Question

Helen borrows $20000 to be repaid over 15 years with level annual payments with an annual effective interest rate of 8%. The first payment is due one year after she takes out the loan. Helen pays an additional $4000 at the end of year 9 (in addition to her normal payment). At that time (the end of year 9) she negotiates to pay off the remaining principal at the end of year 14 with a sinking fund. The sinking fund accumulates at an annual effective interest rate of 7%. Helen will make level annual payments and she will also make annual interest payments at an annual effective interest rate of 10%. You may assume all payments are made at the end of the year. Determine Helen’s total annual outlay starting with year 10.

Answer 1

Formula for calculating Present Value of Annuity is, PV= A*[ (1+r)^n -1]/[(1+r)^n * r]

PV = 20000

I (Interest Rate) = 8%

Repayment Tenure = 15 years

Using the above formula,

20000 = A [((1+.08)^15)-1] / [((1+.08)^15)*.08]

A = 2336.59.

Installement amount Normal Scenario
No of Installment	Value of Install	Interest	Principal	Balance	Interest %
1	$                                     2,336.59	$ 1,600.00	$                                                     736.59	$ 20,000.00	8%
2	$                                     2,336.59	$ 1,541.07	$                                                     795.52	$ 19,263.41	8%
3	$                                     2,336.59	$ 1,477.43	$                                                     859.16	$ 18,467.89	8%
4	$                                     2,336.59	$ 1,408.70	$                                                     927.89	$ 17,608.73	8%
5	$                                     2,336.59	$ 1,334.47	$                                                 1,002.12	$ 16,680.84	8%
6	$                                     2,336.59	$ 1,254.30	$                                                 1,082.29	$ 15,678.72	8%
7	$                                     2,336.59	$ 1,167.71	$                                                 1,168.88	$ 14,596.42	8%
8	$                                     2,336.59	$ 1,074.20	$                                                 1,262.39	$ 13,427.54	8%
9	$                                     2,336.59	$     973.21	$                                                 1,363.38	$ 12,165.16	8%
10	$                                     2,336.59	$     864.14	$                                                 1,472.45	$ 10,801.78	8%
11	$                                     2,336.59	$     746.35	$                                                 1,590.24	$    9,329.33	8%
12	$                                     2,336.59	$     619.13	$                                                 1,717.46	$    7,739.09	8%
13	$                                     2,336.59	$     481.73	$                                                 1,854.86	$    6,021.62	8%
14	$                                     2,336.59	$     333.34	$                                                 2,003.25	$    4,166.76	8%
15	$                                     2,336.59	$     173.08	$                                                 2,163.51	$    2,163.51	8%
			$                                                              -	$            0.00	8%

The payment is computed using the following formula:
Payment = Total Accumulated * (Interest Rate / 100) / ((1 + Interest Rate / 100)** Payment Period) - 1
Using Above formula We receive			932.64

Starting with Year 10	Installment with Addition	Interest	Principal	Balance		Comment
9	$                                     6,336.59	$     973.21	$                                                 5,438.40	$    5,363.38	10%	Paid intallament amount and additional $ 4000
10	paying only interest	$     536.34
11	paying only interest	$     536.34
12	paying only interest	$     536.34
13	paying only interest	$     536.34
14	paying only interest	$     536.34
Sinking Fund Deposit				Interest Earned
10			$                                                     932.64	$          65.28	7%
11			$                                                     932.64	$          69.85	7%
12			$                                                     932.64	$        140.03	7%
13			$                                                     932.64	$        215.12	7%
14			$                                                     932.64	$        295.46	7%
			$                                                 4,663.20	$        785.74
					$ 5,448.94

at the end of the 14th year her sinking fund balance will cover the total oustanding.