Question

You want to buy a car which will cost you $10,000. You do not have sufficient funds to purchase the car. You do not expect the price of the car to change in the foreseeable future. You can either save money or borrow money to buy the car.

• Plan 1: You decide to open a bank account and start saving money. You will purchase the car when you have sufficient savings. The nominal interest rate for the bank account is 6% per annum compounded monthly.

a) You will make regular deposits in your bank account at the start of each month for the next 2.5 years. Calculate the minimum required monthly savings to be deposited into the bank such that you would have sufficient funds to purchase the car in 2.5 years. (1 mark)

b) You will make regular deposits in your bank account at the start of each week for the next 2.5 years. Calculate the minimum required weekly savings to be deposited into the bank such that you would have sufficient funds to purchase the car in 2.5 years.

c) You will make regular deposits of $2,000 at the end of each year. Calculate how long will it take for you to have sufficient funds to purchase the car. (1 mark)

• Plan 2: You decide to borrow $13,000 from the bank and purchase the car now, as well as cover some other expenses. The bank offers two options for the structure of the repayments.

- Option 1: The first repayment will not start until you graduate from university. Therefore, no month-end-instalments will be made for the first 36 months. Then, commencing at the end of the 37^th month, a total of 30 month-end-instalments of $X will be made over the life of the loan. The nominal interest rate is 6% per annum compounded monthly.

d) Calculate X. (2 mark)

e) Your parents agree to help you repay the loan by contributing a lump sum of $1,800 when you successfully graduate from university. Calculate the new value of X. (1 mark)

- Option 2: For the first 36 months (while you are still studying), you will be making month-end-instalments of $Y. Then, commencing at the end of the 37^th month (when you graduate from university), you will double the amount of monthly repayment for the remaining 30 month-end-instalments. The nominal interest rate is 6% per annum compounded monthly.

f) Calculate the value of Y.

Answer 1

Plan 1:

Interest rate per Year = 6% p.a Compounded Monthly

Interest rate per month = 6% /12 = 0.5% per month

a ) Making regular deposits at the start of each month for a period of 2.5 years

No.of Installments = 2.5*12 = 30 Months

Let the installment amount be $1

S.No	Opening	Deposit	Total	Int @ 0.5%	Closing balance
1		$1.0000	$1.00	$0.01	$1.01
2	$1.01	$1.0000	$2.01	$0.01	$2.02
3	$2.02	$1.0000	$3.02	$0.02	$3.03
4	$3.03	$1.0000	$4.03	$0.02	$4.05
5	$4.05	$1.0000	$5.05	$0.03	$5.08
6	$5.08	$1.0000	$6.08	$0.03	$6.11
7	$6.11	$1.0000	$7.11	$0.04	$7.14
8	$7.14	$1.0000	$8.14	$0.04	$8.18
9	$8.18	$1.0000	$9.18	$0.05	$9.23
10	$9.23	$1.0000	$10.23	$0.05	$10.28
11	$10.28	$1.0000	$11.28	$0.06	$11.34
12	$11.34	$1.0000	$12.34	$0.06	$12.40
13	$12.40	$1.0000	$13.40	$0.07	$13.46
14	$13.46	$1.0000	$14.46	$0.07	$14.54
15	$14.54	$1.0000	$15.54	$0.08	$15.61
16	$15.61	$1.0000	$16.61	$0.08	$16.70
17	$16.70	$1.0000	$17.70	$0.09	$17.79
18	$17.79	$1.0000	$18.79	$0.09	$18.88
19	$18.88	$1.0000	$19.88	$0.10	$19.98
20	$19.98	$1.0000	$20.98	$0.10	$21.08
21	$21.08	$1.0000	$22.08	$0.11	$22.19
22	$22.19	$1.0000	$23.19	$0.12	$23.31
23	$23.31	$1.0000	$24.31	$0.12	$24.43
24	$24.43	$1.0000	$25.43	$0.13	$25.56
25	$25.56	$1.0000	$26.56	$0.13	$26.69
26	$26.69	$1.0000	$27.69	$0.14	$27.83
27	$27.83	$1.0000	$28.83	$0.14	$28.97
28	$28.97	$1.0000	$29.97	$0.15	$30.12
29	$30.12	$1.0000	$31.12	$0.16	$31.28
30	$31.28	$1.0000	$32.28	$0.16	$32.44

