Question

(2) Consider two 1-year Treasury bills of face value $100: (i) One has semi-annual coupons at annualized rate 1% and is priced at $98; (i) the other has semi-annual coupons at annualized rate 2% and is priced at $100 Determine the price of the 1-year zero-coupon Treasury bill of face value $100. What is the 1-year spot rate?

Thanks for any help.

Partial solution also welcome

Answer 1

Let the 6-month rate be r1 per annum and 1-year rate be r2 per annum.

Then, price of the first T-bill is:

Semi-annual coupon = coupon rate*face value/2 = 1%*100/2 = 0.5

0.5/(1+r1)^(1/2) + (100+0.5)/(1+r2) = 98 ----- (a)

Price of second T-bill is:

Semi-annual coupon = coupon rate*face value/2 = 2%*100/2 = 1

1/(1+r1)^(1/2) + (100+1)/(1+r2) = 100 ----- (b)

2*(a) - (b) =

1/(1+r1)^(1/2) + 201/(1+r2) - 1/(1+r1)^(1/2) - 101/(1+r2) = 196 - 100

201/(1+r2) - 101/(1+r2) = 96

solving for r2, r2 = 4.17% (This is the 1-year zero spot rate). (Note: Solve this using excel solver otherwise it will be a quadratic equation to solve by hand.)

Price of a 1-year zero coupon T-bill = face value/(1+1-year spot rate) = 100/(1+4.17%) = 96