# Pmt Calculation Formula - DOC by byu58938

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```									PRACTICE MATHS TEST 2 – Solutions & Comments
1.    Compound interest, single payment, FV = \$12,500 i = 1%/qtr, n = 20, calculate PV,
2.    Simple interest, S = \$12,500, (R = 4%, T = 5) OR (R = 1%, T = 20), calculate P,
3.    Compound interest, n = 80, i = 1.6%. Comparing an annuity with the value of a lump sum.
Two ways to solve this:
a. Compare FV of the \$20,000 payments (an annuity due) with FV of prize (a single
payment PV = \$800,000), OR
b. Compare PV of the payments with the value of the prize.
Both methods will give you the same answer – i.e. that the series of payments is more
attractive..
FV payments = \$3,251,685.56                             FV prize = \$2,848,305.87
PV payments = \$913,296.74                 PV prize = \$800,000
(I would make my decision using the PV method because it involves less calculations.)
Furthermore compare the ratio of:
FV payments/FV prize and PV payments/PV prize.
Both are equal to 1.1416209

4.    Hire purchase, this means simple interest.
a. Total costs = repayments plus the deposit
= \$5000 * 0.15 + \$100 * 48                                          Answer = \$5,550
b. How much is interest. Amount borrowed = \$5000 - \$750 deposit = \$4250,
Repayments = \$4800 =>                                  I = \$550
c. I = PRT, therefore R = I/PR                                         Answer = 3.23529% pa
5.    Simple interest, P = \$7500. R = 6%, T = 7, calculate S,                        Answer = \$10,650
6.    Compound interest single payment, PV = 7500, i = 6%, n = 7, calculate FV;
7.    Compound interest single payment. Note; only one payment but a series of different interest
rates
a. FV = PV * (1 + i1)n1 * (1 + i2)n2 etc;                              Answer = \$12,124.06
b. I = FV – PV;                                                        Answer = \$2,124.06
c. Calculate i, FV from part b, PV = \$10,000, n = total number of months = 36.

i = 36 √(FV/PV) - 1

Calculation gives i = 0.536453%/mth
This is a monthly rate that would be expressed as an annual rate of 6.437436 % pa
convertible monthly.
However the effective rate is an annual rate determined using the following:
e = (1 + r)m – 1 = (1 + 0.00536453)12 – 1 =                        Answer = 6.663081% pa

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8.    A compound interest question. There is a series of payments so we are dealing with an
annuity. The lump sum is to fund the series of payments => lump sum before the payments
=> we are dealing with a present value calc. The allowance is paid at the end of the month
meaning an ordinary annuity.

The formula we are dealing with is PV Ordinary Annuity. i = .04/12 = 0.00333333333,
n = 4yrs x 12 months = 48, PMT = \$250/mth.                          Answer = \$11,072.21
9.    Home loans are always compound interest. Again a series of payments so an annuity
question. Payments made in arrears meaning an ordinary annuity. Lump Sum being the
amount borrowed comes before the repayments so a PV calc.

The formula we are dealing with is PV Ordinary Annuity. PV = \$320,000,
i = 0.0725/12 = 0.0060416666, n = 25 yrs x 12 months = 300, Calculate PMT.
10. I = n PMT – PV                                                           Answer = \$373,894
11. Very similar to Q9. A series of payments following a lump sum, therefore dealing with the PV
of an annuity. However first payment occurs immediately so we are dealing with an Annuity
Due.

The formula we are dealing with is PV Ordinary Due. PV = \$430,000, i = 0.08/4 = 0.02,
n = 28 yrs x 4 qtrs = 112, Calculate PMT.                        Answer = \$9,461.10
12. A hire purchase question so Simple Interest
a. Amount Borrowed = Cost – Deposit = 1250 – 125               Answer = \$1,125
b. I = PRT
P = Amount Borrowed
R = 0.14 and T = 3 OR T = 36 and R = 0.14/36                Answer = \$472.50
c. Monthly Payment = S/N
S=P+I
N = 36                                                      Answer = \$44.38/mth
d. E = 2RN/(N + 1)                                             Answer = 27.24% pa
13. A Simple Interest question. P = 8000, S = 16,000, T = 4, calculate R.
Note: The fact that interest is paid quarterly in not relevant for this calculation as we are
dealing with simple interest. However, this is not the case with the next question.
R = I/PT                                                                Answer = 25% pa
14. A Compound Interest question with a single payment so basic formula is FV = PV * (1 + 1)n.
FV = 16000, PV = 8000, n = 16

Rearranging formula to i = n √(FV/PV) - 1 gives a quarterly interest rate of 4.427%/qtr.
15. Regular monthly deposits so an annuity (meaning compound interest). Also we are told that
interest compounds. Payments at start of month so an Annuity Due. Payments are
accumulating, meaning that they occur before the lump sum, hence a FV calculation.

