Question

The following is a list of prices for zero-coupon bonds of various maturities.

a. Calculate the yield to maturity for a bond with a maturity of (i) one year; (ii) two years; (iii) three years; (iv) four years. Assume annual coupon payments. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Maturity (years)	Price of Bond
1	$	978.43
2		924.97
3		840.12
4		784.39

b. Calculate the forward rate for (i) the second year; (ii) the third year; (iii) the fourth year. Assume annual coupon payments. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Maturity (years)	Price of Bond
1	$	978.43
2		924.97
3		840.12
4		784.39

Answer 1

Part A:

YTM :

YTM is the rate at which PV of Cash inflows are equal to Bond price when the bond is held till maturity. Yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures. Yield to maturity is considered a long-term bond yield but is expressed as an annual rate.

YTM of bond matures in 1 Year :

Particulars	Amount
Maturity price	$ 1,000.00
Current Price	$    978.43
Maturity period	1

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1000 / $ 978.43 ] ^ ( 1 / 1) - 1
= [ 1.022 ] ^ ( 1 / 1) - 1
= 1.022 - 1
= 0.022
I.e 2.2 %

YTM of bond matures in 2 Years:

Particulars	Amount
Maturity price	$ 1,000.00
Current Price	$    924.97
Maturity period	2

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1000 / $ 924.97 ] ^ ( 1 / 2) - 1
= [ 1.0811 ] ^ ( 1 / 2) - 1
= 1.0398 - 1
= 0.0398
I.e 3.98 %

YTM of bond matures in 3 Years:

Particulars	Amount
Maturity price	$ 1,000.00
Current Price	$    840.12
Maturity period	3

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1000 / $ 840.12 ] ^ ( 1 / 3) - 1
= [ 1.1903 ] ^ ( 1 / 3) - 1
= 1.0598 - 1
= 0.0598
I.e 5.98 %

YTM of bond matures in 4 Years:

Particulars	Amount
Maturity price	$ 1,000.00
Current Price	$    784.39
Maturity period	4

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1000 / $ 784.39 ] ^ ( 1 / 4) - 1
= [ 1.2749 ] ^ ( 1 / 4) - 1
= 1.0626 - 1
= 0.0626
I.e 6.26 %

Part B:

Second Year forward rate = [ [ (1 + YTM 2 ) ^ 2 / ( 1 + YTM 1 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1 + 0.0398 ) ^ 2 / ( 1 + 0.022 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1.0398 ) ^ 2 / ( 1.022 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.0812 / 1.022 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.0579 ] ^ ( 1 / 1 ) ] - 1
= [ 1.0579 ] - 1
= 0.0579
= I.e 5.79 %

Third year forward rate = [ [ (1 + YTM 3 ) ^ 3 / ( 1 + YTM 2 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1 + 0.0598 ) ^ 3 / ( 1 + 0.0398 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1.0598 ) ^ 3 / ( 1.0398 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.1903 / 1.0812 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.101 ] ^ ( 1 / 1 ) ] - 1
= [ 1.101 ] - 1
= 0.101
= I.e 10.1 %
Fourth year forward rate = [ [ (1 + YTM 4 ) ^ 4 / ( 1 + YTM 3 ) ^ 3 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1 + 0.0626 ) ^ 4 / ( 1 + 0.0598 ) ^ 3 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1.0626 ) ^ 4 / ( 1.0598 ) ^ 3 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.2749 / 1.1903 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.071 ] ^ ( 1 / 1 ) ] - 1
= [ 1.071 ] - 1
= 0.071
= I.e 7.1 %