Question

Question 2

A T-bill quote sheet has 98-day T-bill (Face value: $10,000) quotes with a 3.86% bid and a 3.83% ask rates.

1. (a) Calculate purchase price of the bill.

2. (b) Calculate the corresponding bond equivalent yield.

3. (c) Prices of zero-coupon bonds reveal the following pattern of interest rates:

Years from now	1-year interest rate
0	5.2%
1	7.3%
2	8.6%

Calculate:
i) 2-year interest rate on today and 3-year interest rate on today.

ii) 1-year forward rate 3 years from now if 4-year interest rate on today is 7.8%.

Answer 1

a) Calculate the Purchase Price of T-Bill

Purchase Price of T-bill = Face Value * ( 1- Ask Rate/Discount Rate )

Answer =$10,000(1-0.01042) = $ 9895.80

Working Notes

As per the question , we have treasury bill having 98 days maturity , so first of all we need to convert ask rate percentage with 98 days maturity

For 360 Days - 3.83 %

For 98 days = 0.0383/360*98= 0.01042

b) Calculation of  Bond Equivalent Yield

Bond Equivalent Yield =[( Face Value- Purchase Price)/Purchase Price\Time)]

where time we need to convert for based on 98 days T bill maturity   

= [($10,000-$9895.8)/9895.8]*360/98

Answer :  3.86 % ( Rounding up to two decimals )

i) 2-year interest rate on today and 3-year interest rate on today.

Based on expectation theory , it reveals the future long term intrest rate , based on that we need to calculate short term interest rate for zero coupon bonds

1st Year	5.20%
2nd year	7.30%
3rd Year	8.60%

Calculation of 2 Years Intrest Rate on Today

(1+R)²= (1+r1) ( 1+r2) where R is forecasting intrest long term rate and R1 is the rate 1 year from now and R2 needs to be calculated

(1+0.073)²=(1+0.052) ( 1+r2) = 1.151329/1.052-1=R2 ( 2nd Year Intrest Rate on Today )

Answer : 9.44 % ( 2 Years Intrest Rate on Today )

Calculation of 3 Years Intrest Rate on Today

(1+R)³= (1+R1)(1+R2)(1+R3) where R is forecating long term rate for 3 Year and R1 is the rate for Year 1 from now and R2 is the 2year intrest rate ad R3 needs to be calculated based on expectation theory

(1+0.086)³= (1+0.052)(1+0.073)(1+R3)

1.2808 = 1.1288 ( 1+R3)

Answer : 13.46 % ( 3 Years Intrest Rate on Today )

ii) 1-year forward rate 3 years from now if 4-year interest rate on today is 7.8%.

Calculation of 1 year forward rate 3 Year from now

Given in the Question

when 4-Year intrest rate on today is 7.8 % where R3 is the forward rate 3 years from now

(1+0.078)(1.078)(1.078)(1.078)(= (1+ 0.052) (1+0.073)(1+0.086) (1+R4)

Forward Rate for the last year = 1.3504= 1.2259 (1 + R4)

Answer : 10.15 %