Question

You find a zero coupon bond with a par value of $10,000 and 23 years to maturity. The yield to maturity on this bond is 4.5 percent. Assume semiannual compounding periods.

What is the price of the bond? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

Price of the bond = 3593.76

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Since the compounding is done semi-annually, let’s calculate effective interest first.

The effective annual interest rate is the interest rate that is actually earned or paid on an investment, loan or other financial product due to the result of compounding over a given time period. It is also called the effective interest rate, the effective rate or the annual equivalent rate

So if nominal interest rate (i), number of compounding in a year is (m), effective interest will be

Effective interest rate = (1 + i/m) ^m -1

Where,

Nominal interest rate (i) = 4.5% per year

Number of compounding in a year (m) = 2

Let's put all the values in the formula

Effective interest rate = (1 + 0.045/2) ^2 - 1

                                              = (1 + 0.0225) ^2 - 1

                                              = (1.0225) ^2 - 1

                                              = 1.04551 - 1

                                              = 0.0455

So annual effective interest rate is 4.55% per year

Price of the bond

If the amount to be received in the future is FV, rate of interest is r and time period is n then PV of that amount can be calculated using below formula

PV = FV/ (1+r)^n

Where,

FV = $10000

r = 4.55%

n = 23

Let's put all the values in the formula to find PV

PV = 10000/ (1 + 0.0455) ^23

       = 10000/ (1.0455) ^23

       = 10000/ 2.7826

       = 3593.76

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