Question

1. Suppose today a one year ZCB is priced at 0.97, a two year ZCB is priced at 0.95 and a three year ZCB is priced at 0.92. What is the implied forward rate for a one year loan starting one year from now? What is the implied forward rate for a one year loan starting two years from now?

2. Suppose the same info as in question 1. Suppose you wanted to lend $1 at the end of year 1 for two years, so that the loan matured at the end of year 3. Describe how you would buy and sell the ZCBs to accomplish this goal, and the amount of money that you would collect when the loan matured.

Please answer question 2

Answer 1

Part 1:

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 1 year Bond:

Particulars	Amount
Maturity price	$           1.00
Current Price	$           0.97
Maturity period	1

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1 / $ 0.97 ] ^ ( 1 / 1) - 1
= [ 1.0309 ] ^ ( 1 / 1) - 1
= 1.0309 - 1
= 0.0309
I.e 3.09 %

YTM of 2 year Bond:

Particulars	Amount
Maturity price	$           1.00
Current Price	$           0.95
Maturity period	2

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1 / $ 0.95 ] ^ ( 1 / 2) - 1
= [ 1.0526 ] ^ ( 1 / 2) - 1
= 1.026 - 1
= 0.026
I.e 2.6 %

YTM of 3 year Bond:

Particulars	Amount
Maturity price	$           1.00
Current Price	$           0.92
Maturity period	3

YTM = [ Maturity Value / Current Price ] ^ ( 1 / n ) - 1
= [ $ 1 / $ 0.92 ] ^ ( 1 / 3) - 1
= [ 1.087 ] ^ ( 1 / 3) - 1
= 1.0282 - 1
= 0.0282
I.e 2.82 %

One Year Rate after 1 Year:

1 Year after 1 year from Today Rate = [ [ (1 + YTM 2 ) ^ 2 / ( 1 + YTM 1 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1 + 0.026 ) ^ 2 / ( 1 + 0.0309 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1.026 ) ^ 2 / ( 1.0309 ) ^ 1 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.0527 / 1.0309 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.0211 ] ^ ( 1 / 1 ) ] - 1
= [ 1.0211 ] - 1
= 0.0211
= I.e 2.11 %

One Year Rate after 2 Years:
1 Year after 2 year from Today Rate = [ [ (1 + YTM 3 ) ^ 3 / ( 1 + YTM 2 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1 + 0.0282 ) ^ 3 / ( 1 + 0.026 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ ( 1.0282 ) ^ 3 / ( 1.026 ) ^ 2 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.087 / 1.0527 ] ^ ( 1 / 1 ) ] - 1
= [ [ 1.0326 ] ^ ( 1 / 1 ) ] - 1
= [ 1.0326 ] - 1
= 0.0326
= I.e 3.26 %

Part 2:

2 Year after 1 year from Today Rate = [ [ (1 + YTM 3 ) ^ 3 / ( 1 + YTM 1 ) ^ 1 ] ^ ( 1 / 2 ) ] - 1
= [ [ ( 1 + 0.0282 ) ^ 3 / ( 1 + 0.0309 ) ^ 1 ] ^ ( 1 / 2 ) ] - 1
= [ [ ( 1.0282 ) ^ 3 / ( 1.0309 ) ^ 1 ] ^ ( 1 / 2 ) ] - 1
= [ [ 1.087 / 1.0309 ] ^ ( 1 / 2 ) ] - 1
= [ [ 1.0544 ] ^ ( 1 / 2 ) ] - 1
= [ 1.0269 ] - 1
= 0.0269
= I.e 2.69 %
If amount $ 1 is given after 1 year for 2 Years:

Future Value:

Future Value is Value of current asset at future date grown at given int rate or growth rate.

FV = PV (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Present Value	$                     1.00
Int Rate	2.6900%
Periods	2

Future Value = Present Value * ( 1 + r )^n
= $ 1 ( 1 + 0.0269) ^ 2
= $ 1 ( 1.0269 ^ 2)
= $ 1 * 1.0545
= $ 1.05

Amount that can be collected after 2 years is $ 1.05