Question

[The information presented here applies to questions 7, 8 and 9.] The Smith family has borrowed $320,000 using a 5/1 ARM with an initial fully-indexed rate of 4.25% and paid no points at origination. The fully-indexed rate is determined by the yield on the LIBOR index plus a margin of 250 basis points.

7. What was the yield on the LIBOR index at the time when the Smith family purchased their house?

8. What is the balance remaining of their loan at the first reset date?

9. If, at the first reset date, the yield on the LIBOR index is 2.25%, what is the scheduled monthly payment for the sixth year of the loan?

Answer 1

7.

The Smiths have a 5/1 ARM. This is a hybrid structure. ARM suggests that this is an adjustable rate mortgage. It is linked to the LIBOR index and has a margin of 250 bps or 2.5%. This means that the adjustable rate of interest on this margin will be 2.5% higher than the LIBOR rate at reset date. The hybrid nature comes from the 5/1 part. This indicates that the mortgage will have a fixed rate of interest for first 5 years. after 5 years, the rate of interest will be reset every year for a 0ne-year period. the reference rate will be the one-year LIBOR at the date of reset.

The rate is fully indexed initially. This implies that the interest rate for the first 5 years will be based on LIBOR. The loan had no points for origination. Hence, there only margin over LIBOR in the effective interest rate on the mortgage. The prevalent LIBOR rate on the origination date = 4.25%- 2.5%= 1.75%.

8.

First reset date will be at the end of 5 years or beginning of 6^th year. The balance calculation is as follows-

Assume that the loan tenure is 15 years and the Smiths make a monthly payment as no information is provided.

PMT=

$2,407.29

=PMT(4.25%/12,15*12,-320000,0,0)

