Question

The yield to maturity of a 10-year 4% annual coupon bond is 4%.

a.Suppose that you buy the bond today and hold it for 10 years.Assume that the interest rates go up to 5% (100 basis points increase) after the bond is purchased and before the first coupon is received. What is your realized rate of return?

b.Suppose that you buy the bond today and hold it for 6years.Assume that the interest rates go up to 5% (100 basis points increase) after the bond is purchased and before the first coupon is received. What is your realized rate of return?

Answer 1

a. Assume that the bond value is 100

Given coupon rate and yied to maturity rate is same so the bond is trading at par

if the interest rate increases then the couon will be reinvested at revised interest rates

Realised rate of return is the return that the bond holder gain for the time period he hold the bonds and later on by selling it

given coupon rate = 4% so coupon received will be on par value=100*4%=4

So calculationg the cashflows he receives if he holds for 10 years

Each coupon is assumed to have received at the end of the year so the time period that coupon would have invested is remaining years-1

Year	Coupon received	Reinvested @ 5%	Explanation
1	4	6.205	remaining years 9 =4*(1.05)^9
2	4	5.910	remaining years 8 =4*(1.05)^8
3	4	5.628	remaining years 7 =4*(1.05)^7
4	4	5.360	remaining years 6 =4*(1.05)^6
5	4	5.105	remaining years 5 =4*(1.05)^5
6	4	4.862	remaining years 4 =4*(1.05)^4
7	4	4.631	remaining years 3 =4*(1.05)^3
8	4	4.410	remaining years 2=4*(1.05)^2
9	4	4.200	remaining years 1 =4*(1.05)^1
10	4	4.000	remaining years 0 =4*(1.05)^0
	Total	50.31

So by the end of 10 years he received interest amount of 50.31

Since the bond is held for maturity so he gets the par value = 100

So total he receives in the 10 th year=100+50.31=150.3116

We know that future value=present value*(1+r)^n

Here future value= 150.3116 , present value= 100

r=? n=10 years

=150.3116=(100)*(1+r)^10

=1.503116=(1+r)^10

1+r=1.503116^(1/10)

by using excel or calcualtor we can find that r= 4.160%

so the realised yield is 4.160%

b.

here the bond is held for 6 years so

if the interest rate increases then the couon will be reinvested at revised interest rates

Realised rate of return is the return that the bond holder gain for the time period he hold the bonds and later on by selling it

given coupon rate = 4% so coupon received will be on par value=100*4%=4

So calculationg the cashflows he receives if he holds for 6 years

calculating the interest amount he receives after 6 years considering the reinvestment rate as 5%

Year	Coupon received	Reinvested @ 5%	Explanation
1	4	5.105	remaining years 5 =4*(1.05)^5
2	4	4.862	remaining years 4 =4*(1.05)^4
3	4	4.631	remaining years 3 =4*(1.05)^3
4	4	4.410	remaining years 2 =4*(1.05)^2
5	4	4.200	remaining years 1 =4*(1.05)^1
6	4	4.000	remaining years 0 =4*(1.05)^0
	Total amount received	27.208

So by the end of 10 years he received interest amount of 27.208

Since the bond is held for 6 years the price of the bond at the end of the 6 th year=

Year	Coupon (A)	Present value factor @ 5%(B)	Present value(A*B)
1	4	0.952381=(1/(1.05)^1	3.809524
2	4	0.907029=(1/(1.05)^2	3.628118
3	4	0.863838=(1/(1.05)^3	3.45535
4	104	0.822702=(1/(1.05)^4	85.56106
		Total	96.45405

so the bond will be sold at 96.454 so the capital loss will be sale value-purchase price =96.454-100=-3.54595 capital loss

so in the 6 th year end= interest + sale value of the bond=27.208+96.45405=123.6617

We know that future value=present value*(1+r)^n

Here future value= 123.6617 , present value= 100

r=? n=6 years

=123.6617=(100)*(1+r)^6

=1.236617=(1+r)^10

1+r=1.236617^(1/10)

by using excel or calcualtor we can find that r= 3.603%

so the realised yield is 3.603%

Because of the capital loss the realised yield is less than the coupon rate