1. Time Value of Money
Present Value
Future Value
Sinking Fund
CA (DR) Prithvi Ranjan Parhi
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Concept of Interest
Simple Compounded Effective
On Original
Principal
On Original
Principal
+
All previously
earned Interest
Actual equivalent
Annual rate of
interest when
interest is credited
more than once in
a year.
E={(1+Int/Freq) Freq } -1
FVn= P(1+i)n
FVn= P0+SI
=P0+P0 (I )(n)
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Simple interest
• Simple interest is interest paid on the original principal only.
• For example, Rs.4,000 is deposited into a bank account and
the annual interest rate is 8%. How much is the interest after
4 years?
•
Use the following simple interest formula:
I = p× r × t
where p is the principal or money deposited, r is the rate of
interest ,t is time
We get:
I = p× r × t
I = 4000× 8% × 4
I = 4000× 0.08 × 4
I = Rs. 1,280 3
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Compound interest
• Compound interest is the interest earned not only on the
original principal, but also on all interests earned previously.
• A = P (1+r)n
• In other words, at the end of each year/ period , the interest
earned is added to the original amount and the money is
reinvested
If we use compound interest for the situation above, the
interest will be computed as follow:
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Compound interest
• Interest at the end of the first year: I = 4000× 0.08 × 1 = Rs.320
Your new principal per say is now 4000 + 320 = 4320
• Interest at the end of the second year: I = 4320× 0.08 × 1 = Rs.345.6
Your new principal is now 4320 + 345.6 = 4665.6
• Interest at the end of the third year:
I = 4665.6× 0.08 × 1 = Rs.373.248
Your new principal is now 4665.6 + 373.248 = 5038.848
• Interest at the end of the fourth year:
I = 5038.848 × 0.08 × 1 = 403.10784
Your new principal is now 5038.848 + 403.10784 = 5441.95584
• Total interest earned = 5441.95584 − 4000 = 1441.95584
• The difference in money between compound interest and simple interest is
1441.96 - 1280 = 161.96
As you can see, compound interest yield better result, so you make more money.
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Effective interest rate (EIR)
• Actual equivalent Annual rate of interest when
interest is credited more than once in a year.
• It Is the interest rate on a loan or financial product
restated from the nominal interest rate as an
interest rate with annual compound interest
payable in arrears.
• It is used to compare the annual interest between
loans with different compounding terms (daily,
monthly, annually, or other).
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Calculation
• The effective interest rate is calculated as if compounded annually. The
effective rate is calculated in the following way,
• E={(1+Int/Freq) Freq } -1
• For example, a nominal interest rate of 6% compounded monthly is
equivalent to an effective interest rate of 6.17%. 6% compounded
monthly is credited as 6% /12 = 0.5% every month. After one year, the
initial capital is increased by the factor (1 + 0.005)12 ≈ 1.0617.
• N.B.: Percentage figures must always be divided by 100, as the
percent sign is a notational convenience; e.g. 6% = 0.06.
• The yield depends on the frequency of compounding:
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Annual Percentage Rate (APR)
• The effective interest rate differs in two important respects from the
annual percentage rate (APR):
1. the effective interest rate generally does not incorporate one-time
charges such as front-end fees;
2. the effective interest rate is (generally) not defined by legal or regulatory
authorities (as APR is in many jurisdictions).
• By contrast, the effective APR is used as a legal term, where front-fees
and other costs can be included, as defined by local law
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FVn= P0+SI
=P0+P0 (I )(n)
Excel Sheet
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FVn= P(1+i)n
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E={(1+Int/Freq) Freq } -1
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12. Continuous
Compounding
of Interest
(Time Value of Money :
Part –V)
Classroom Deliberations
CA. Dr Prithvi Ranjan Parhi
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Continuous Compounding Interest
• Interest that is, hypothetically, computed and added to the
balance of an account every instant.
• This is not actually possible, but continuous compounding is
well-defined nevertheless as the upper bound of "regular"
compound interest.
• A= Pert
• where, P = principal amount (initial investment)
r = annual interest rate (as a decimal)
t = number of years
A = amount after time t
e stands for the Napier's number (base of the natural logarithm) which is
approximately 2.7183.
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• An amount of Rs.2,340.00 is deposited in a bank paying an
annual interest rate of 3.1%, compounded continuously. Find
the balance after 3 years.
Solution
• Use the continuous compound interest formula,
• A = Pe rt,
• with P = 2,340, r = 3.1/100 = 0.031, t = 3.
• e stands for a value, which is approximately 2.7183.
• The exponential constant (e) is an important mathematical constant Its value is
approximately 2.718. It has been found that this value occurs so frequently when
mathematics is used to model physical and economic phenomena that it is
convenient to write simply e.
• Therefore the balance after 3 years is approximately
Rs.2,568.06.
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1. P = Rs. 2,340
2. r = 0.031
3. E = 2.7183
4. t = 3
5. r*t = 0.093
6. FV = Pe^rt
7. FV = Rs. 2,568.06
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18. TVM Concepts
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A bird in hand…..
