Question

Given the following table of spot and coupon rates for Treasury Securities. Calculate the theoretical spot rate for the 2 years Treasury

Period	Years	Yearly Spot Rate	Yearly Coupon Rate
1	0.5	4.50%	4.50%
2	1	4.75%	4.75%
3	1.5		5%
4	2		5.25%

Answer 1

We can use the above table to solve for the spot rates where S is spot rate and C is coupon rate

0.5 year spot rate, S1 = 4.5%

1 year spot rate,S2= 4.75%

C1 = 4.5% , C2 = 4.75%, C3= 5%, C4 = 5.25%

We can use the above data to calculate 1,5 year spot rate. Since, 1.5 year bond is selling at par its coupon will be 5%. The three cashflows are (assuming par value = 100):

Cashflow at 0.5 year = 100*0.05*0.5 = 2.5

Cashflow at 1 year = 100*0.05*0.5 = 2.5

Cashflow at 1.5 year = 100 + 100*0.05*0.5 = 102.5

we calculate the 1.5 year spot rate such that present value of all cashflows discounted by their respective spot rates is equal to bonds price i.e., 100

100 = 2.5/ (1 +4.5%/2)¹ + 2.5 / (1 + 4.75%/2)² + 102.5 / (1+S₃ /2)³

Solving for S₃ , we get S₃ = 4.94 = 5% (rounding off to nearest whole number)

Similarly using the same process, we calculate cashflows for 2 year spot rate and the S₄

Cashflow at 0.5 year = 100*0.0525*0.5 = 2.625

Cashflow at 1 year = 100*0.0525*0.5 = 2.625

Cashflow at 1.5 year = 100*0.0525*0.5 = 2.625

Cashflow at 2 year = 100 + 100*0.0525*0.5 = 102.625

we calculate the 2 year spot rate such that present value of all cashflows discounted by their respective spot rates is equal to bonds price i.e., 100

100 = 2.625/ (1 +4.5%/2)¹ + 2.625 / (1 + 4.75%/2)² + 2,625 / (1+4.9% /2)³ + 102.625/ (1 +5.25%/2)⁴ = 5.25%

Hence, the 2 year Treasury theoretical spot rate is 5.25%