Question

You observe that the current three-year discount factor for default-risk free cash flows is 0.68. Remember, the t-year discount factor is the present value of $1 paid at time t, i.e. ???? = (1 + ????)−??, where ???? is the t-year spot interest rate (annual compounding). Assume all bonds have a face value of $100 and that all securities are default-risk free. All cash flows occur at the end of the year to which they relate. a) What is the price of a zero-coupon bond maturing in exactly 3 years? b) Your friend makes the following observation about the above bond: “Since there is no risk of default and there are no coupons to re-invest, buying the 3-year zero coupon bond today is a risk-free investment; that is, you are guaranteed to earn an annual return of 13.72% (i.e. 3-year spot rate)”. Explain why your friend is not entirely correct and how you would modify the statement to make it correct. c) In addition to the bond in (a), you observe the following: a 2-year coupon bond paying 10% annual coupons with a market price of $97, and two annuities that are trading at the same market price as each other. The first annuity matures in 3 years and pays annual cash flows of $20, while the second annuity pays annual cash flows of $28 and matures in 2 years. Using this information: i. Complete the term structure of interest rates, i.e. determine the one- and two-year discount factors, d1 and d2, respectively.

Answer 1

a)A zero coupon bond is somthing where face value is paid at the end of maturity period.

Face Value(FV) =$100

Duration(n) = 3 years

Discounting factor for 3 years(r)= 0.68

Current Price= FV/(1+r)^n = (100*0.68)

= $68

  

b) The Yield to Maturity(YTM) of the zero coupon bond is also known as the spot rate.So, the statement related to realizing the 13.72% return which is 3 years spot rate is correct.

Yield to Maturity(YTM) = (FV/Price)^(1/n)-1

=(100/68)^(1/3)-1

=1.1375-1=0.1372

=13.72%

However, the statement is incorrect in terms of the risk. Zerocoupon bonds are not risk free investment and they are subjected to multiple risks such as default risk and interest rate risk.A Zero coupon bond held till maturity is subject to default risk as there is always a chance that the issuer of the bonds fails to repay the principal.

Also if Zero coupon bond is sold prior to maturity it is subject to interest rate risk.

c)

Annual coupon payment= 110%*100=$10

d1= 1 yr discount rate

d2=2yr discount rate

Market price(2 yr bond)= $97

Price of 2 year(Annual coupon bond)=(10*d1)+(110*d2)

So, 97=10*d1+110*d2........Equation 1

Given the price of Annuity 1 that would mature in 3 years $20 Annual cash flow= (20*d1 +20*d2 + 20*d3)

price of Annuity 2 that would mature in 2 years $28 Annual cash flow= (28*d1 +28*d2)

And since the price of both annuities are equal so,

(20*d1 +20*d2 + 20*d3)=(28*d1 +28*d2)

Since, d3= 0.68

So,20*0.68=(28*d1- 20*d1) +(28*d2- 20*d2)

13.60=8*d1+8*d2

13.60=8(d1+d2)

13.60/8=(d1+d2)

1.70=d1+d2...........equation 2

By solving equation 1 and 2, we get

97=10*(1.7-d2)+110*d2

97= 17-1.7*d2+ 110*d2

80=108.3*d2

So, d2=(80/108.3)=0.738688

d2= 0.74

So, 2 year discount factor =0.74

On solving equation 2 we get d1=(1.7- 0.74)=0.96

cii)Price of the annuity 2=(28*d1+28*d2)

=(28*0.96+28*0.74)

=26.88+20.72= $47.6

Hence price of annuity 1= $47.60 =Priceof annuity 2