Question

Mordecai bought a 3-year 15% Treasury bond on 8 May 2020 at a yield of j2 = 18.6% p.a. Coupons can be reinvested at j2 = 14.0% p.a. The bond will be redeemed at par on the maturity date (face value $100).

a. Calculate the total accumulated value at maturity generated by this bond if Mordecai holds it to maturity and reinvests all coupon payments received at the available rate.

b. Calculate the total realised compound yield (TRCY) of this bond.

c. Decompose the total accumulated value generated by this bond into: original purchase price, coupons, interest on coupons, and capital gain/loss.

d. If Mordecai holds the bond for 2 years and sells it for a yield of j2 = 18.8% p.a., calculate the holding period yield (HPY).

e. Calculate duration of this bond if it is held to maturity.

f.Use the concept of modified duration to estimate the price of the bond if the yield to maturity increases to j2 = 18.7% p.a. im- mediately after Mordecai buys the bond.

g. ]What fixed liability could Mordecai be reasonably confident of paying off in 2 1/2 years’ time? Why? I don't really know how to do part E,F and G, other parts are done

Answer 1

Purchase price of the bond can be calculated using the formula

PV=(Coupon pmt.*(1-(1+Yield)^-n)/Yield)+(FV/(1+Yield)^n)

where, coupon pmt.=100*15%/2=$ 7.5 per semi-annual period

Yield= semi-annual yield = 18.6%/2=9.3% per s/a period

n= no.of periods to maturity, ie. 3*2=6 s/a periods

& FV= Face value=$ 100

so, PV/Price=(7.5*(1-1.093^-6)/0.093)+(100/1.093^6)=

92.00

S/a period	Coupons recd.	Interest income on re-inv. Of coupons	Face value	Total	FV F at 9.3% per period	FV at end yr. 3
0
1	7.5			7.5	1.093^5=	1.559915	11.69936
2	7.5	0.525		8.025	1.093^4=	1.427186	11.45317
3	7.5	0.525		8.025	1.093^3=	1.305751	10.47865
4	7.5	0.525		8.025	1.093^2=	1.194649	9.587058
5	7.5	0.525		8.025	1.093^1=	1.093	8.771325
6	7.5	0.525	100	108.025	1.093^0=	1	108.025
	45	2.625					160.0146

b. Realised compound yield= (Total accumulation/Purchase price)^(1/holding period)-1

ie. (160.01/92)^(1/3)-1=

20.26%

c..Total accumulated value	160.01
Coupons	45
Int. on coupons	2.625
Total accumulated value	207.64
Original purchase price	92.00
Capital gain	115.64

d.To find selling price at end of 2 yrs. At the given yield of 18.8% p.a.
92=(7.5*(1-1.094^-4)/0.094)+(SP/1.094^4)=
97.28
so, $ holding period amts.=
Coupons = $ 7.5*4)	30
Inv. Interests(0.525*3)	1.575
Sale value	97.28
Total return	128.86
Purchase price	92.00
Holding period gain=	36.86
Holding period Yield= 36.86/92=	40.07%

Duration of the bond, if held till maturity

S/a period	CF s	PV F at 9.3%	PV of CFs	S/a period * CFs	PV of(S/a period*CFs)
1	2	3=1/1.093^ yr.n	4=2*3	5=1*2	6=5*3
1	7.5	0.91491	6.8618	7.5	6.8618
2	7.5	0.83707	6.2780	15	12.5560
3	7.5	0.76584	5.7438	22.5	17.2315
4	7.5	0.70068	5.2551	30	21.0204
5	7.5	0.64106	4.8080	37.5	24.0398
6	107.5	0.58651	63.0503	645	378.3021
			91.9971		460.0115

(Macaulay)Duration of the bond=PV of Time- weighted CFs/Current Market value

ie. 460.0115/91.9971=

5.000

f. Using the concept of modified duration to estimate the price of the bond if the yield to maturity increases to 18.7%, that is by (18.7%-18.6%=) 0.1% or 0.001

Modified duration=Macaulay duration/(1+0.093)=

ie. 5.00/1.093=

4.5746

Approximate change in price= 4.5746*0.001=

0.0045746

So, when the yield increases to 18.7%, the estimated price will be

92*(1-0.004575)=

91.58

g. Fixed liability Mordecai could be reasonably confident of paying off in 2 1/2 years’ time=

By looking at the duration & modified duration, we can see that weighted average time when all the cash flows of the bond are received is 5 semi-annual periods, ie 2.5 years.

So, Mordecai could reasonably be confident of paying off   $ 160 of fixed liability (as in a. above) ---in this 2 1/2 years’ time----as , by that time all monies from the bond would have been received---along with his investment income.