Question

You have just sold your house for $1,000,000 in cash. Your mortgage was originally a 30-year mortgage with monthly payments and had an initial balance of $800,000. The mortgage is currently exactly 18.5 years old, and you have just made a payment. If the interest rate on the mortgage is 5.25% p.a. (with monthly compounding), how much cash will you have from the sale once you pay off the mortgage? (Round to the nearest dollar).

	a.	$540.661
	b.	$459,340
	c.	$456,931
	d.	$626,635
	e.	$543,069

Answer 1

In the given case, first we calculate monthly payments required for $ 800000 with 5.25% p.a. for 30 years.

Monthly Payments= Loan Amount/ PVAF(r%, n period)

PVAF(r%, n period) is the annuity factor where r% is interest rate, which is converted into monthly rate and n refers to no. of months i.e. 360 months as there as monthly payments.

PVAF(0.4375%, 360 months) = (1+r%)ⁿ-1/ (r% *(1+r%)ⁿ)

= (1+0.4375%)³⁶⁰-1/ (0.4375% * (1+0.4375%)³⁶⁰)

= (1.004375)³⁶⁰-1/ (0.004375 *(1.004375)³⁶⁰)

= 4.81417506-1/ (0.004375*4.81417506)

= 3.81417506/ 0.021062016

= 181.0926

Therefore:

Monthly Payment = 800000/181.0926

= $ 4417.63

Now remaining amount of loan remain after 18.5 years i.e. after 222 months is as follow :