If we invest Rs 1 per month it became $ 32.44 within a period of 2.5 years

To become Rs 10000 we have to invest following amount per month

= $ 10000/$ 32.44

= $308.2614

Therfore if we invest $ 308.2614 per month it will become $ 10000 after 2.5 years.

b)  Making regular deposits at the start of each week  for a period of 2.5 years

No.of months = 2.5*12 = 30 Months

No.of weeks = 30*4 = 120 weeks

Assuming 4 weeks per month
Let the installment amount be $1

Interest per month= 0.5%

Effective interest rate per week = [( 1+i/12)^12n-1]*100

Here n = No.of years

n= 1/12*1/4

n = 1/48 year

Effective interest rate per week = [ ( 1+6/1200) ^12*1/48 ) -1 ] *100

= [ ( 1.005)^0.25 - 1 ] *100

= [ ( 1.0012477-1)]*100

= 0.12477%

Let the installment amount be $1

S.No	Opening	Deposit	Total	Int @ 0.12477%	Closing balance
1		$1.00000	$1.00000	$0.00125	$1.00125
2	$1.00125	$1.00000	$2.00125	$0.00250	$2.00374
3	$2.00374	$1.00000	$3.00374	$0.00375	$3.00749
4	$3.00749	$1.00000	$4.00749	$0.00500	$4.01249
5	$4.01249	$1.00000	$5.01249	$0.00625	$5.01875
6	$5.01875	$1.00000	$6.01875	$0.00751	$6.02626
7	$6.02626	$1.00000	$7.02626	$0.00877	$7.03502
8	$7.03502	$1.00000	$8.03502	$0.01003	$8.04505
9	$8.04505	$1.00000	$9.04505	$0.01129	$9.05633
10	$9.05633	$1.00000	$10.05633	$0.01255	$10.06888
11	$10.06888	$1.00000	$11.06888	$0.01381	$11.08269
12	$11.08269	$1.00000	$12.08269	$0.01508	$12.09777
13	$12.09777	$1.00000	$13.09777	$0.01634	$13.11411
14	$13.11411	$1.00000	$14.11411	$0.01761	$14.13172
15	$14.13172	$1.00000	$15.13172	$0.01888	$15.15060
16	$15.15060	$1.00000	$16.15060	$0.02015	$16.17075
17	$16.17075	$1.00000	$17.17075	$0.02142	$17.19217
18	$17.19217	$1.00000	$18.19217	$0.02270	$18.21487
19	$18.21487	$1.00000	$19.21487	$0.02397	$19.23885
20	$19.23885	$1.00000	$20.23885	$0.02525	$20.26410
21	$20.26410	$1.00000	$21.26410	$0.02653	$21.29063
22	$21.29063	$1.00000	$22.29063	$0.02781	$22.31844
23	$22.31844	$1.00000	$23.31844	$0.02909	$23.34754
24	$23.34754	$1.00000	$24.34754	$0.03038	$24.37792
25	$24.37792	$1.00000	$25.37792	$0.03166	$25.40958
26	$25.40958	$1.00000	$26.40958	$0.03295	$26.44253
27	$26.44253	$1.00000	$27.44253	$0.03424	$27.47677
28	$27.47677	$1.00000	$28.47677	$0.03553	$28.51230
29	$28.51230	$1.00000	$29.51230	$0.03682	$29.54912
30	$29.54912	$1.00000	$30.54912	$0.03812	$30.58724
