Question

Mark​ Goldsmith's broker has shown him two bonds issued by different companies. Each has a maturity of 4 ​years, a par value of ​$1000​, and a yield to maturity of 7.20 %.   The first bond is issued by Crabbe Waste Disposal and has a coupon interest rate of 6.329​% paid annually. The second ​ bond, issued by Malfoy​ Enterprises, has a coupon interest rate of 8.80​% paid annually. a.  Calculate the selling price for each of the bonds. b.  Mark has ​$21000 to invest. If he wants to invest only in bonds issued by Crabbe Waste​ Disposal, how many of those bonds could he​ buy? What if he wants to invest only in bonds issued by Malfoy​ Enterprises? c.  What is the total interest income that Mark could earn each year if he invested only in Crabbe​ bonds? How much interest would he earn each year if he invested only in Malfoy​ bonds? d.  Assume that Mark will reinvest all the interest he receives as it is paid and that his rate of return on the reinvested interest will be 9​%. Calculate the total dollars that Mark will accumulate over 4 years if he invests in Crabbe bonds or Malfoy bonds. Your total calculation will include the interest Mark​ gets, the principal he receives when the bonds​ mature, and all the additional interest he earns from reinvesting the coupon payments he receives. e.  The bonds issued by Crabbe and Malfoy might appear to be equally good investments because they offer the same yield to maturity of 7.20​%. ​Notice, however, that your answers to part d are not the same for each​ bond, suggesting that one bond is a better investment than the other. Why is that the​ case?

Answer 1

Price of the bond (P0) = I*(1-(1+r)^(-p))/r + FV/(1+r)^p

Where

I = Coupon amout per year = coupon rate ×face value

r= Yield to maturity

P= period to maturity

FV = face value

For first bond

I = .06329 × 1000$ =63.29$

P=4 years

r = 7.2% or .072

FV= 1000

So price P01= 63.29×(1-(1.072)^-4)/.072 + 1000/1.072^4

P0= 213.41 + 757.22 = 970.63$

for second bond we have

I = .088 × 1000$ =88

r = .072 as the Yields are same for both the bonds

FV=1000$ AS the FV is same for both

P = 4 years same for both

Hence price for second bond P02 = 88(1- (1.072)^-4)/.072 + 1000/1.072^4

=296.73 + 757.22 = 1053.95$

Part b

If he wants to invest only in first bond then he can buy $21000/$970.63 = 21.63 i.e. 21 bonds in full ( as he can not buy bonds in part)

And if he wants to invest in second bond only then he would can buy $21000/$1053.95 = 19.92 i.e. 19 bonds in full

Part c

Total interest from earnings if he invests in first bond only = .06329 ×1000= 63.29 $

Total interest earnings of he invests in second bond only = .088 × 1000 = 88 $

Part d

Our cash flows per bond from first bond are

At Time(t) cash flows in $

1 63.29

2 63.29

3 63.29

4 63.29 +1000= 1063.29

And our reinvestment( i.e. earnings rate) rate =9%

If we invest C0 now at k% for(p- t) years then it accumulate to C0(1+k)^(p-t)

K is 9%

P-t is time for which they are reinvested which

So accumulation per bond from fist bond= 63.29(1.09)^3 +63.29(1.09)^2 +63.29(1.09)^1 + 1063.29(1.09)^0

=1385.13 $ per bond

Total accumulation = 1385.13 ×21( because he can buy 21 bonds at max.)= 29087.73 $

Now cashflows per bond from second bond are

Years (t) cashflows in $

1 88

2 88

3 88

4 88 +1000 =1088

Accumulation per bond = 88×1.09^3 +88×1.09^2 + 88 ×1.09^1+1088

=1402.44$ per bond

Hence our total accumulation = 1402.44× 19 =26646.36$

Part e

Though the yield from both the bonds are same but we can't take any decision just on the basis of yield because yield doesn't consider reinvestment opportunities niether we can take decisions on the basis of accumulation as it doesn't consider the price effect now thankfully there is another tool for this which is modified yield( M) which considers both reinvestment opportunities and price it is given by

Price per bond = accumulation per bond/ (1+M)^p

Hence for first bond M is given by

970.63 =1385.13/(1+M)^4

Solving for M we get M =9.30%

For second bond M is given by

1053.95 =1402.44/(1+M)^4

SO M = 7.40%

Since modified yield for first bond is higher we can go with that