Question

Consider a 2-year $1000 par value bond that pays semi-annual coupons at a rate of 42) purchased for $1058.82 6%. Suppose that the bond was (a) Use the method of averages to approximate the effective yield rate compounded semi-annually. State the result as a percent to 1 decimal place % compounded semi-annually (b) Complete the chart below by performing 2 iterations of the bisection method to approximate the effective yield rate compounded semi-annually [Note: For the initial interval [a(0),b(0) use the percentage result y2) in part (a) as follows: a(0)-y)-0.5 and b(0))0.5] a(n)(as a %) b(n)(as a %) c(n)(as a %) P(c(n))-1058.82 (6 decimals) 2 c) Use the last line in the table above to find the bracket a compounded semi-annually. State the final result as a percent to 4 decimal places (2) c % compounded semi-annually

Answer 1

The cash flows from the bond (denoted as CFt for cash flow at time t in years from today) are:

CF0= -1058.82

CF0.5 = 6% of 1000 = 60

CF1 = 60

CF1.5 = 60

CF2 = 60 + 1000 = 1060

Discounting the above cash flows at rate = r% (compounded semi annually) is represented as

(CF0 / (1+r)^0) + (CF0.5 / (1+r)^0.5) +(CF1 / (1+r)^1) +(CF1.5 / (1+r)^1.5) +(CF2 / (1+r)^2)

= (-1058.82 / (1+r)^0) + (60 / (1+r)^0.5) +(60 / (1+r)^1) +(60 / (1+r)^1.5) +(1060 / (1+r)^2)

a) In the above equation assume r = 5% then the present value is :

-1058.52/1 + 60/1.02 + 60/1.05 + 60/1.08 + 1060/1.10 = 74.1

Now assume r=10%, the present value is:

1058.52/1 + 60/1.05 + 60/1.10 + 60/1.15 + 1060/1.21 = -19.0

Now the yield is that percentage at which the present value is zero.

As the present value is positive at r=5% and negative at r=10%, the present value is zero for 5%<r<10%

Using the method of averages: consider r=(5%+10%)/2 = 7.5%

Then present value is 25.9

Since this is positive, we have 7.5%<r<10%

Again taking average of 7.5% & 10%, we have r=8.8% (one place decimal)

At r=8.8%, the present value is 2.2

Therefore we have 8.8%<r<10%

Again taking average of 8.8% & 10%, we have r=9.4% (one place decimal)

At r=9.4%, the present value is -8.5

Therefore we have 8.8%<r<9.4%

Again taking average of 8.8% & 9.4%, we have r=9.1% (one place decimal)

At r=9.1%, the present value is -3.2

Therefore we have 8.8%<r<9.1%

Again taking average of 8.8% & 9.1%, we have r=9.0% (one place decimal)

At r=9.0%, the present value is -1.4

Therefore we have 8.8%<r<9.0%

Again taking average of 8.8% & 9.0%, we have r=8.9% (one place decimal)

At r=8.9%, the present value is 0.4

Therefore we have 8.9%<r<9.0%

Therefore r at one place decimal is 8.9%

b.

The table using bisection method is :

n	a(n) (as a %)	b(n) (as a %)	c(n) (as a %)	P(c(n)) - 1058.82 (6 decimals)
0	8.40%	9.40%	8.90%	0.389476
1	8.90%	9.40%	9.15%	-4.073657
2	8.90%	9.15%	9.03%	-1.845838

c. for n=3, the table is :

n	a(n) (as a %)	b(n) (as a %)	c(n) (as a %)	P(c(n)) - 1058.82 (6 decimals)
0	8.40%	9.40%	8.90%	0.389476
1	8.90%	9.40%	9.15%	-4.073657
2	8.90%	9.15%	9.03%	-1.845838
3	8.90%	9.03%	8.9625%	-0.729120

Therefore, c(3) = 8.9625%