Month	EMI	Interest component	Principal Component	Outstanding Balance
0	-	-	-	800000.00
1	4417.63	3500.00	917.63	799082.37
2	4417.63	3495.99	921.64	798160.73
3	4417.63	3491.95	925.68	797235.05
4	4417.63	3487.90	929.73	796305.32
5	4417.63	3483.84	933.79	795371.53
6	4417.63	3479.75	937.88	794433.65
7	4417.63	3475.65	941.98	793491.67
8	4417.63	3471.53	946.10	792545.56
9	4417.63	3467.39	950.24	791595.32
10	4417.63	3463.23	954.40	790640.92
11	4417.63	3459.05	958.58	789682.35
12	4417.63	3454.86	962.77	788719.58
13	4417.63	3450.65	966.98	787752.60
14	4417.63	3446.42	971.21	786781.38
15	4417.63	3442.17	975.46	785805.92
16	4417.63	3437.90	979.73	784826.19
17	4417.63	3433.61	984.02	783842.18
18	4417.63	3429.31	988.32	782853.86
19	4417.63	3424.99	992.64	781861.21
20	4417.63	3420.64	996.99	780864.23
21	4417.63	3416.28	1001.35	779862.88
22	4417.63	3411.90	1005.73	778857.15
23	4417.63	3407.50	1010.13	777847.02
24	4417.63	3403.08	1014.55	776832.47
25	4417.63	3398.64	1018.99	775813.48
26	4417.63	3394.18	1023.45	774790.04
27	4417.63	3389.71	1027.92	773762.11
28	4417.63	3385.21	1032.42	772729.69
29	4417.63	3380.69	1036.94	771692.76
30	4417.63	3376.16	1041.47	770651.28
31	4417.63	3371.60	1046.03	769605.25
32	4417.63	3367.02	1050.61	768554.65
33	4417.63	3362.43	1055.20	767499.44
34	4417.63	3357.81	1059.82	766439.62
35	4417.63	3353.17	1064.46	765375.17
36	4417.63	3348.52	1069.11	764306.05
37	4417.63	3343.84	1073.79	763232.26
38	4417.63	3339.14	1078.49	762153.77
39	4417.63	3334.42	1083.21	761070.57
40	4417.63	3329.68	1087.95	759982.62
41	4417.63	3324.92	1092.71	758889.92
42	4417.63	3320.14	1097.49	757792.43
43	4417.63	3315.34	1102.29	756690.14
44	4417.63	3310.52	1107.11	755583.03
45	4417.63	3305.68	1111.95	754471.08
46	4417.63	3300.81	1116.82	753354.26
47	4417.63	3295.92	1121.70	752232.56
48	4417.63	3291.02	1126.61	751105.94
49	4417.63	3286.09	1131.54	749974.40
50	4417.63	3281.14	1136.49	748837.91
51	4417.63	3276.17	1141.46	747696.45
52	4417.63	3271.17	1146.46	746549.99
53	4417.63	3266.16	1151.47	745398.52
54	4417.63	3261.12	1156.51	744242.00
55	4417.63	3256.06	1161.57	743080.43
56	4417.63	3250.98	1166.65	741913.78
57	4417.63	3245.87	1171.76	740742.02
58	4417.63	3240.75	1176.88	739565.14
59	4417.63	3235.60	1182.03	738383.11
60	4417.63	3230.43	1187.20	737195.91
61	4417.63	3225.23	1192.40	736003.51
62	4417.63	3220.02	1197.61	734805.89
63	4417.63	3214.78	1202.85	733603.04
64	4417.63	3209.51	1208.12	732394.92
65	4417.63	3204.23	1213.40	731181.52
66	4417.63	3198.92	1218.71	729962.81
67	4417.63	3193.59	1224.04	728738.77
68	4417.63	3188.23	1229.40	727509.37
69	4417.63	3182.85	1234.78	726274.59
70	4417.63	3177.45	1240.18	725034.42
71	4417.63	3172.03	1245.60	723788.81
72	4417.63	3166.58	1251.05	722537.76
73	4417.63	3161.10	1256.53	721281.23
74	4417.63	3155.61	1262.02	720019.21
75	4417.63	3150.08	1267.55	718751.66
76	4417.63	3144.54	1273.09	717478.57
77	4417.63	3138.97	1278.66	716199.91
78	4417.63	3133.37	1284.26	714915.66
79	4417.63	3127.76	1289.87	713625.78
80	4417.63	3122.11	1295.52	712330.26
81	4417.63	3116.44	1301.18	711029.08
82	4417.63	3110.75	1306.88	709722.20
83	4417.63	3105.03	1312.59	708409.61
84	4417.63	3099.29	1318.34	707091.27
85	4417.63	3093.52	1324.11	705767.16
86	4417.63	3087.73	1329.90	704437.27
87	4417.63	3081.91	1335.72	703101.55
88	4417.63	3076.07	1341.56	701759.99
89	4417.63	3070.20	1347.43	700412.56
90	4417.63	3064.30	1353.32	699059.24
91	4417.63	3058.38	1359.25	697699.99
92	4417.63	3052.44	1365.19	696334.80
93	4417.63	3046.46	1371.16	694963.63
94	4417.63	3040.47	1377.16	693586.47
95	4417.63	3034.44	1383.19	692203.28
96	4417.63	3028.39	1389.24	690814.04
97	4417.63	3022.31	1395.32	689418.72
98	4417.63	3016.21	1401.42	688017.30
99	4417.63	3010.08	1407.55	686609.75
100	4417.63	3003.92	1413.71	685196.03
101	4417.63	2997.73	1419.90	683776.14
102	4417.63	2991.52	1426.11	682350.03
103	4417.63	2985.28	1432.35	680917.68
104	4417.63	2979.01	1438.61	679479.06
105	4417.63	2972.72	1444.91	678034.16
106	4417.63	2966.40	1451.23	676582.93
107	4417.63	2960.05	1457.58	675125.35
108	4417.63	2953.67	1463.96	673661.39
109	4417.63	2947.27	1470.36	672191.03
110	4417.63	2940.84	1476.79	670714.23
111	4417.63	2934.37	1483.25	669230.98
112	4417.63	2927.89	1489.74	667741.24
113	4417.63	2921.37	1496.26	666244.97
114	4417.63	2914.82	1502.81	664742.17
115	4417.63	2908.25	1509.38	663232.78
116	4417.63	2901.64	1515.99	661716.80
117	4417.63	2895.01	1522.62	660194.18
118	4417.63	2888.35	1529.28	658664.90
119	4417.63	2881.66	1535.97	657128.93
120	4417.63	2874.94	1542.69	655586.24
121	4417.63	2868.19	1549.44	654036.80
122	4417.63	2861.41	1556.22	652480.58
123	4417.63	2854.60	1563.03	650917.55
124	4417.63	2847.76	1569.87	649347.69
125	4417.63	2840.90	1576.73	647770.95
126	4417.63	2834.00	1583.63	646187.32
127	4417.63	2827.07	1590.56	644596.76
128	4417.63	2820.11	1597.52	642999.24
129	4417.63	2813.12	1604.51	641394.73
130	4417.63	2806.10	1611.53	639783.21
131	4417.63	2799.05	1618.58	638164.63
132	4417.63	2791.97	1625.66	636538.97
133	4417.63	2784.86	1632.77	634906.20
134	4417.63	2777.71	1639.92	633266.28
135	4417.63	2770.54	1647.09	631619.19
136	4417.63	2763.33	1654.30	629964.90
137	4417.63	2756.10	1661.53	628303.36
138	4417.63	2748.83	1668.80	626634.56
139	4417.63	2741.53	1676.10	624958.46
140	4417.63	2734.19	1683.44	623275.02
141	4417.63	2726.83	1690.80	621584.22
142	4417.63	2719.43	1698.20	619886.02
143	4417.63	2712.00	1705.63	618180.39
144	4417.63	2704.54	1713.09	616467.30
145	4417.63	2697.04	1720.59	614746.72
146	4417.63	2689.52	1728.11	613018.61
147	4417.63	2681.96	1735.67	611282.93
148	4417.63	2674.36	1743.27	609539.67
149	4417.63	2666.74	1750.89	607788.77
150	4417.63	2659.08	1758.55	606030.22
151	4417.63	2651.38	1766.25	604263.97
152	4417.63	2643.65	1773.97	602490.00
153	4417.63	2635.89	1781.74	600708.26
154	4417.63	2628.10	1789.53	598918.73
155	4417.63	2620.27	1797.36	597121.37
156	4417.63	2612.41	1805.22	595316.15
157	4417.63	2604.51	1813.12	593503.02
158	4417.63	2596.58	1821.05	591681.97
159	4417.63	2588.61	1829.02	589852.95
160	4417.63	2580.61	1837.02	588015.93
161	4417.63	2572.57	1845.06	586170.87
162	4417.63	2564.50	1853.13	584317.73
163	4417.63	2556.39	1861.24	582456.49
164	4417.63	2548.25	1869.38	580587.11
165	4417.63	2540.07	1877.56	578709.55
166	4417.63	2531.85	1885.78	576823.78
167	4417.63	2523.60	1894.03	574929.75
168	4417.63	2515.32	1902.31	573027.44
169	4417.63	2507.00	1910.63	571116.80
170	4417.63	2498.64	1918.99	569197.81
171	4417.63	2490.24	1927.39	567270.42
172	4417.63	2481.81	1935.82	565334.60
173	4417.63	2473.34	1944.29	563390.31
174	4417.63	2464.83	1952.80	561437.51
175	4417.63	2456.29	1961.34	559476.17
176	4417.63	2447.71	1969.92	557506.25
177	4417.63	2439.09	1978.54	555527.71
178	4417.63	2430.43	1987.20	553540.51
179	4417.63	2421.74	1995.89	551544.62
180	4417.63	2413.01	2004.62	549540.00
181	4417.63	2404.24	2013.39	547526.61
182	4417.63	2395.43	2022.20	545504.41
183	4417.63	2386.58	2031.05	543473.36
184	4417.63	2377.70	2039.93	541433.43
185	4417.63	2368.77	2048.86	539384.57
186	4417.63	2359.81	2057.82	537326.75
187	4417.63	2350.80	2066.83	535259.92
188	4417.63	2341.76	2075.87	533184.06
189	4417.63	2332.68	2084.95	531099.11
190	4417.63	2323.56	2094.07	529005.03
191	4417.63	2314.40	2103.23	526901.80
192	4417.63	2305.20	2112.43	524789.37
193	4417.63	2295.95	2121.68	522667.69
194	4417.63	2286.67	2130.96	520536.73
195	4417.63	2277.35	2140.28	518396.45
196	4417.63	2267.98	2149.65	516246.81
197	4417.63	2258.58	2159.05	514087.76
198	4417.63	2249.13	2168.50	511919.26
199	4417.63	2239.65	2177.98	509741.28
200	4417.63	2230.12	2187.51	507553.77
201	4417.63	2220.55	2197.08	505356.68
202	4417.63	2210.94	2206.69	503149.99
203	4417.63	2201.28	2216.35	500933.64
204	4417.63	2191.58	2226.04	498707.60
205	4417.63	2181.85	2235.78	496471.81
206	4417.63	2172.06	2245.57	494226.25
207	4417.63	2162.24	2255.39	491970.86
208	4417.63	2152.37	2265.26	489705.60
209	4417.63	2142.46	2275.17	487430.43
210	4417.63	2132.51	2285.12	485145.31
211	4417.63	2122.51	2295.12	482850.19
212	4417.63	2112.47	2305.16	480545.03
213	4417.63	2102.38	2315.25	478229.79
214	4417.63	2092.26	2325.37	475904.41
215	4417.63	2082.08	2335.55	473568.87
216	4417.63	2071.86	2345.77	471223.10
217	4417.63	2061.60	2356.03	468867.07
218	4417.63	2051.29	2366.34	466500.74
219	4417.63	2040.94	2376.69	464124.05
220	4417.63	2030.54	2387.09	461736.96
221	4417.63	2020.10	2397.53	459339.43
222	4417.63	2009.61	2408.02	456931.41

However calculating remaining balance of loan by this method is very tedious.

The formula to calculate the remaning balance of loan is as follow

FV of original balance - FV of annuity

i.e. (PV*(1+r)ⁿ) - P* {((1+r)ⁿ-1)/r}

FV signifies future value

Here PV is present value of loan i.e. $ 800000

P is the monthly repayment amount i.e. $ 4417.63 and r is 0.4375%

=    (800000 *(1+0.004375)²²²) - 4417.63 * {((1+0.004375)²²²-1)/0.004375}

= (800000*2.635654576) - 4417.63 *((2.635654576-1)/0.004375)

= (2108523.661) - 4417.63 * (1.635654576/0.004375)

= 2108523.661 - (4417.63 *373.8639)

= 2108523.661-1651592.381

   =$ 456931.28

Therefore cash remain after mortgage payment = 1000000-456931.28

= $ 543069 round off