31	$30.58724	$1.00000	$31.58724	$0.03941	$31.62665
32	$31.62665	$1.00000	$32.62665	$0.04071	$32.66736
33	$32.66736	$1.00000	$33.66736	$0.04201	$33.70937
34	$33.70937	$1.00000	$34.70937	$0.04331	$34.75267
35	$34.75267	$1.00000	$35.75267	$0.04461	$35.79728
36	$35.79728	$1.00000	$36.79728	$0.04591	$36.84319
37	$36.84319	$1.00000	$37.84319	$0.04722	$37.89041
38	$37.89041	$1.00000	$38.89041	$0.04852	$38.93893
39	$38.93893	$1.00000	$39.93893	$0.04983	$39.98877
40	$39.98877	$1.00000	$40.98877	$0.05114	$41.03991
41	$41.03991	$1.00000	$42.03991	$0.05245	$42.09236
42	$42.09236	$1.00000	$43.09236	$0.05377	$43.14613
43	$43.14613	$1.00000	$44.14613	$0.05508	$44.20121
44	$44.20121	$1.00000	$45.20121	$0.05640	$45.25761
45	$45.25761	$1.00000	$46.25761	$0.05772	$46.31532
46	$46.31532	$1.00000	$47.31532	$0.05904	$47.37436
47	$47.37436	$1.00000	$48.37436	$0.06036	$48.43471
48	$48.43471	$1.00000	$49.43471	$0.06168	$49.49639
49	$49.49639	$1.00000	$50.49639	$0.06300	$50.55940
50	$50.55940	$1.00000	$51.55940	$0.06433	$51.62373
51	$51.62373	$1.00000	$52.62373	$0.06566	$52.68939
52	$52.68939	$1.00000	$53.68939	$0.06699	$53.75637
53	$53.75637	$1.00000	$54.75637	$0.06832	$54.82469
54	$54.82469	$1.00000	$55.82469	$0.06965	$55.89435
55	$55.89435	$1.00000	$56.89435	$0.07099	$56.96533
56	$56.96533	$1.00000	$57.96533	$0.07232	$58.03766
57	$58.03766	$1.00000	$59.03766	$0.07366	$59.11132
58	$59.11132	$1.00000	$60.11132	$0.07500	$60.18632
59	$60.18632	$1.00000	$61.18632	$0.07634	$61.26266
60	$61.26266	$1.00000	$62.26266	$0.07769	$62.34035
61	$62.34035	$1.00000	$63.34035	$0.07903	$63.41938
62	$63.41938	$1.00000	$64.41938	$0.08038	$64.49975
63	$64.49975	$1.00000	$65.49975	$0.08172	$65.58148
64	$65.58148	$1.00000	$66.58148	$0.08307	$66.66455
65	$66.66455	$1.00000	$67.66455	$0.08443	$67.74898
66	$67.74898	$1.00000	$68.74898	$0.08578	$68.83475
67	$68.83475	$1.00000	$69.83475	$0.08713	$69.92189
68	$69.92189	$1.00000	$70.92189	$0.08849	$71.01038
69	$71.01038	$1.00000	$72.01038	$0.08985	$72.10022
70	$72.10022	$1.00000	$73.10022	$0.09121	$73.19143
71	$73.19143	$1.00000	$74.19143	$0.09257	$74.28400
72	$74.28400	$1.00000	$75.28400	$0.09393	$75.37793
73	$75.37793	$1.00000	$76.37793	$0.09530	$76.47323
74	$76.47323	$1.00000	$77.47323	$0.09666	$77.56989
75	$77.56989	$1.00000	$78.56989	$0.09803	$78.66792
76	$78.66792	$1.00000	$79.66792	$0.09940	$79.76732
