fadhilconsult.files.wordpress.com · web viewinterest can either be simple or compound simple...
TRANSCRIPT
2
Compound Interestis based
on the fact thatas
interest is calculated
(EVERY COMPOUNDING PERIOD)
it isadded
to the
balance
Simple Interest is calculated on
the original principle
Therefore, the amount of interest each period remains
the same
SIMPLE INTEREST IS SUCH THAT
COMPOUNDING DOES NOT OCCUR
The amount of simple
FINANCIALMATHEMATICS
Interest can either be simple or compound
Simple Interest Calculated only on the original principle.
Compound Interest Calculated on the principle and on
3
oraccumulated sum
interest is directly proportional to time.
ANNUITY
An ANNUITYrepresentsa stream
ofregular payments
usually of
equal amount
MostORDINARY ANNUITIES
AndANNUITIES DUEfall under the heading
ofSIMPLE ANNUITIES
which means thatINTEREST
is compoundedor
charged at the same frequencyas the
PAYMENTS
ORDINARY ANNUITIES
describe the situationwhere
INTERESTis
compoundedor
chargedat
DIFFERENT TIMESto when
PAYMENTSare made
ORDINARY ANNUITYAn Ordinary Annuity isone whose payments aremade at the end of each
period.
The value at time zero is the present value
4
Ordinary Annuities
Annuities mean same amount of cash flows for a limited number of periods,
Ordinary annuity means that receipts or payments are made at the end of the period.
Discount Rate = r 0.0083Cash Flow = C 300No of Periods = n 30
10.0083 0.0083 1 + 0.0083 30
PV Annuity = 7,938.02$
1PV
Annuity= 300 -
Discount Rate = r 0.0083Cash Flow = C 300No of Periods = n 30
1 - 1 + -30
PV Annuity = 7,938.02$
PVAnnuity
= 300 0.00830.0083
5
Growth Annuity
Discount Rate = r 0.1Growth Rate = g 0.04Cash Flow = C 100000No of Periods = n 20
1 + 20
0.1 - 0.04 0.1 - 0.04 x 1 + 0.1 20
PV Annuity = 1123839.178
PVAnnuity
= 100000 1 - 0.04
8.3 I30 n
300 +/- PMTCOMP PV
= 7938.02
6
ANNUITY DUE
An Annuity Due is one whose payments are made at the
beginning of each period.
The first payment is due immediately
Place calculator in BEGIN mode
Discount Rate = rCash Flow = CNo of Periods = n
1 - 1 + -2
PV Annuity =
0.10.1
273.55$
0.1100
3
PVAnnuity
= 100 + 100
7
10 I3 n
100 +/- PMTCOMP PV
= 273.55
DEFERRED ANNUITY
A Deferred Annuity Due is one whose payments commence only after a certain number of periods have elapsed.
A Deferred Annuity due means that receipts or payments begin at some time in the future.
1Discount Rate Discount Rate 1 + Discount Rate no of periods
1 + Discount Rate no of periods
-1
PVAnnuity
= Cash Flow
Discount Rate = r 0.1Cash Flow = C 100000No of Periods = n 20No of Periods before beginning 4
10.1 0.1 1 + 0.1 20
1 + 0.1 4
PV Annuity = 581,487.86$
= 100000 -1
PVAnnuity
8
GENERAL ANNUITIES
Note that the frequency with which the annuity payments are made does affect the present value of the investment. For example, payments are made monthly, but the interest may be compounded daily. Such annuities are called General Annuities. In order to value such annuities, the payment intervals and the compounding intervals should be the same.
Two MethodsExample
Suppose you plan to buy a car and finance it with a loan.
10 I20 n
100000 +/- PMTCOMP PV
= 851356.37
10 I4 n
851356.37 +/- FVCOMP PV
= 581487.86
Step 1
Step 2
9
You believe you can afford repayments of $100 per month. The bank will give you a three-year loan at an interest rate of 14 per cent per year
compounded daily with payments monthly. What sort of car can you afford to buy? Assume that ‘monthly’ loan payments means a payment every four weeks (28
days), which totals 39 payments over the three years.
10
Convert the interest rate to the effective rate for the payment period.
The effective interest rate per day is
Effective daily rate = 0.14/365 = 0.0003836, or 0.03836%
We convert this to an effective ‘monthly” rate as
Effective ‘monthly’ rate = (1 + 0.0003836)28 -1 = 0.010796, or 1.0796%
To find out how much you can borrow we simply calculate the present value of an annuity for 39 payments at 1.0796 per cent interest per ‘month’.
10.010796 0.010796 1 + 0.010796 39
PV Annuity = 3,169.28$
PVAnnuity
= 1001
-
Convert the payments to EACs that correspond with the compounding period.
Convert the $100 per month to an equivalent annuity cash flow per day.
Treat the $100 monthly payment as the future value of a 28-day annuity, with daily payments and a daily interest rate of 0.03836 per cent.
Solving for the daily payment in the following equation gives us the daily EAC:
1Discount Rate Discount Rate 1 + Discount Rate no of periodsPV
Annuity= -
1Cash Flow
11
TO CALCULATE EACAnnual Days
Discount Rate = r 0.00038 0.14 365Payment = C 100No of Periods = n 28
1Discount Rate Discount Rate 1 + Discount Rate no of periods
10.00038 0.000383562 1 + 0.000383562 28
= EAC 27.84$
= 3.5913$ EAC
100
-1
= EAC -1
Payment
100
= Cash Flow
Now we find the value of the loan that runs for 1092 days (39 x 28 days)
1Discount Rate Discount Rate 1 + Discount Rate no of periods
10.000384 0.000384 1 + 0.000384 1092
PV Annuity = 3,203.53$
-1
PVAnnuity
= 3.59133 -1
PVAnnuity
= Cash Flow
12
PERPETUIY
AnANNUITY
whosePAYMENTS
areintended
toCONTINUEFOREVER
PERPETUITYPRESENT VALUE
(Simple ordinary Perpetuity)
R = Payment (PMT)i = Interest
Note*** There is no Future Value for Perpetuities
The present value of perpetuities that are not growing is calculated using this formula:
Cash FlowDiscount RatePerpetuity
PV =
Cash Flow 100Discount Rate 0.07
1000.07
PV Perpetuity = 1,428.57$
PV Perpetuity
=
The present value of perpetuities that are growing is calculated using this formula, that is, cash flow (C) is constant.
Discount Rate - Growth RateCash FlowPV
Perpetuity=
Cash Flow 6500Discount Rate 0.12Growth Rate 0.05
0.12 - 0.05
PV Perpetuity = 92,857$
6500PV Perpetuity
=
15
The formula used to convert nominal to AER is as follows:
NOMINAL INTEREST RATE Nominal interest rates are interest rates before taking out the effects of inflation.
16
Real Nominal RateInterest = Interest - of
Rate Rate Inflation
Nominal Real RateInterest = Interest + of
Rate Rate Inflation
therefore
Example
17
When the bank quotes you a 10 per cent interest rate it is quoting you a nominal interest rate. The rate tells you how rapidly your money will grow:
11001.06
= 1,037.74$
The bank account offers a l0 per cent nominal rate of return or the bank offers a 3.774 per cent expected real rate of return.
The formula that relates the nominal interest rate to the real rate is:
In our example
1.10 = (1.03774)(1.06)
However, with an inflation rate of 6 per cent you are only 3.774 per cent better off at the end of the year than
at the start: