Question

How do you calculate the price of a coupon bond from the prices of zero-coupon bonds? How would you calculate the price from the yields of zero-coupon bonds? Why could two coupon bonds with the same maturity each have a different yield to maturity?

Answer 1

With zero-coupon bonds, there isn’t a coupon payment until maturity. As a result, the present value of annuity formula is unnecessary. Instead, you calculate the present value of the par value at maturity. Here’s an example, assuming a zero-coupon bond that matures in five years, with a face value of $1,000 and a required yield of 6%.

Determine the number of periods. The required yield of zero-coupon bonds is usually based on a semi-annual coupon payment. We have to adjust the 6% required yield to its equivalent semi-annual coupon rate. As such, the number of periods will be doubled to 10 periods (5 X 2).
Determine the yield. We have to divide the 6% yield by two because the number of periods has doubled. The yield for this bond is 3% (.06 / 2).
Enter the amounts into the formula:

Zero Coupon bond price = M/(1+I)^n

=$1000/(1.03)^10=$744.09

It’s important to note that zero-coupon bonds are always priced at a discount. If they were sold at par, you would have no way to make money from them, and no incentive to buy them.

2)

The formula for calculating the yield to maturity on a zero-coupon bond is:

Yield to Maturity = (Face Value / Current Price of Bond) ^ (1 / Years to Maturity) - 1

Consider a $1,000 zero-coupon bond with two years until maturity. The bond is currently valued at $925. To get its yield to maturity, divide its $1,000 face value by its current value of $925, to the 1 over years until maturity. Then subtract by 1. The equation produces a value of .03975, which is rounded and listed as a yield of 3.98%.

If economic uncertainty makes an investor more willing to hold a bond, the bond’s price will probably rise, which will increase the denominator in the formula. As a result, the yield will decrease.

3)

Two coupon bonds with the same year of maturity can have different yield of maturity that is depending upon their coupon rates. If coupon rate increases cash flow per coupon payment increases, this increases earlier cash flow than later in the calculation of present value. And resulting yield to maturity decreases with increase in coupon rate.

If coupon rate decreases, per coupon cash flow also reduced. This will increase later cash flows in the calculation of present value. And resulting yield to maturity increases.

Let us understand with the help of an example:

Suppose, we have a coupon bond with face value $ 1000 and coupon payment at the rate of 10% per annum with three years of maturity period.

Face value of Coupon payments=$1000

Coupon rate =10%

Years to Maturity =3 years

We calculate coupon payments (CPN) for the all three years

CPN= Coupon rate *Face value / No.of coupon payments per year

Coupon payment for year1, year 2 and year3

We have,

CPN=10%*$1000/1

= $100

Now we have coupon payment for year one is $100, for year two is $100 and for year three is sum of face value and coupon rate $1100.

Thanks