77	$79.76732	$1.00000	$80.76732	$0.10077	$80.86810
78	$80.86810	$1.00000	$81.86810	$0.10215	$81.97024
79	$81.97024	$1.00000	$82.97024	$0.10352	$83.07377
80	$83.07377	$1.00000	$84.07377	$0.10490	$84.17867
81	$84.17867	$1.00000	$85.17867	$0.10628	$85.28494
82	$85.28494	$1.00000	$86.28494	$0.10766	$86.39260
83	$86.39260	$1.00000	$87.39260	$0.10904	$87.50164
84	$87.50164	$1.00000	$88.50164	$0.11042	$88.61206
85	$88.61206	$1.00000	$89.61206	$0.11181	$89.72387
86	$89.72387	$1.00000	$90.72387	$0.11320	$90.83707
87	$90.83707	$1.00000	$91.83707	$0.11459	$91.95165
88	$91.95165	$1.00000	$92.95165	$0.11598	$93.06763
89	$93.06763	$1.00000	$94.06763	$0.11737	$94.18500
90	$94.18500	$1.00000	$95.18500	$0.11876	$95.30376
91	$95.30376	$1.00000	$96.30376	$0.12016	$96.42392
92	$96.42392	$1.00000	$97.42392	$0.12156	$97.54547
93	$97.54547	$1.00000	$98.54547	$0.12296	$98.66843
94	$98.66843	$1.00000	$99.66843	$0.12436	$99.79279
95	$99.79279	$1.00000	$100.79279	$0.12576	$100.91854
96	$100.91854	$1.00000	$101.91854	$0.12716	$102.04571
97	$102.04571	$1.00000	$103.04571	$0.12857	$103.17428
98	$103.17428	$1.00000	$104.17428	$0.12998	$104.30426
99	$104.30426	$1.00000	$105.30426	$0.13139	$105.43564
100	$105.43564	$1.00000	$106.43564	$0.13280	$106.56844
101	$106.56844	$1.00000	$107.56844	$0.13421	$107.70266
102	$107.70266	$1.00000	$108.70266	$0.13563	$108.83829
103	$108.83829	$1.00000	$109.83829	$0.13705	$109.97533
104	$109.97533	$1.00000	$110.97533	$0.13846	$111.11380
105	$111.11380	$1.00000	$112.11380	$0.13988	$112.25368
106	$112.25368	$1.00000	$113.25368	$0.14131	$113.39499
107	$113.39499	$1.00000	$114.39499	$0.14273	$114.53772
108	$114.53772	$1.00000	$115.53772	$0.14416	$115.68187
109	$115.68187	$1.00000	$116.68187	$0.14558	$116.82746
110	$116.82746	$1.00000	$117.82746	$0.14701	$117.97447
111	$117.97447	$1.00000	$118.97447	$0.14844	$119.12292
112	$119.12292	$1.00000	$120.12292	$0.14988	$120.27279
113	$120.27279	$1.00000	$121.27279	$0.15131	$121.42410
114	$121.42410	$1.00000	$122.42410	$0.15275	$122.57685
115	$122.57685	$1.00000	$123.57685	$0.15419	$123.73104
116	$123.73104	$1.00000	$124.73104	$0.15563	$124.88667
117	$124.88667	$1.00000	$125.88667	$0.15707	$126.04374
118	$126.04374	$1.00000	$127.04374	$0.15851	$127.20225
119	$127.20225	$1.00000	$128.20225	$0.15996	$128.36221
120	$128.36221	$1.00000	$129.36221	$0.16141	$129.52361

