1. You have taken out a loan of $300 000 at an interest rate of 3%...

1. You have taken out a loan of $300 000 at an interest rate of 3% per annum compounded quarterly. You plan to repay the loan in 240 reducing-principal payments made at the end of each quarter and starting at the end of the first quarter.

(a) How much is the principal reduced by after each payment?

(b) Write down an expression for the loan balance before repayment Am at the end of period m. This means that A1 should give the initial loan balance.

Solutions

Expert Solution

Effective annual interest rate = =(1+3%/4)^4-1 = 3.0339%

Quarterly interest rate = 0.758%

Period	Loan at the beginning (A1)	Installment paid every quarter	Interest	Principal	Loan balance at the end
1	300,000	2,718.84	2,275	443	299,557
2	299,557	2,718.84	2,272	447	299,110
3	299,110	2,718.84	2,269	450	298,660
4	298,660	2,718.84	2,265	454	298,206
5	298,206	2,718.84	2,262	457	297,749
6	297,749	2,718.84	2,258	460	297,289
7	297,289	2,718.84	2,255	464	296,825
8	296,825	2,718.84	2,251	467	296,357
9	296,357	2,718.84	2,248	471	295,886
10	295,886	2,718.84	2,244	475	295,412
11	295,412	2,718.84	2,241	478	294,933
12	294,933	2,718.84	2,237	482	294,451
13	294,451	2,718.84	2,233	485	293,966
14	293,966	2,718.84	2,230	489	293,477
15	293,477	2,718.84	2,226	493	292,984
16	292,984	2,718.84	2,222	497	292,487
17	292,487	2,718.84	2,218	500	291,987
18	291,987	2,718.84	2,215	504	291,483
19	291,483	2,718.84	2,211	508	290,975
20	290,975	2,718.84	2,207	512	290,463
21	290,463	2,718.84	2,203	516	289,947
22	289,947	2,718.84	2,199	520	289,428
23	289,428	2,718.84	2,195	524	288,904
24	288,904	2,718.84	2,191	528	288,376
25	288,376	2,718.84	2,187	532	287,845
26	287,845	2,718.84	2,183	536	287,309
27	287,309	2,718.84	2,179	540	286,770
28	286,770	2,718.84	2,175	544	286,226
29	286,226	2,718.84	2,171	548	285,678
30	285,678	2,718.84	2,167	552	285,126
31	285,126	2,718.84	2,163	556	284,570
32	284,570	2,718.84	2,158	560	284,009
33	284,009	2,718.84	2,154	565	283,445
34	283,445	2,718.84	2,150	569	282,876
35	282,876	2,718.84	2,146	573	282,302
36	282,302	2,718.84	2,141	578	281,725
37	281,725	2,718.84	2,137	582	281,143
38	281,143	2,718.84	2,132	586	280,556
39	280,556	2,718.84	2,128	591	279,965
40	279,965	2,718.84	2,123	595	279,370
41	279,370	2,718.84	2,119	600	278,770
42	278,770	2,718.84	2,114	604	278,166
43	278,166	2,718.84	2,110	609	277,557
44	277,557	2,718.84	2,105	614	276,943
45	276,943	2,718.84	2,101	618	276,325
46	276,325	2,718.84	2,096	623	275,702
47	275,702	2,718.84	2,091	628	275,074
48	275,074	2,718.84	2,086	632	274,442
49	274,442	2,718.84	2,082	637	273,804
50	273,804	2,718.84	2,077	642	273,162
51	273,162	2,718.84	2,072	647	272,515
52	272,515	2,718.84	2,067	652	271,863
53	271,863	2,718.84	2,062	657	271,207
54	271,207	2,718.84	2,057	662	270,545
55	270,545	2,718.84	2,052	667	269,878
56	269,878	2,718.84	2,047	672	269,206
57	269,206	2,718.84	2,042	677	268,529
58	268,529	2,718.84	2,037	682	267,847
59	267,847	2,718.84	2,032	687	267,160
60	267,160	2,718.84	2,026	692	266,467
61	266,467	2,718.84	2,021	698	265,770
62	265,770	2,718.84	2,016	703	265,066
63	265,066	2,718.84	2,010	708	264,358
64	264,358	2,718.84	2,005	714	263,644
65	263,644	2,718.84	2,000	719	262,925
66	262,925	2,718.84	1,994	725	262,201
67	262,201	2,718.84	1,989	730	261,471
68	261,471	2,718.84	1,983	736	260,735
69	260,735	2,718.84	1,978	741	259,994
70	259,994	2,718.84	1,972	747	259,247
71	259,247	2,718.84	1,966	753	258,494
72	258,494	2,718.84	1,961	758	257,736
73	257,736	2,718.84	1,955	764	256,972
74	256,972	2,718.84	1,949	770	256,202
75	256,202	2,718.84	1,943	776	255,427
76	255,427	2,718.84	1,937	781	254,645
77	254,645	2,718.84	1,931	787	253,858
78	253,858	2,718.84	1,925	793	253,064
79	253,064	2,718.84	1,919	799	252,265
80	252,265	2,718.84	1,913	805	251,460
81	251,460	2,718.84	1,907	812	250,648
82	250,648	2,718.84	1,901	818	249,830
83	249,830	2,718.84	1,895	824	249,006
84	249,006	2,718.84	1,889	830	248,176
85	248,176	2,718.84	1,882	836	247,340
86	247,340	2,718.84	1,876	843	246,497
87	246,497	2,718.84	1,870	849	245,648
88	245,648	2,718.84	1,863	856	244,792
89	244,792	2,718.84	1,857	862	243,930
90	243,930	2,718.84	1,850	869	243,061
91	243,061	2,718.84	1,844	875	242,186
92	242,186	2,718.84	1,837	882	241,304
93	241,304	2,718.84	1,830	889	240,415
94	240,415	2,718.84	1,824	895	239,520
95	239,520	2,718.84	1,817	902	238,618
96	238,618	2,718.84	1,810	909	237,709
97	237,709	2,718.84	1,803	916	236,793
98	236,793	2,718.84	1,796	923	235,870
99	235,870	2,718.84	1,789	930	234,940
100	234,940	2,718.84	1,782	937	234,004

Excel formula:

jeff jeffy answered 6 months ago

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