Fixed Income Bonds – Part 3

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In bonds
Feb 19th, 2012
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Fixed Income BondsThis is our third and final post on Bonds. Till now we have covered quite a lot on bonds. What remains are roughly the following topics:
  • Eurobonds
  • Bond Pricing
  • Yield to maturity
  • Reinvestment Risk 
We will quickly see what a Eurobond is and try to discuss bond pricing and yield to maturity in detail. In the end we will have a brief on reinvestment risk.

Eurobond: A Eurobond is a debt instrument like all other bonds. It is an international bond which can be issued in any country around the world but the currency in which the bond is issued cannot be the same as the native currency where the bonds are issued.
  • Foreign Bonds: These bonds are issued by foreign domiciled issuers but in the native currency of the country where they are issued.
  • Domestic Bonds: As is apparent, these bonds are issued in the native currency and by a domestic borrower.
Bond Pricing: Now if you intend to buy a bond how much you should be willing to pay for it? Next we will be answering this question.
We have already read about time value of money. So we know that if the interest rates are 10% then Rs 100/- today are of the same worth as Rs 110/- after a year. Or we can say that Rs110/- after one year has a present value (PV) of Rs 100/-.
Hence we follow the same logic while calculating the price for a bond. Like in the above example, we will discount all the cash flows that we are expecting in the future during the bond’s life time to their present value. Then we will add these values to calculate the price.

Let’s do the math.  Take a simple example of a bond.

Example:
Face Value
Rs 100/-
Annual Coupon
10%
Maturity
5 Years
Interest Rates(Prevailing   currently)
8%
Cash flows Expected:
1st Year
Rs 10/- (Coupon)
2nd Year
Rs 10/- (Coupon)
3rd Year
Rs 10/- (Coupon)
4th Year
Rs 10/- (Coupon)
5th Year
Rs 110/- (FV + Coupon)
Total
Rs 150/-
This shows that after investing a certain amount to purchase a bond you would get Rs 150/- in a span of five years. All we need to do is to calculate the present value of the individual cash flows at the end of each year to calculate the price.

Calculation:   



This is a simple formula taken from compound interest. 
PV, present value
FV, face value
I, interest rate by which we discount the future cash flow
T, length of each time period
N, total number of periods

PV = Rs 107.9854/-
Hence this bond will be trading at premium to its face value in the market, i.e. at a higher price than its par value which is Rs 100/- in this case.

This was a simple technique to calculate the price of a bond. Next we move on to another important term related to bonds.

Yield to Maturity: First we will again have a look at the bond in our previous example.

Face Value
Rs 100/-
Annual Coupon
10%
Maturity
5 Years
Interest Rates(Prevailing   currently)
8%
However the bond is offering a 10% coupon it is only giving 8% annual return to the investor because it is priced at a premium to par i.e. Rs 107.9854/- This annual interest rate is the Yield to Maturity in case of bonds.
Yield to Maturity (YTM) is the annual return rate that an investor would earn purchasing the bond and holding it till maturity. It takes into account different coupon rates and the fact that the bond might be purchased at different prices.

A particular bond pays a regular fixed coupon rate throughout its lifetime which cannot be adjusted to match the current interest rates. The only way that a bond can match the current market yields i.e. the current interests is by adjusting its price.

To understand it better we will compare two bonds now which differ only in coupons they pay.

Bond 1
Bond 2
Face Value
Rs100
Face Value
Rs100
Coupon
10%
Coupon
6%
Maturity
5 Years
Maturity
5 Years
YTM
8%
YTM
8%
Price(Calculate above)
Rs 107.9854/-
Price
??
We will quickly calculate the price for Bond 2 using the formula discussed above.

PV = Rs 92.01458/-


Hence we see Bond 2 will be available at a price of Rs 92.0146/-
Now the question is, does the Bond 2 even after offering a coupon of just 6% yields 8% in its life time? Let us validate this.

Purchase Price
Rs 92.01458/-
Coupon
6%
YTM
8%
Time Line
Calculation
Bond Value
After 1st Year
According to YTM
1.08 * 92.0146
99.37575
After coupon payment of Rs 6/-
99.37575 – 6
93.37575
After 2nd Year
According to YTM
1.08*93.37575
100.8458
After coupon payment of Rs 6/-
100.8458 – 6
94.8458
After3rd Year
According to YTM
1.08*94.8458
102.4335
After coupon payment of Rs 6/-
102.4335 – 6
96.4335
After 4th Year
According to YTM
1.08*96.4335
104.1481
After coupon payment of Rs 6/-
104.1481 – 6
98.1481
After 5th Year
According to YTM
1.08*98.1481
106
After coupon payment of Rs 6/-
106-6
100

The price of the Bond will always move towards the par value as it approaches maturity. The above calculation shows that even with a coupon of 6% the bond is actually yielding 8% as its price is adjusted to a discount at par.
NOTE: During the above calculation we have ignored the coupon that an investor would receive. We are deducting the coupon amount and then calculating the YTM over the remaining amount. In other words we assume that the coupons are reinvested at the same rate of YTM.
Here we summarize our understanding now:
  • If a bond is offering higher coupon than the YTM it will be available at a premium.
  • If a bond offers same coupon as the YTM it will be available at par. (Can be calculated by keeping coupon as 8% in the above examples.)
  • If it offers lower coupon than the YTM, the bond will be available at a discount to par.
The last topic in our discussion today is reinvestment risk, quite a simple one.

Reinvestment Risk: The investors face a reinvestment risk as he does not know what will be the interest rates at the time he receives a coupon. If the interest rates remain same as YTM throughout the life of a bond the investor would actually be able to effectively earn YTM.

In other cases if the interest rates at the time of coupon payments are less than the YTM, effective yield would decline, as the investor would not be able to reinvest the coupons at the same level of YTM.

Here we come to an end of our last post on bonds with a hope that the information was useful and easy to understand.

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