If we invest Rs 1 per week it became $ 129.52361 within a period of 2.5 years

To become Rs 10000 we have to invest following amount per week

= $ 10000/$ 129.52361

=$77.206

Therfore if we invest $ 77.2060 week then it will become $ 10000 after 2.5 years.

c) Interest rate per month = 0.5%

S.No	Opening	Interest@ 0.5%	Total	Amount deposited	Closing balance
1
2
3
4
5
6
7
8
9
10
11
12				$2,000	$2,000
13	$2,000	$10.0	$2,010.0	$0	$2,010.0
14	$2,010.0	$10.1	$2,020.1	$0	$2,020.1
15	$2,020.1	$10.1	$2,030.2	$0	$2,030.2
16	$2,030.2	$10.2	$2,040.3	$0	$2,040.3
17	$2,040.3	$10.2	$2,050.5	$0	$2,050.5
18	$2,050.5	$10.3	$2,060.8	$0	$2,060.8
19	$2,060.8	$10.3	$2,071.1	$0	$2,071.1
20	$2,071.1	$10.4	$2,081.4	$0	$2,081.4
21	$2,081.4	$10.4	$2,091.8	$0	$2,091.8
22	$2,091.8	$10.5	$2,102.3	$0	$2,102.3
23	$2,102.3	$10.5	$2,112.8	$0	$2,112.8
24	$2,112.8	$10.6	$2,123.4	$2,000	$4,123.4
25	$4,123.4	$20.6	$4,144.0	$0	$4,144.0
26	$4,144.0	$20.7	$4,164.7	$0	$4,164.7
27	$4,164.7	$20.8	$4,185.5	$0	$4,185.5
28	$4,185.5	$20.9	$4,206.4	$0	$4,206.4
29	$4,206.4	$21.0	$4,227.5	$0	$4,227.5
30	$4,227.5	$21.1	$4,248.6	$0	$4,248.6
31	$4,248.6	$21.2	$4,269.9	$0	$4,269.9
32	$4,269.9	$21.3	$4,291.2	$0	$4,291.2
33	$4,291.2	$21.5	$4,312.7	$0	$4,312.7
34	$4,312.7	$21.6	$4,334.2	$0	$4,334.2
35	$4,334.2	$21.7	$4,355.9	$0	$4,355.9
36	$4,355.9	$21.8	$4,377.7	$2,000	$6,377.7
37	$6,377.7	$31.9	$6,409.6	$0	$6,409.6
38	$6,409.6	$32.0	$6,441.6	$0	$6,441.6
39	$6,441.6	$32.2	$6,473.8	$0	$6,473.8
40	$6,473.8	$32.4	$6,506.2	$0	$6,506.2
41	$6,506.2	$32.5	$6,538.7	$0	$6,538.7
42	$6,538.7	$32.7	$6,571.4	$0	$6,571.4
43	$6,571.4	$32.9	$6,604.3	$0	$6,604.3
44	$6,604.3	$33.0	$6,637.3	$0	$6,637.3
45	$6,637.3	$33.2	$6,670.5	$0	$6,670.5
46	$6,670.5	$33.4	$6,703.8	$0	$6,703.8
47	$6,703.8	$33.5	$6,737.3	$0	$6,737.3
48	$6,737.3	$33.7	$6,771.0	$2,000	$8,771.0
49	$8,771.0	$43.9	$8,814.9	$0	$8,814.9
50	$8,814.9	$44.1	$8,859.0	$0	$8,859.0
51	$8,859.0	$44.3	$8,903.3	$0	$8,903.3
52	$8,903.3	$44.5	$8,947.8	$0	$8,947.8
53	$8,947.8	$44.7	$8,992.5	$0	$8,992.5
54	$8,992.5	$45.0	$9,037.5	$0	$9,037.5
55	$9,037.5	$45.2	$9,082.7	$0	$9,082.7
56	$9,082.7	$45.4	$9,128.1	$0	$9,128.1
57	$9,128.1	$45.6	$9,173.7	$0	$9,173.7
58	$9,173.7	$45.9	$9,219.6	$0	$9,219.6
59	$9,219.6	$46.1	$9,265.7	$0	$9,265.7
60	$9,265.7	$46.3	$9,312.0	$2,000	$11,312.0

We deposited $ 2000 at the end of each year. So By the end of the 5th year we are already having $ 9312 balance.