• Re 1 today ≠ Re 1 tomorrow
• Re 1 today > Re 1 tomorrow
• B’coz :-
1. Postponement of consumption : Preference for present
consumption
2. Opportunity of use : Investment Opportunities
3. Risk : Uncertainty about receipt
4. Inflation : Re today represents greater purchasing power
• Money received today can be put to use immediately. By
not putting money to use immediately, there takes
postponement of consumption & postponement of
satisfaction that is derived from consumption.
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TVM
• The compensation that is required for
postponement of consumption is called TVM.
• TVM = Rate of Inflation + Real rate of return
from Risk free investment+ Risk Premium.
= Ri + Rf + Rp
• Risk free investment is one which carries no risk.
• Risk premium is the return in excess of Rf rate of
return that would make / enthuse an investor to
invest in that investment.
• Higher the risk, higher the risk premium & higher
the TVM.
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Barometer of Reward
Inflation
(Say 6.5%)
Real rate of return
of Risk free
investment
(Return – Inflation)
(Say 9% )
Risk Premium (Say 2.5%)
15.5%
18%
Ri
Rf
Rp
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TVM
• TVM for a particular investment can be different for
different people b’coz their expectation of Rp can b
different.
• TVM for different investments in respect of same person
can b different b’coz the Rp associated with different
investment can b different.
• Present Value is the today’s value of tomorrows money.
• Future value is tomorrow’s value of today’s money.
• PV & FV are inverse of each other.
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Principal & Interest Formula
A = P (1+r)n
Where A = Amount or Future Value
P = Principal or Present Value
r = Rate of Interest or TVM
n = No. of years
(1+r)n is referred to as Future Value Factor.
1/ (1+r)n is referred to as Present Value Factor.
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Link between PV & FV : 1 yr.
Present Value Interest / TVM Future Value
100 10% 110
100 X 1.10
110 X 1 / 1.10
t0 t1
Compounding
Discounting
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Link between PV & FV : 3 yrs.
Present Value Interest / TVM Future Value
100 10% 133
100 X (1.10)3
133X (1 / 1.10)3
t0 t3
Compounding
Discounting
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Link between PV & FV
• Its TVM which links PV & FV.
• One can move from PV to FV by
compounding the cash flows by the given
no. of years @TVM.
• One can move from FV to PV by
discounting the future cash flows by the
given no. of years @TVM.
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Cash Flow Structure
Year Cash Flow (Rs.)
0 Investment
1 100
2 90
3 110
• Can we add these cash flows ?
• No, They r not addable as they are realized @
different point of time.
• To be brought to PV in order to have a
comparison.
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Evaluating Cash Flows
• Cash flows realised @ different points of
time are not addable ‘coz they do not
represent the value of money for any
common year.
Hence CFs must be converted either to
the today’s value (PV) or to the terminal
value (FV) before they are added.
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CASH FLOW PATTERNS
Single sum Annuity Perpetuity Uneven
Annuity Regular Annuity Immediate
‘n’ Payments ‘n+1’ Payments
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1. Single Sum
• PV & FV tables are to be referred.
• Future Value of a single sum
=Single sum X Future Value Factor for single sum
• Present Value of a single sum
=Single sum X Present Value Factor for single sum
pvtables.xls
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2. Annuity
• The term annuity means equal annual ( Periodic)
cash flow.
Annuity
Regular
E.g. Renting a
house
Annuity
Immediate
‘n’ Payments
E.g. Post Office RD
A/c
Annuity
Immediate
‘n+1’ Payments
E.g. PPF A/c
1st Payment 1 year latter Immediate Immediate
Last Payment is
blocked for
0 years 1 year 0 years
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Annuity
Years 0 1 2 3 Annuity Type
(3 years)
Annuity
Cash flows X √ √ √ Regular
E.g.. Renting a house
Annuity
Cash flows √ √ √ X Immediate with ‘n’
payments
E.g.. Post Office RD
A/c
Annuity
Cash flows √ √ √ √ Immediate with
‘n+1’ payments
E.g.. PPF A/c
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Annuity
• Annuity PV/FV tables are for Annuity regular
or ordinary annuity.
• Future Value of an annuity
= Annuity X Future Value Annuity Factor
• Present Value of an annuity
= Annuity X Present Value Annuity Factor
Link
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3. Perpetuity
• Perpetuity is an annuity which lasts for ever.
• Present Value of Perpetuity
=Perpetuity / TVM
E.g. = 500 / 10% = 5,000/-
• Since perpetuity is never ending, there is no
future value or terminal value of perpetuity.
• Present Value of growing Perpetuity
=Perpetuity / (TVM - growth rate)
E.g. = 500 / (10% - 2%) = 6,250/-
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4. Uneven Cash flow
• Cash flows are uneven at different years
• Each cash flow shall be treated as a single
cash flow.
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Year 0 1 2 3 4
Cash flow 500 950 100 250 1900

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Sinking Fund
• Fund created for a specified purpose by way of sequence of
periodic payments over a time period at a specified interest
rate.
• Size of the sinking fund deposit is computed at TVM.
• Maturity Value of Sinking Fund =
Future Value of an annuity
= Annuity X Future Value Annuity Factor
• Periodic Investment on sinking fund = Annuity = Maturity
Value/ FVAF
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Thank you
prithvi.baps@gmail.com
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