Question

Consider two bonds: 1) a zero-coupon bond having a face value F and maturity 1 year; 2) a coupon bond with face value F, coupons C = 15 paid annually and maturity 3 years. Assume that the continuously compounded interest rate is 10%.

a) If F = 100, find at the end of which year the price of the second bond will be for the first time below 110?

b) If the price of the second bond is equal to 1.20 times the price of the first bond, find the (common) face value F?

Please show work/explain steps

Answer 1

Formula sheet

A1	B	C	D	E	F	G	H	I	J	K	L
2		a)
3		Coupon rate	0.15
4		Par value (F)	100
5		YTM	0.1
6		Original Time to maturity	3	Year
7		Annual coupon (C)	=D4*D3
8
9
10		Current Value of the bond can be calculated by finding the present value of cash flows of bonds.
11		Cash Flow of Bonds can be written as follows:
12		Semiannual Period	0	1	2	3	…	=D8
13		Cash Flow of Bonds		=$D7	=$D7	=$D7	=$D7	=D7+D4
14
15		Current Value of Bond	=C*(P/A,i,n)+F*(P/F,i,n)
16		Where, C is annual coupon, F is par value of bond, i is annual market rate and n is total annual periods.
17
18		Current Value of Bond	=C*(P/A,i,n)+F*(P/F,i,n)
19			=15*(P/A,10%,3)+100*(P/F,10%,3)
20			=D7*PV(D5,D6,-1,0)+D4*(1/((1+D5)^D6))	=D7*PV(D5,D6,-1,0)+D4*(1/((1+D5)^D6))
21		Hence current price of bond is	=D20
22
23		Alternative method:
24		Price of the bond can also be calculated by finding the present value of cash flows of the bond using PV formula of excel as follows:
25		RATE	=D5
26		NPER	=D6
27		PMT	=D7
28		FV	=D4
29		TYPE	0	(End of the period Cash Flow)
30
31		Price of the Bond	=-PV(D25,D26,D27,D28,0)	=-PV(D25,D26,D27,D28,0)
32
33		Hence Current Price of Bond is	=D31
34
35		Simillarly price in other years in future can be calculated as follows:
36		Years	Original Time to maturity	Remaining Time to maturity	Coupon Rate	YTM	Price of Bond
37		1	=$D$6	=D37-C37	=$D$3	=$D$5	=-PV(G37,E37,$D$4*F37,$D$4)	=-PV(G37,E37,$D$4*F37,$D$4)
38		2	=$D$6	=D38-C38	=$D$3	=$D$5	=-PV(G38,E38,$D$4*F38,$D$4)
39
40		Since price of bond in year 1 is $108.68 thus price of bond decreases below $110 in year 1.
41
42		b)
43
44		As Calculated Above,
45		Price of Coupon Bond (P1)	=C*(P/A,i,n)+F*(P/F,i,n)
46			=15*(P/A,10%,3)+F*(P/F,10%,3)
47
48		Price of Zero coupon Bond(P2)	=F*(P/F,i,n)
49			=F*(P/F,10%,3)
50
51		(P/A,10%,3)	=PV(10%,3,-1,0)
52		(P/F,10%,3)	=1/((1+10%)^3)
53
54		Price of Coupon Bond (P1)	= 15*(P/A,10%,3)+F*(P/F,10%,3)
55			= 15*2.46+F*0.75
56			= 0.75F+37.30
57
58		Price of Zero Coupon Bond (P2)	=F*(P/F,10%,3)
59			=0.75F
60
61		Price of Coupon bond is 1.20 times the price of coupon bond i.e.
62		P1 = 1.20 P2
63		or
64		0.75F+37.30 = 1.20*(0.75F)
65		or
66		F=	=D7*D51/(1.2*D52-D52)	=D7*D51/(1.2*D52-D52)
67
68		Hence Face value	=D66
69