Question

A bondholder purchased an 8 percent coupon, $1,000 par three-year bond at a 7 percent yield. Interest rates then immediately fell to 6 percent and his bond was called at a price of $1,040. He reinvested his money and earned 6 percent on the $1,040 for three years. a) Did the call help or hurt the bondholder? (20 marks) b) What was his three-year rate of return on his original investment? (5 marks)

Answer 1

Price of the bond at the time of purchase is given as =

CF1/(1+ Yield )^1+CF2/(1+ Yield )^2+CF3/(1+ Yield )^3

CF1= 8% of 1000= 80

CF2= 8% of 1000= 80

CF3= 8% of 1000+1000= 1080

=80/(1+7%)^1+80/(1+7%)^2+1080/(1+7%)^3

=74.76636+69.8751+881.6017

=1026.24316

Hence the purchase price of the bond = 1026.24316

Now the Yield falls to 6%

Hence the price should be=

=80/(1+6%)^1+80/(1+6%)^2+1080/(1+6%)^3

=75.4717+71.19972+906.7888

=1053.4602

However the bond was called at 1040

Hence the call option had limited the gain due to fall in yield and the investor had suffered a notional loss of 13.4602

Hence the call option has hurt the investor

b) Gain on Call of bond = 1040- 1026.24316=13.75684

hence net cash outflow today = Inital investment in callable bond - Gain on call immediately

=1026.24316-12.75684=1012.48632

Now the proceeds from the callbale bond i.e 1040 has bee invested at 6%

Hence CF1= 6% of 1040=62.4

CF2= 6% of 1040=62.4

CF3= 6% of 1040+1040=1102.4

Now let rate of return for 3 years be x

hence,

1012.48632=62.4/(1+x%)^1+62.4/(1+x%)^2+1102.4/(1+x%)^3

Using trial and error method in RHS

let x= 6.5%

Hence RHS=62.4/(1+6.5%)^1+62.4/(1+6.5%)^2+1102.4/(1+6.5%)^3

=58.59155+55.01554+912.6208

=1026.228

let x= 7.5%

Hence RHS=62.4/(1+7.5%)^1+62.4/(1+7.5%)^2+1102.4/(1+7.5%)^3

=58.04651+53.99676+887.3885

=999.4318

Hence using the interpolation method the rate of return is calclulated as =

Lower interest rate+(Value at lower interest rate- Required value)/(Value at lower interest rate-Value at Higher interest rate)*(Upper interest rate - lower interest rate)

=6.5%+(1026.228-1012.48632)/(1026.228-999.4318)*(7.5%-6.5%)=6.5%+(13.74168/26.7962)*1%

=7.0128%

hence the rate of return for 3 years is 7.0128%(YTM)

(Note the rate of return in excel is calculated as = 7.008%

The difference between the excel and normal computation is due to rounding off)