Question

A savings plan is set up so that $100 is placed into an account on the 1st and 15th of each month. The first payment was made on January 1, 2010 and this continued until December 1, 2013. Then no money was deposited for the rest of December 2013 until March 1, 2014 when the $100 payments were continued until the last payment was made on February 1, 2017. Interest is 2.5% effective annual interest. a. Draw a careful number line and indicate the number of payments that were made before the missed payment, the number of missed payments and the number of payments after the missed payments. Remember that the payments are twice per month. b. Find the annual rate compounded twice per month. c. Find the amount in the account after the payment is made on December 1, 2013. d. Find the amount in the account after the payment is made on March 15, 2014. e. Find the amount in the account on February 15, 2017

Answer 1

Part (a)

Timeline and payment

Payment no.	Date	Payment
1	1-Jan-10	100
2	15-Jan-10	100
3	1-Feb-10	100
4	15-Feb-10	100
5	1-Mar-10	100
6	15-Mar-10	100
7	1-Apr-10	100
8	15-Apr-10	100
9	1-May-10	100
10	15-May-10	100
11	1-Jun-10	100
12	15-Jun-10	100
13	1-Jul-10	100
14	15-Jul-10	100
15	1-Aug-10	100
16	15-Aug-10	100
17	1-Sep-10	100
18	15-Sep-10	100
19	1-Oct-10	100
20	15-Oct-10	100
21	1-Nov-10	100
22	15-Nov-10	100
23	1-Dec-10	100
24	15-Dec-10	100
25	1-Jan-11	100
26	15-Jan-11	100
27	1-Feb-11	100
28	15-Feb-11	100
29	1-Mar-11	100
30	15-Mar-11	100
31	1-Apr-11	100
32	15-Apr-11	100
33	1-May-11	100
34	15-May-11	100
35	1-Jun-11	100
36	15-Jun-11	100
37	1-Jul-11	100
38	15-Jul-11	100
39	1-Aug-11	100
40	15-Aug-11	100
41	1-Sep-11	100
42	15-Sep-11	100
43	1-Oct-11	100
44	15-Oct-11	100
45	1-Nov-11	100
46	15-Nov-11	100
47	1-Dec-11	100
48	15-Dec-11	100
49	1-Jan-12	100
50	15-Jan-12	100
51	1-Feb-12	100
52	15-Feb-12	100
53	1-Mar-12	100
54	15-Mar-12	100
55	1-Apr-12	100
56	15-Apr-12	100
57	1-May-12	100
58	15-May-12	100
59	1-Jun-12	100
60	15-Jun-12	100
61	1-Jul-12	100
62	15-Jul-12	100
63	1-Aug-12	100
64	15-Aug-12	100
65	1-Sep-12	100
66	15-Sep-12	100
67	1-Oct-12	100
68	15-Oct-12	100
69	1-Nov-12	100
70	15-Nov-12	100
71	1-Dec-12	100
72	15-Dec-12	100
73	1-Jan-13	100
74	15-Jan-13	100
75	1-Feb-13	100
76	15-Feb-13	100
77	1-Mar-13	100
78	15-Mar-13	100
79	1-Apr-13	100
80	15-Apr-13	100
81	1-May-13	100
82	15-May-13	100
83	1-Jun-13	100
84	15-Jun-13	100
85	1-Jul-13	100
86	15-Jul-13	100
87	1-Aug-13	100
88	15-Aug-13	100
89	1-Sep-13	100
90	15-Sep-13	100
91	1-Oct-13	100
92	15-Oct-13	100
93	1-Nov-13	100
94	15-Nov-13	100
95	1-Dec-13	100
96	15-Dec-13
97	1-Jan-14
98	15-Jan-14
99	1-Feb-14
100	15-Feb-14
101	1-Mar-14	100
102	15-Mar-14	100
103	1-Apr-14	100
104	15-Apr-14	100
105	1-May-14	100
106	15-May-14	100
107	1-Jun-14	100
108	15-Jun-14	100
109	1-Jul-14	100
110	15-Jul-14	100
111	1-Aug-14	100
112	15-Aug-14	100
113	1-Sep-14	100
114	15-Sep-14	100
115	1-Oct-14	100
116	15-Oct-14	100
117	1-Nov-14	100
118	15-Nov-14	100
119	1-Dec-14	100
120	15-Dec-14	100
121	1-Jan-15	100
122	15-Jan-15	100
123	1-Feb-15	100
124	15-Feb-15	100
125	1-Mar-15	100
126	15-Mar-15	100
127	1-Apr-15	100
128	15-Apr-15	100
129	1-May-15	100
130	15-May-15	100
131	1-Jun-15	100
132	15-Jun-15	100
133	1-Jul-15	100
134	15-Jul-15	100
135	1-Aug-15	100
136	15-Aug-15	100
137	1-Sep-15	100
138	15-Sep-15	100
139	1-Oct-15	100
140	15-Oct-15	100
141	1-Nov-15	100
142	15-Nov-15	100
143	1-Dec-15	100
144	15-Dec-15	100
145	1-Jan-16	100
146	15-Jan-16	100
147	1-Feb-16	100

the number of payments that were made before the missed payment = 95

the number of missed payments = 5

and the number of payments after the missed payments = 47

Part (b)

Let r be the annual rate compounded twice per month

EAR = 2.5% = (1 + r / 24)²⁴ - 1

Hence, r = [(1 + 2.5%)^1/24 - 1] x 24 = 0.1029%

Part (c)

the amount in the account after the payment is made on December 1, 2013 = FV (Rate, Nper, PMT, PV) = FV (0.1029%, 94, -100, -100) = $9,974.64

Part (d)

the amount in the account after the payment is made on March 1, 2014 = $9,974.64 x (1 + 0.1029%)⁶ + 100 = $10,136.41

Part (e)

the amount in the account on the last payment date of February 1, 2016 = FV (Rate, Nper, PMT, PV) = FV (0.1029%, 46, -100, -10136.41) = $15,335.84