3.      The par bond  

From the above we can see that as the maturity date approaches, the bond value converges on par ($1,000). This is also expressed by the following graph:

Time Path to Maturity at different discount rates  

This shows the relationship between price and the remaining time to maturity at a given discount rate.

Interest Rate Risk/Reinvestment Rate Risk/Default Risk

Interest Rate Risk

Recall from our earlier lectures that the likelihood that interest rate will rise is referred to as interest rate risk. Let us now examine what the impact on a bond's price will be given an increase in interest rates.

Using the premium bond our example, we see that the bond was originally priced and issued at $1,307.19 (when the interest rate, k, was 10%). Now let's assume that immediately after the issue (e.g. the very next day), interest  rates rose to 12% (i.e. k is now 12%). The price of the bond will now be:

Therefore if you were to sell these bonds you would effectively lose $137.66 (i.e. $1,307.19 - $1,169.53).

Note - Longer term bonds are exposed to greater interest rate risks as the magnitude of the fall in the price will be much greater than for  short term bonds of similar class.

Reinvestment Rate Risk

On the other hand a decrease in interest rates would lead to an increase in your bond's value. However, such a decline in interest rates will bring about reinvestment rate risk. This is the risk that the cash flows received from existing bonds, when reinvested at the lower rates, will result in reduced income for the bondholders. Shorter term bonds are more exposed to reinvestment rate risks than longer term bonds. This is so because not only the interest earned will be reinvested at these lower rates but the principal will also be reinvested at these lower rates when the bonds mature. Only the interest on long term bonds will be exposed to reinvestment rate risks as the principal will normally not mature in the near future.

Default Risk

Recall from our earlier lectures that a default risk premium (DRP) as added to k^* for corporations. However none was added for Government instruments. The reason was that corporations are more likely to default on payment than the Government. As far as corporations are concerned, the more likely it is that a company will default on payment, then the higher the DRP will be and in turn the higher the interest rate, k will be.

Yield/Rate of Return 

The total yield or total rate of return on a bond a any point in time is equal to the sum of its current yield and its capital gains yield. The current yield is equal to the annual interest payment divided by the   previous price and the capital gains yield is equal to the current price - the previous price, all divided by the previous price i.e.

Therefore using our previous example of the discount bond, the total yield (k) over the first year of its life would be computed as follows :

Yield to maturity(YTM) /  Internal rate of return (IRR) 

This is similar to the preceding discussion and captures the total yield of the investment. That is, It is the true rate of return if the investment is held to maturity and each interest payment is reinvested at the YTM . Assuming that previous statement always holds, a bond's yield to maturity is that interest rate when used to discount the bond's future cash flows equates them to the current price.

Illustration

Let's use one of our previous examples again, the premium bond:  

CIR=15%, k=10%, n=10 years, M=$1,000 and Price = $1,307.19

  Given the current price of $1,307.19, the bonds YTM is 10%, so long as each interest payment is reinvested at 10% and the bond is held until maturity.

  Proof:  

There are two ways of estimating the yield to maturity

1.      The Rodriques formula: (Note - this formula can be used irrespective of how often interest is paid per annum)

Note - the version below could also be used (assuming that interest is paid once per year). In fact, I have used it quite a lot in recent times. Also see  http://www.e-analytics.com/fp22.htm  for its application.

  2.      Trial and error

  Two discount rates are applied in an attempt to arrive at two prices . One above and one below the current price. A process called linear interpolation is then used to arrive at YTM. E.g.  

Class Question

Find the yield to maturity on a bond with 4 years to maturity 8% coupon, $1,000 par and currently sells at $890.00.

Note - the Rodriques formula would have given:

Yield to Call

You will recall that earlier in our discussion we mentioned Callable Bonds. Now bonds are likely to be called if interest rates fall, thereby allowing the issuer to to issue new bonds paying a lower interest. The issuer will use the proceeds of these new bonds to pay-off (retire) the older callable bonds. Now if an investor bought a callable bond and it is likely that interest rates will fall, and therefore likely that the bonds will be called, he is more likely to earn a Yield to Call (YTC) rather than a Yield to Maturity (YTM). This YTC is determined in a manner similar to the YTM, except that the maturity value will not be $1000 but will be the call price. In addition, you will not automatically have as a starting point, a price of $1000 if you were to use k = CIR. (Note - the Rodriques formula could still be used).

Importance of Bond Ratings

Bond Ratings are important both to firms and to investors. First, because a bond's rating is an indicator of its default risk, the rating has a direct, measurable influence on the bond's interest rate and the firm's cost of debt. Second, most bonds are purchased by institutional investors rather than individuals, and many institutions are restricted to investment-grade securities. Thus if a firm's bonds fall below the BBB, it will have a difficult time selling new bonds because many potential purchasers will not be allowed to buy them^.[1]

[1] The section has been taken directly from the text, page 290. However, please read pages 286 - 294 for a better understanding.