Formula = FV Annuity Due with PMT = \$80, i = 0.03/12, n = 16 * 12 + 7 = 199. Calculate FV.

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16. A series of payments made at the start of each two month period. Hence an Annuity Due.
Payments are made to accumulate to the lump sum, so lump sum is after the payments
meaning FV calculation.

Formula = FV Annuity Due with FV = \$53,000, n = 8 x 6 (there are 6 periods of two months
each year) and i = 0.05/6. Calculate PMT                   Answer = \$895.05/two mths
17. Home loan means compound interest and with a series of repayments an annuity. Payments
made at the start of the month so an Annuity Due. Borrowing occurs before repayments
hence a PV calculation.

Formula = PV Annuity Due, PV = \$390,000, i = 0.084/12, n = 25 x 12, calculate PMT.
18. Two annuity questions in one here.
a. A series of payments (\$700) made in advance. Hence annuity due. Payments occur
from the lump sum, i.e., lump sum before payments meaning a PV calc.
Formula = PV Annuity Due with i = 0.06/12, n = 18 mths, PMT = \$700, calculate FV.
b. A series of payments required to accumulate to the \$12,081.04 from part a.
Payments made at end of fortnight => Ordinary Annuity. Lump Sum occurs after the
fortnightly deposits hence a FV calculation.
Formula = FV Ordinary Annuity with i = 0.0525/26, n = 3 * 26 = 78, FV = \$12,081.04,
19. Very similar in principle to Q Error! Reference source not found.3. We have to compare
\$48,000 with the value of the series of payments (or annuity). Note that the comparisons are
NOT with \$60,000. Whilst this is the amount that is owed to us now, it is NOT the lump sum
being offered in settlement of the debt. Payments to be made in arrears, hence an ordinary
annuity.

i = 0.1/2, n = 10, PMT = \$10,000

Compare either
PV Ordinary Annuity (\$46,330.41) with \$48,000, OR
FV Ordinary Annuity (\$75,467.36) with FV of \$48,000 (\$78,186.94)
The lump sum settlement of \$48,000 now is more attractive than the series of payments.
20. Two separate calculations here.
a. Simple Interest, S = P + I, P = \$30,650, R = 0.09, T = 10, Calculate S,
b. Compound Interest, a single payment so FV = PV (1 + i)n., PV = \$30,650, i = 0.05/4,
n = 10 x 40, Calculate FV.                                Answer = \$50,376.94
 Answer from a is greater than answer from b, so take the simple interest
option
 NOTE: Be careful to use S = P + I for part A, Using I = PRT and
comparing this with the answer from part B means you are not comparing
the same thing.
21. Simple Interest P = 10000, S = 15000, R = 0.04 (note this is a half yearly rate)

S = P (1 + RT), solve for T gives 12.5 half years.                     Answer = 6.25 years.
22. Compound Interest, single payment, PV = 10000, FV = 15000, i = 0.04 (again this is the half
yearly rate). Calculate n = log(FV/PV)/log(1 + i) 10.338 half years.

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23. Hire purchase, therefore simple interest.
a. Amount Paid under HP = deposit plus payments = \$2,148.        Cash Price = \$1,800
b. E = 2NR/(N + 1) To calculate effective interest rate (E) we need N (= 104) and R. To
do this need to solve for R. Pay’ts = S/N, S = P (1 + RT); S = \$19.50 x 104,
P = \$1800 - \$120, Use T = 2 years to give an annual R = 10.357%
24. The Consumer Price Index has increased from 147.6 to 161.5 over a 3 year period. What has
been the average annual inflation rate.
Remember compound interest calculations do not just involve money – they can be used for
any situation where there is compound growth. This question is an adaptation of
FV = PV (1 + i)n. FV = 161.5, PV = 147.6, n = 3, inflation rate = i Answer = 3.045% pa
25. This is an annuity question. Annuity Due because deposits made at start of the month with
i = 0.5%.
To work out interest earned in 9th year calculate FV at end of 8th year and FV at end of 9th
year. Interest earning during 9th year = FV9 – FV8 less payments made during the 9th year.
FV9, (n = 108) = \$14,345.36, FV8 (n = 96) = \$12,344.27, Payments during the year = 1200
26. This question centres around the PV of an annuity due.
a. With i = 0.096/12 = 0.008, n = 30 x 12 = 360, Reshma will make 12 payments of
\$2,356.00 during the year.                                 Answer = \$28,272.00
b. The reduction in principal during the 5th year will be PV after loan has been in force 5
years, ie PV25 less PV after loan has been in force for 4 years, PV26. Note; we use
25 and 26 years here because this is the amount of the remaining obligation to the
bank. (Original contract was that you pay the bank off by means of 30 years of
monthly instalments.)
The amount of interest paid during the year
=     total payments made during the year
less the principal reduction during the year