Question

An annuity immediate pays $500 per month for the first three years. After that the annuity payments increase by $50 per month for five years and then remain level for an additional six years. At a nominal rate of annual interest of 12% convertible monthly what is the present value of this annuity?

The answer in the back is: 140339.571

Answer 1

	.A. $500 per month for first three years:
	Present Value (PV) of payment=(Payment)/((1+i)^N)
	i=monthly interest=(12/12)%=1%=0.01
	N=Month of payment
	Payment and PV of payments are given below:
		N	A	B=A/(1.01^N)
		Month	Payment	PV of payment
		0	$500	500
		1	$500	495.049505
		2	$500	490.148025
		3	$500	485.295074
		4	$500	480.490172
		5	$500	475.732844
		6	$500	471.022618
		7	$500	466.359027
		8	$500	461.741611
		9	$500	457.169912
		10	$500	452.643477
		11	$500	448.161859
		12	$500	443.724613
		13	$500	439.3313
		14	$500	434.981485
		15	$500	430.674737
		16	$500	426.410631
		17	$500	422.188744
		18	$500	418.008657
		19	$500	413.869958
		20	$500	409.772235
		21	$500	405.715084
		22	$500	401.698103
		23	$500	397.720894
		24	$500	393.783064
		25	$500	389.884221
		26	$500	386.023982
		27	$500	382.201962
		28	$500	378.417784
		29	$500	374.671073
		30	$500	370.961459
		31	$500	367.288573
		32	$500	363.652053
		33	$500	360.051537
		34	$500	356.486671
		35	$500	352.9571
			TOTAL	$ 15,204.29
	.B. Increasing by $50 per month for next five years
		N	A	B=A/(1.01^N)
		Month	Payment	PV of payment
		36	$550	384.408722
		37	$600	415.20294
		38	$650	445.349689
		39	$700	474.858769
		40	$750	503.739854
		41	$800	532.002486
		42	$850	559.656081
		43	$900	586.709927
		44	$950	613.173192
		45	$1,000	639.054916
		46	$1,050	664.364021
		47	$1,100	689.10931
		48	$1,150	713.299466
		49	$1,200	736.943056
		50	$1,250	760.048531
		51	$1,300	782.62423
		52	$1,350	804.678378
		53	$1,400	826.21909
		54	$1,450	847.254371
		55	$1,500	867.792118
		56	$1,550	887.84012
		57	$1,600	907.406064
		58	$1,650	926.497528
		59	$1,700	945.121991
		60	$1,750	963.286828
		61	$1,800	980.999316
		62	$1,850	998.26663
		63	$1,900	1015.09585
		64	$1,950	1031.49396
		65	$2,000	1047.46784
		66	$2,050	1063.0243
		67	$2,100	1078.17002
		68	$2,150	1092.91162
		69	$2,200	1107.25561
		70	$2,250	1121.20843
		71	$2,300	1134.77641
		72	$2,350	1147.9658
		73	$2,400	1160.78278
		74	$2,450	1173.23342
		75	$2,500	1185.32372
		76	$2,550	1197.0596
		77	$2,600	1208.44689
		78	$2,650	1219.49134
		79	$2,700	1230.19862
		80	$2,750	1240.57433
		81	$2,800	1250.62399
		82	$2,850	1260.35303
		83	$2,900	1269.76682
		84	$2,950	1278.87064
		85	$3,000	1287.66972
		86	$3,050	1296.16919
		87	$3,100	1304.37412
		88	$3,150	1312.28952
		89	$3,200	1319.92031
		90	$3,250	1327.27135
		91	$3,300	1334.34744
		92	$3,350	1341.15329
		93	$3,400	1347.69357
		94	$3,450	1353.97286
		95	$3,500	1359.9957
			TOTAL	$ 59,554.86
	.C. $3500 per month for next six years
		N	A	B=A/(1.01^N)
		Month	Payment	PV of payment
		96	$3,500	1346.5304
		97	$3,500	1333.19841
		98	$3,500	1319.99843
		99	$3,500	1306.92914
		100	$3,500	1293.98924
		101	$3,500	1281.17747
		102	$3,500	1268.49254
		103	$3,500	1255.93321
		104	$3,500	1243.49823
		105	$3,500	1231.18636
		106	$3,500	1218.9964
		107	$3,500	1206.92713
		108	$3,500	1194.97736
		109	$3,500	1183.1459
		110	$3,500	1171.43158
		111	$3,500	1159.83325
		112	$3,500	1148.34975
		113	$3,500	1136.97995
		114	$3,500	1125.72272
		115	$3,500	1114.57696
		116	$3,500	1103.54154
		117	$3,500	1092.61539
		118	$3,500	1081.79741
		119	$3,500	1071.08655
		120	$3,500	1060.48173
		121	$3,500	1049.98191
		122	$3,500	1039.58605
		123	$3,500	1029.29312
		124	$3,500	1019.1021
		125	$3,500	1009.01198
		126	$3,500	999.02176
		127	$3,500	989.130455
		128	$3,500	979.337084
		129	$3,500	969.640678
		130	$3,500	960.040275
		131	$3,500	950.534926
		132	$3,500	941.123689
		133	$3,500	931.805632
		134	$3,500	922.579834
		135	$3,500	913.44538
		136	$3,500	904.401367
		137	$3,500	895.446898
		138	$3,500	886.581087
		139	$3,500	877.803056
		140	$3,500	869.111937
		141	$3,500	860.506868
		142	$3,500	851.986998
		143	$3,500	843.551483
		144	$3,500	835.199488
		145	$3,500	826.930187
		146	$3,500	818.742759
		147	$3,500	810.636395
		148	$3,500	802.610292
		149	$3,500	794.663656
		150	$3,500	786.795699
		151	$3,500	779.005642
		152	$3,500	771.292715
		153	$3,500	763.656153
		154	$3,500	756.095201
		155	$3,500	748.60911
		156	$3,500	741.197139
		157	$3,500	733.858553
		158	$3,500	726.592627
		159	$3,500	719.398641
		160	$3,500	712.275882
		161	$3,500	705.223645
		162	$3,500	698.241233
		163	$3,500	691.327954
		164	$3,500	684.483122
		165	$3,500	677.706062
		166	$3,500	670.996101
		167	$3,500	664.352575
			TOTAL	$ 69,564.31
				A	PV of $500 per month for first three years:	$    15,204.29
				B	PV of payment Increasing by $50 per month for next five years	$    59,554.86
				C	PV of $3500 per month for next six years	$    69,564.31
					Total Present Value of Payments for(3+5+6)=14 years	$ 144,323.46