Question

Exercise 2. The 6-month, 12-month. I 8-month, and 24-month zero rates are 4%, 4.5%, 4.75% and 5%, with continuous compounding (a) What are the rates with semi-annual compounding? (c) Forward rates are rates of interest implied by current zero rates for periods of time in the future. Calculate the forward rate for year 2, i.e. the rate for the period of time between the end of 12-month and the end of 24-month. (d) Consider a 2-year bond providing semiannual coupon of 10% per annum. Estimate the price of this bond using the given zero rates. (e) Estimate the (continuous compounded) yield of this bond using the result in (c).

Answer 1

We convert the continuous compounding in to semiannualby using the following equation.

R= ( e^(r * .50) - 1) * 2

Where

R = p.a. rate compounding semi annually =??

r = continueous zero rates

Now lets convert the rates

Time effective rates semiannual rates

6 month 4% (e^(.04 *.50) -1) * 2 = 4.04%

12 months 4.5%   (e^(.045 * .50) -1) * 2 = 4.55%

18 months 4.75% (e^(.0475 * .50) -1) * 2= 4.807%

24 months 5% ( e^(.05 * .50) -1) * 2 = 5.06%

Part b

Since zero rates are continuous the forward rate should also be calculated in lines with zero rates i.e. in continuous compounding.

Continuous forward rate is given by

F(1,2) =* r(0,2) * 2 - r(0,1)* 1)/(2-1)

Where

F(1,2) = continuous forward rate from year 1 to 2 i.e. forward rate for year 2

r(0,1) =continuous zero rate for year 1

r(0,2)= continuous zero rate for year 1 and 2

F(1,2) = (5 *2 - 4.5 * 1)/1 = 5.5%

Part c

Let us calculate the cashflows of the bond (assuming the face value is $100)

Time cashflows

6m interest = 100 * 10% * .5 = $5

12m 5

18m 5

24m 5 + 100 =105 interest and face value

Price = present value of all these cashflows @ zero rates as applicable

P0 = 5e^-(.04 * .5) +5e^-(.045 * 1) +5e^-(.0475 * 1.5) +105e^(.05 * 2) = $109.32

Part d

First we need to calculate the effective yield which is given by

Y/2= ( I + ( F -P0)/2n)/((F+P0)/2)

Where

Y = effective yield

I = coupon amout =$5

F= Face value =$100

P0 =Price =$128.2

Y/2 = (10 +(100 -109.32)/2 * 2)/((109.32 +100)/2

Y =5.10%

Now let us convert it into continuous by using

r=ln(1+Y)

r = ln(1.051) =4.97%