Question

The price of a one-year zero-coupon bond is $943.396, the price of a two-year zero is $873.439, and the price of a three-year zero-coupon bond is $793.832. The bonds (each) have a face value of $1,000. Assume annual compounding.

a) Come the yield to maturity (YTM) on the one-year zero, the two-year zero, and the three-year zero.

b) Compute the implied forward rates for year 2 and for year 3.

c) Assume that the expectations hypothesis is correct. Based on your answers to parts a) and b), can you conclude that interest rates are expected to rise? Explain.

d) If the expectations hypothesis is correct, what will the pure yield curve be next year? [In other words, compute the yields on both a one-year zero and a two-year zero in one year from today.] Use the data from your answers to parts a) and b).

Answer 1

(a) Computation of Rate of Return of Zero Coupon Bond:

One Year Zero Coupon Bond:

Time to Maturity = 1 Years

Face Value = $ 1,000

Issue Price = $943.396

Issue Price = Present Value of Future Expected Payments

Issue Price = Present Value of Redemption Value(in case of a Zero Coupon Bond, as no Coupon Payments)

$943.396 = $1,000 * PV(r1%, 1 Year)

(1+r1)^1 = $1,000 / $943.396

1+r1 = 1.06

YTM (r1) = 0.06 (or) 6%

Two Year Zero Coupon Bond:

Time to Maturity = 2 Years

Face Value = $ 1,000

Issue Price = $873.439

Issue Price = Present Value of Future Expected Payments

Issue Price = Present Value of Redemption Value(in case of a Zero Coupon Bond, as no Coupon Payments)

$873.439 = $1,000 * PV(r2%, 2 Years)

(1+r2)^2 = $1,000 / $873.439

1+r2 = 1.1449^(1/2)

YTM (r2) = 0.07 (or) 7%

Three Year Zero Coupon Bond:

Time to Maturity = 3 Years

Face Value = $ 1,000

Issue Price = $793.832

Issue Price = Present Value of Future Expected Payments

Issue Price = Present Value of Redemption Value(in case of a Zero Coupon Bond, as no Coupon Payments)

$793.832 = $1,000 * PV(r3%, 3 Years)

(1+r3)^3 = $1,000 / $793.832

1+r3 = 1.2597^(1/3)

YTM (r3) = 0.08 (or) 8%

Maturity (n)	Spot Rate (S0n)
1 year	6.00%
2 year	7.00%
3 year	8.00%

(b) 1f1 represents Forward rate after 1 year for 1 year, i.e., for Year 2

Forward Rates between years 1 and 2

(1 + S02)^2 = (1+S01)(1+1f1)

(1+0.07)^2 = (1 + 0.06)(1 + 1f1)

(1.07)^2/(1.06) = (1 + 1f1)

(1 + 1f1) = 1.1449 / 1.06

1 + 1f1 = 1.0801

1f1 = 1.0801 - 1

Implied forward rate for Year 2 (1f1) = 8.01%

2f1 represents Forward rate after 2nd year for 1 year, i.e., for Year 3

Forward Rates between years 2 and 3

(1 + S03)^3 = (1+S02)^2 * (1+2f1)

(1+0.08)^3 = (1 + 0.07)^2 * (1 + 2f1)

(1.08)^3/(1.07)^2 = (1 + 2f1)

(1 + 2f1) = 1.2597 / 1.1449

1 + 2f1 = 1.1003

2f1 = 1.1003 - 1

Implied forward rate for Year 3 (2f1) = 10.03%

(c) Assume that the expectations hypothesis is correct, from the above we can conclude that Interest rates are expected to raise.

Implied forward rate for Year 2 (1f1) = 8.01%

Implied forward rate for Year 3 (2f1) = 10.03%

(d)

Forward Rates between years 1 and 3 (1f2)

(1 + S03)3 = (1+S01)1 (1+1f2)2

(1+0.08)3 = (1 + 0.06)(1 + 1f2)2

(1.08)^3/(1.06) = (1 + 1f2)2

(1 + 1f2)2 = 1.259712 / 1.06

1 + 1f2 = 1.0901

1f2 = 1.0901 - 1

1f2 = 9.01% (approx)

Implied Yield for One Year Zero Coupon Bond after one Year from today (1f1) = 8.01%

Implied Yield for Two Year Zero Coupon Bond after one Year from today (1f2) = 9.01%