Question

A loan of $10,000 is to be repaid with 10 semi-annual payments. The first payment is X in 6 months time. i(2) = 4%. Find X if

a) Payments increase by $100 every 6 months.

b) Payments increase by 10% every 6 months.

Answer 1

a.. Equating the PV to the semi-annual cash flows

10000=(x/1.04^1)+((x+100)/1.04^2)+((x+200)/1.04^3)+((x+300)/1.04^4)+((x+400)/1.04^5)+((x+500)/1.04^6)+((x+600)/1.04^7)+((x+700)/1.04^8)+((x+800)/1.04^9)+((x+900)/1.04^10)

Solving the above in an online calculator,

we get X=

815.18

Amortisation table No. Semi-annual payments Tow. Int. at 4% semi-annual Tow.Principal Prin. Bal. 10000 1 815.18 400.00 415.18 9584.82 2 915.18 383.39 531.79 9053.03 3 1015.18 362.12 653.06 8399.96 4 1115.18 336.00 779.18 7620.78 5 1215.18 304.83 910.35 6710.43 6 1315.18 268.42 1046.77 5663.66 7 1415.18 226.55 1188.64 4475.03 8 1515.18 179.00 1336.18 3138.84 9 1615.18 125.55 1489.63 1649.22 10 1715.18 65.97 1649.21 0.00 12651.83 2651.83 10000.00

b...Using Pv of growing annuity formula,

PV(GA)=(X/(r-g))*(1-((1+g)/(1+r))^n)

10000=(X/(0.04-0.1))*(1-(1.1/1.04)^10)

797.618

ie. X= 798

	Amortisation table
No.	Semi-annual payments	Tow. Int.(4%)	Tow.Principal	Prin. Bal.
				10000
1	797.62	400.00	397.62	9602.38
2	877.38	384.10	493.28	9109.10
3	965.12	364.36	600.75	8508.34
4	1061.63	340.33	721.30	7787.05
5	1167.79	311.48	856.31	6930.74
6	1284.57	277.23	1007.34	5923.39
7	1413.03	236.94	1176.09	4747.30
8	1554.33	189.89	1364.44	3382.86
9	1709.77	135.31	1574.45	1808.41
10	1880.74	72.34	1808.41	0.01
	12711.98	2711.98	9999.99