Question

Paul takes out a 15-year loan of 250,000 from his bank. The bank charges interest at 4% p.a. compounded half-yearly. During the first 10 years, Paul repays $11,000 at the end of each 6 months. After that period, Paul will repay $X at the end of each year for the remaining 5 years. Which of the following can be used to calculate $X.

Answer 1

Loan Amortization Schedule:

Period	Opening Balance	Instalment	Int	Principal Repay	Closing Balance
1	$ 2,50,000.00	$ 11,000.00	$ 5,000.00	$ 6,000.00	$ 2,44,000.00
2	$ 2,44,000.00	$ 11,000.00	$ 4,880.00	$ 6,120.00	$ 2,37,880.00
3	$ 2,37,880.00	$ 11,000.00	$ 4,757.60	$ 6,242.40	$ 2,31,637.60
4	$ 2,31,637.60	$ 11,000.00	$ 4,632.75	$ 6,367.25	$ 2,25,270.35
5	$ 2,25,270.35	$ 11,000.00	$ 4,505.41	$ 6,494.59	$ 2,18,775.76
6	$ 2,18,775.76	$ 11,000.00	$ 4,375.52	$ 6,624.48	$ 2,12,151.27
7	$ 2,12,151.27	$ 11,000.00	$ 4,243.03	$ 6,756.97	$ 2,05,394.30
8	$ 2,05,394.30	$ 11,000.00	$ 4,107.89	$ 6,892.11	$ 1,98,502.19
9	$ 1,98,502.19	$ 11,000.00	$ 3,970.04	$ 7,029.96	$ 1,91,472.23
10	$ 1,91,472.23	$ 11,000.00	$ 3,829.44	$ 7,170.56	$ 1,84,301.67
11	$ 1,84,301.67	$ 11,000.00	$ 3,686.03	$ 7,313.97	$ 1,76,987.71
12	$ 1,76,987.71	$ 11,000.00	$ 3,539.75	$ 7,460.25	$ 1,69,527.46
13	$ 1,69,527.46	$ 11,000.00	$ 3,390.55	$ 7,609.45	$ 1,61,918.01
14	$ 1,61,918.01	$ 11,000.00	$ 3,238.36	$ 7,761.64	$ 1,54,156.37
15	$ 1,54,156.37	$ 11,000.00	$ 3,083.13	$ 7,916.87	$ 1,46,239.50
16	$ 1,46,239.50	$ 11,000.00	$ 2,924.79	$ 8,075.21	$ 1,38,164.29
17	$ 1,38,164.29	$ 11,000.00	$ 2,763.29	$ 8,236.71	$ 1,29,927.57
18	$ 1,29,927.57	$ 11,000.00	$ 2,598.55	$ 8,401.45	$ 1,21,526.13
19	$ 1,21,526.13	$ 11,000.00	$ 2,430.52	$ 8,569.48	$ 1,12,956.65
20	$ 1,12,956.65	$ 11,000.00	$ 2,259.13	$ 8,740.87	$ 1,04,215.78

Opening Balance = Previous month closing balance
EMI = Instalment calculated
Int = Opening Balance * Int Rate
Principal repay = Instalment - Int
Closing Balance = Opening balance - Principal Repay

Balance to be repaid after 10 Years is 104215.78

Let x be the CF for balance 10 periods ( 5 * 2 periods )

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
PV Annuity	$      1,04,215.78
Int Rate	2.000%
Periods	10

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 104215.78 / [ 1 - [(1+0.02)^-9]] /0.02
= $ 104215.78 / [ 1 - [(1.02)^-9]] /0.02
= $ 104215.78 / [ 1 - 0.8203 ] /0.02
= $ 104215.78 / [0.1797 / 0.02 ]
= $ 104215.78 / 8.9826
= $ 11601.98
Balance to be paid for each period during balance 10 periods is $11601.98