Ii is enough if we deposit the $ 10000-$ 9312= $ 688

So By 5 years we can accumulate $ 10000.

d) Plan 2: Option 1

S.No	Opening Balance	Interest @ 0.5%	Closing balance
1	$13,000	$65.0	$13,065
2	$13,065	$65.3	$13,130
3	$13,130	$65.7	$13,196
4	$13,196	$66.0	$13,262
5	$13,262	$66.3	$13,328
6	$13,328	$66.6	$13,395
7	$13,395	$67.0	$13,462
8	$13,462	$67.3	$13,529
9	$13,529	$67.6	$13,597
10	$13,597	$68.0	$13,665
11	$13,665	$68.3	$13,733
12	$13,733	$68.7	$13,802
13	$13,802	$69.0	$13,871
14	$13,871	$69.4	$13,940
15	$13,940	$69.7	$14,010
16	$14,010	$70.0	$14,080
17	$14,080	$70.4	$14,150
18	$14,150	$70.8	$14,221
19	$14,221	$71.1	$14,292
20	$14,292	$71.5	$14,364
21	$14,364	$71.8	$14,435
22	$14,435	$72.2	$14,508
23	$14,508	$72.5	$14,580
24	$14,580	$72.9	$14,653
25	$14,653	$73.3	$14,726
26	$14,726	$73.6	$14,800
27	$14,800	$74.0	$14,874
28	$14,874	$74.4	$14,948
29	$14,948	$74.7	$15,023
30	$15,023	$75.1	$15,098
31	$15,098	$75.5	$15,174
32	$15,174	$75.9	$15,250
33	$15,250	$76.2	$15,326
34	$15,326	$76.6	$15,402
35	$15,402	$77.0	$15,479
36	$15,479	$77.4	$15,557

Outstanding Balance of the loan after 36 months is $ 15557

Given, Monthly Installment = 30

We know that Present value of future cash outflows is equal to the loan amount.

S.No	Disc @ 0.5%	Discounting factor
1	1/( 1.005)^1	0.9950
2	1/( 1.005)^2	0.9901
3	1/( 1.005)^3	0.9851
4	1/( 1.005)^4	0.9802
5	1/( 1.005)^5	0.9754
6	1/( 1.005)^6	0.9705
7	1/( 1.005)^7	0.9657
8	1/( 1.005)^8	0.9609
9	1/( 1.005)^9	0.9561
10	1/( 1.005)^10	0.9513
11	1/( 1.005)^11	0.9466
12	1/( 1.005)^12	0.9419
13	1/( 1.005)^13	0.9372
14	1/( 1.005)^14	0.9326
15	1/( 1.005)^15	0.9279
16	1/( 1.005)^16	0.9233
17	1/( 1.005)^17	0.9187
18	1/( 1.005)^18	0.9141
19	1/( 1.005)^19	0.9096
20	1/( 1.005)^20	0.9051
21	1/( 1.005)^21	0.9006
22	1/( 1.005)^22	0.8961
23	1/( 1.005)^23	0.8916
24	1/( 1.005)^24	0.8872
25	1/( 1.005)^25	0.8828
26	1/( 1.005)^26	0.8784
27	1/( 1.005)^27	0.8740
28	1/( 1.005)^28	0.8697
29	1/( 1.005)^29	0.8653
30	1/( 1.005)^30	0.8610
		27.7941

X * PVAF ( 0.5%,30) = $ 155

X * 27.7941 = $ 15557

X = $ 15557/27.7941

X = $ 559.7231

Hence the Monthly installment is $ 559.7231 or $ 560.

e) If parents Contribute $ 1800 then the loan outstanding balance after 36 months

Outstanding balance = $ 15557-$ 1800= $ 13757

We know that Present value of future cash outflows is equal to the loan amount.