Year	Month	Periods	Monthly payment	Interest portion	Principal portion	Principal outstanding
1	1	1	$2,407.29	$      1,133.33	$      1,273.96	$ 318,726.04
	2	2	$2,407.29	$      1,128.82	$      1,278.47	$ 317,447.57
	3	3	$2,407.29	$      1,124.29	$      1,283.00	$ 316,164.58
	4	4	$2,407.29	$      1,119.75	$      1,287.54	$ 314,877.03
	5	5	$2,407.29	$      1,115.19	$      1,292.10	$ 313,584.93
	6	6	$2,407.29	$      1,110.61	$      1,296.68	$ 312,288.26
	7	7	$2,407.29	$      1,106.02	$      1,301.27	$ 310,986.99
	8	8	$2,407.29	$      1,101.41	$      1,305.88	$ 309,681.11
	9	9	$2,407.29	$      1,096.79	$      1,310.50	$ 308,370.60
	10	10	$2,407.29	$      1,092.15	$      1,315.15	$ 307,055.46
	11	11	$2,407.29	$      1,087.49	$      1,319.80	$ 305,735.65
	12	12	$2,407.29	$      1,082.81	$      1,324.48	$ 304,411.18
2	1	13	$2,407.29	$      1,078.12	$      1,329.17	$ 303,082.01
	2	14	$2,407.29	$      1,073.42	$      1,333.88	$ 301,748.13
	3	15	$2,407.29	$      1,068.69	$      1,338.60	$ 300,409.53
	4	16	$2,407.29	$      1,063.95	$      1,343.34	$ 299,066.19
	5	17	$2,407.29	$      1,059.19	$      1,348.10	$ 297,718.10
	6	18	$2,407.29	$      1,054.42	$      1,352.87	$ 296,365.22
	7	19	$2,407.29	$      1,049.63	$      1,357.66	$ 295,007.56
	8	20	$2,407.29	$      1,044.82	$      1,362.47	$ 293,645.09
	9	21	$2,407.29	$      1,039.99	$      1,367.30	$ 292,277.79
	10	22	$2,407.29	$      1,035.15	$      1,372.14	$ 290,905.65
	11	23	$2,407.29	$      1,030.29	$      1,377.00	$ 289,528.65
	12	24	$2,407.29	$      1,025.41	$      1,381.88	$ 288,146.77
3	1	25	$2,407.29	$      1,020.52	$      1,386.77	$ 286,760.00
	2	26	$2,407.29	$      1,015.61	$      1,391.68	$ 285,368.32
	3	27	$2,407.29	$      1,010.68	$      1,396.61	$ 283,971.71
	4	28	$2,407.29	$      1,005.73	$      1,401.56	$ 282,570.15
	5	29	$2,407.29	$      1,000.77	$      1,406.52	$ 281,163.63
	6	30	$2,407.29	$         995.79	$      1,411.50	$ 279,752.12
	7	31	$2,407.29	$         990.79	$      1,416.50	$ 278,335.62
	8	32	$2,407.29	$         985.77	$      1,421.52	$ 276,914.10
	9	33	$2,407.29	$         980.74	$      1,426.55	$ 275,487.55
	10	34	$2,407.29	$         975.69	$      1,431.61	$ 274,055.94
	11	35	$2,407.29	$         970.61	$      1,436.68	$ 272,619.27
	12	36	$2,407.29	$         965.53	$      1,441.76	$ 271,177.50
4	1	37	$2,407.29	$         960.42	$      1,446.87	$ 269,730.63
	2	38	$2,407.29	$         955.30	$      1,451.99	$ 268,278.64
	3	39	$2,407.29	$         950.15	$      1,457.14	$ 266,821.50
	4	40	$2,407.29	$         944.99	$      1,462.30	$ 265,359.20
	5	41	$2,407.29	$         939.81	$      1,467.48	$ 263,891.72
	6	42	$2,407.29	$         934.62	$      1,472.67	$ 262,419.05
	7	43	$2,407.29	$         929.40	$      1,477.89	$ 260,941.16
	8	44	$2,407.29	$         924.17	$      1,483.12	$ 259,458.04
	9	45	$2,407.29	$         918.91	$      1,488.38	$ 257,969.66
	10	46	$2,407.29	$         913.64	$      1,493.65	$ 256,476.01
	11	47	$2,407.29	$         908.35	$      1,498.94	$ 254,977.07
	12	48	$2,407.29	$         903.04	$      1,504.25	$ 253,472.83
5	1	49	$2,407.29	$         897.72	$      1,509.57	$ 251,963.25
	2	50	$2,407.29	$         892.37	$      1,514.92	$ 250,448.33
	3	51	$2,407.29	$         887.00	$      1,520.29	$ 248,928.04
	4	52	$2,407.29	$         881.62	$      1,525.67	$ 247,402.37
	5	53	$2,407.29	$         876.22	$      1,531.07	$ 245,871.30
	6	54	$2,407.29	$         870.79	$      1,536.50	$ 244,334.80
	7	55	$2,407.29	$         865.35	$      1,541.94	$ 242,792.86
	8	56	$2,407.29	$         859.89	$      1,547.40	$ 241,245.46
	9	57	$2,407.29	$         854.41	$      1,552.88	$ 239,692.58
	10	58	$2,407.29	$         848.91	$      1,558.38	$ 238,134.20
	11	59	$2,407.29	$         843.39	$      1,563.90	$ 236,570.30
	12	60	$2,407.29	$         837.85	$      1,569.44	$ 235,000.87

From this table, it can be seen that the balance principal remaining at the first reset date i.e. at the end of 5^th year is $ 253,472.83.

9.

The LIBOR at reset date is 2.25%. The margin on LIBOR is 2.5%. Hence, the applicable rate of interest on mortgage is 2.25%+ 2.5%= 4.75%. The monthly payment can be calculated by PMT formula as below-

Assume that the loan tenure is 15 years and now remaining life of loan is 10 years as no information is provided.

PMT=

$2,463.93

=PMT(4.75%/12,10*12,-235000.867,0,0)

This amount is slightly larger than the previous amount as the LIBOR and consequently the rate of interest on mortgage has increased in the ARM.