A five-year 2.4% defaultable coupon bond is selling to yield 3% (Annual Percent Rate and semi-annual compounding). The bond pays interest semi-annually. The risk-free yield is 2.4%. Therefore, its current credit spread is 3% - 2.4% = 0.6%. Two years later its credit spread increases from 0.6% to 1% while the risk-free yield doesn't change. Assuming the face value of the coupon bond and risk-free bond is 100.

 

a) What is the return of investing in this bond over the two year?

rate of return on bonds is 2.44% per annum

Calculation of purchase price:

face value =100

coupon amount semiannual =100*2.4%*1/2 = 1.2

YTM at time of purchase = 3%

semiannual YTM = 3%/2 =1.5%

number of years to maturity =5

semiannual periods (n) =5*2 =10

Bond price formula = Coupon amount * (1 - (1/(1+i)^n)/i + face value/(1+i)^n

=(1.2*(1-(1/(1+1.5%)^10))/1.5%) + (100/(1+1.5%)^10)

=97.23334463

Calculation of selling price:

after 2 years, Yield spread is 1%

credit yield spread = risky bond Yield - Risk free rate

1% = Risky bond yield -2.4%

Risky bond yield =1+2.4% = 3.4%

So YTM = 3.4%

semiannual YTM (i) = 3.4%/2 = 1.7%

semiannual coupon amount = 1.2

number of years to maturity = 5-2 =3

semiannual periods (n) =3*2 = 6

Bond price formula = Coupon amount * (1 - (1/(1+i)^n)/i + face value/(1+i)^n

=(1.2*(1-(1/(1+1.7%)^6))/1.7%) + (100/(1+1.7%)^6)

=97.17070731

Calculation of rate of return or Realized YTM

Sale price = 97.17070731

coupon received =1.2

number of coupon receipt in 2 years (n) =2*2 =4

Purchase price = 97.23334463

Rate of return is that return (i) which equates present value of sale price and coupon received equal to Purchase price

Purchase price of bond = Coupon amount * (1 - (1/(1+i)^n)/i + sale value/(1+i)^n

97.23334463=(1.2*(1-(1/(1+i)^4))/i) + (100/(1+i)^4)

We will calculate i by trial and error method.

Assume i is 1.2%

PV =(1.2*(1-(1/(1+1.2%)^4))/1.2%) + ( 97.17070731/(1+1.2%)^4)

=97.30253494

Assume i is 1.25%

PV =(1.2*(1-(1/(1+1.25%)^4))/1.25%) + ( 97.17070731/(1+1.25%)^4)

=97.11395642

interpolation formula = lower rate +((uper rate - lower rate)*(Uper price - bond actual price)/(uper price - lower price))

1.2% +((1.25%-1.2%)*(97.30253494- 97.23334463)/(97.30253494-97.11395642))

=0.01218345226

This is semiannual return

annual return =0.01218345226*2 =0.02436690452

or 2.44%

So rate of return on bonds is 2.44% per annum