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%
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 ar...
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