Question

There are four zero-coupon Treasury bonds as follows: Maturity (years) Price ($) 0.5 979.43 1.0 955.54 1.5 928.60 2.0 897.17 Assume that the face values are $1000 for all the bonds. (a) Determine the quasi-modified duration for the given 1.0-year zero-coupon bond. (Keep 2 decimal places, e.g. xx.12) (b) The price for a 2-year Treasury note with semi-annual coupon payments is $ 987.42. Find the annual coupon rate for the note, and hence determine its quasi-modified duration. Coupon rate: % (Keep it in percentage format with 2 decimal places, e.g. xx.12%) Qusi-modified duration: (Keep 2 decimal places, e.g. xx.12)

Answer 1

If S_i is the spot rate for the period i then price of the zero coupon bond (ZCB) with maturity period i will be:

P_i = FV / (1 + S_i / 2)ⁱ

Hence, spot rate, S_i = [(FV / P_i)^1/i - 1] x 2 = [(1000 / P_i)^1/i - 1] x 2

Please see the table below:

Period	Year	Price of the ZCB	Spot rate
i		P	Si = [(1000 / Pi)1/i - 1] x 2
1	0.50	979.43	0.0420
2	1.00	955.54	0.0460
3	1.50	928.6	0.0500
4	2.00	897.17	0.0550

Part (a)

the quasi-modified duration for the given 1.0-year zero-coupon bond = 1 / (1 + S₁ / 2) = 1 / (1 + 0.0420 / 2) =  0.98

Part (b)

Let C be the coupon per period, i.e. the semi annual coupon.

Hence, Price = 987.42 = PV of all the future coupons + PV of redemption = C / (1 + 0.0420 / 2) + C / (1 + 0.0460 / 2)² + C / (1 + 0.0500 / 2)³ + (1,000 + C) / (1 + 0.0550 / 2)⁴ =  0.9794C + 0.9555C + 0.9286C + 0.8972 x (1,000 + C) = 3.7607C +  897.17

Hence, C = (987.42 - 897.17) / 3.7607 =  24.00

Hence, annual coupon rate = 2 x C / Face Value = 2 x 24 / 1000 = 4.80%

Yield of this bond = 2 x RATE (Period, PMT, PV, FV) = 2 x RATE (2 x 2, 24, -987.42, 1000) = 5.47%

Quasi modified duration can be calculated using the MDURATION formula of excel.