X * PVAF ( 0.5%,30) = $ 13757

X * 27.7941 = $ 13757

X = $ 13757/27.7941

X = $ 494.9611

Hence the Monthly installment will be $ 494.9611

f) Option 2

S.No	Disc @ 0.5%	Discounting factor	Cummulative Discountinng Factor
1	1/( 1.005)^1	0.9950	0.9950
2	1/( 1.005)^2	0.9901	1.9851
3	1/( 1.005)^3	0.9851	2.9702
4	1/( 1.005)^4	0.9802	3.9505
5	1/( 1.005)^5	0.9754	4.9259
6	1/( 1.005)^6	0.9705	5.8964
7	1/( 1.005)^7	0.9657	6.8621
8	1/( 1.005)^8	0.9609	7.8230
9	1/( 1.005)^9	0.9561	8.7791
10	1/( 1.005)^10	0.9513	9.7304
11	1/( 1.005)^11	0.9466	10.6770
12	1/( 1.005)^12	0.9419	11.6189
13	1/( 1.005)^13	0.9372	12.5562
14	1/( 1.005)^14	0.9326	13.4887
15	1/( 1.005)^15	0.9279	14.4166
16	1/( 1.005)^16	0.9233	15.3399
17	1/( 1.005)^17	0.9187	16.2586
18	1/( 1.005)^18	0.9141	17.1728
19	1/( 1.005)^19	0.9096	18.0824
20	1/( 1.005)^20	0.9051	18.9874
21	1/( 1.005)^21	0.9006	19.8880
22	1/( 1.005)^22	0.8961	20.7841
23	1/( 1.005)^23	0.8916	21.6757
24	1/( 1.005)^24	0.8872	22.5629
25	1/( 1.005)^25	0.8828	23.4456
26	1/( 1.005)^26	0.8784	24.3240
27	1/( 1.005)^27	0.8740	25.1980
28	1/( 1.005)^28	0.8697	26.0677
29	1/( 1.005)^29	0.8653	26.9330
30	1/( 1.005)^30	0.8610	27.7941
31	1/( 1.005)^31	0.8567	28.6508
32	1/( 1.005)^32	0.8525	29.5033
33	1/( 1.005)^33	0.8482	30.3515
34	1/( 1.005)^34	0.8440	31.1955
35	1/( 1.005)^35	0.8398	32.0354
36	1/( 1.005)^36	0.8356	32.8710

S.No	Disc @ 0.5%	Discounting factor	Cummulative Discountinng Factor
37	1/( 1.005)^37	0.8315	0.8315
38	1/( 1.005)^38	0.8274	1.6588
39	1/( 1.005)^39	0.8232	2.4821
40	1/( 1.005)^40	0.8191	3.3012
41	1/( 1.005)^41	0.8151	4.1163
42	1/( 1.005)^42	0.8110	4.9273
43	1/( 1.005)^43	0.8070	5.7343
44	1/( 1.005)^44	0.8030	6.5372
45	1/( 1.005)^45	0.7990	7.3362
46	1/( 1.005)^46	0.7950	8.1312
47	1/( 1.005)^47	0.7910	8.9222
48	1/( 1.005)^48	0.7871	9.7093
49	1/( 1.005)^49	0.7832	10.4925
50	1/( 1.005)^50	0.7793	11.2718
51	1/( 1.005)^51	0.7754	12.0472
52	1/( 1.005)^52	0.7716	12.8187
53	1/( 1.005)^53	0.7677	13.5864
54	1/( 1.005)^54	0.7639	14.3503
55	1/( 1.005)^55	0.7601	15.1104
56	1/( 1.005)^56	0.7563	15.8667
57	1/( 1.005)^57	0.7525	16.6193
58	1/( 1.005)^58	0.7488	17.3681
59	1/( 1.005)^59	0.7451	18.1132
60	1/( 1.005)^60	0.7414	18.8545
61	1/( 1.005)^61	0.7377	19.5922
62	1/( 1.005)^62	0.7340	20.3262
63	1/( 1.005)^63	0.7304	21.0566
64	1/( 1.005)^64	0.7267	21.7833
65	1/( 1.005)^65	0.7231	22.5064
66	1/( 1.005)^66	0.7195	23.2260

We know that Present value of future cash outflows is equal to the loan amount.

Let the Monthly installment during first 36 months is Y and next 30 months is 2Y

Y * PVAF ( 0.5% ,36) + 2Y * PVAF( 0.5% , ( 37-66) ) = $ 13000

Y * 32.8710 + 2Y * 23.2260 = $ 13000

32.8710Y + 46.452Y = $ 13000

79.323Y = $ 13000

Y = $ 163.8868

f) Hence the Value of Y is $ 163.8868  

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