Topic: Time Value of Money
BY
KAJAL VIPANI
TIME VALUE OF MONEY
FUTURE VALUE OF SINGLE AMOUNT
FUTURE VALUE OF AN ANNUITY
PRESENT VALUE OF AN ANNUITY
PRESENT VALUE OF SINGLE AMOUNT
Present Value of Perpetuity
Impact of Compounding Frequency on
Effective Rate of Interest
Rule of 72 And Rule of 69
Year-0 Year-1 Year-2 Year-3
Amount 1000 1000 1000 1000
Market 50 40 100 250
Price
Units 20 25 10 4
you can
purchase
Simple Interest and Compound Interest
Suppose Principal = P = 1000, r= 8%
Simple Interest Compound Interest
Year Starting Ending Starting Ending
Interest Interest
balance Balance balance Balance
0 1000 0 1000 1000 0 1000
1 1000 80 1080 1000 80 1080
2 1080 80 1160 1080 86.4 1166.4
5 1320 80 1400 1360.4890 108.8391 1469.3281
10 1720 80 1800 1999.0046 159.9204 2158.9250
15 2120 80 2200 2937.1936 234.9755 3172.1691
20 2520 80 2600 4315.7011 345.2561 4660.9571
25 2920 80 3000 6341.1807 507.2945 6848.4752
12000
10000
8000
6000 compound interest
simple interest
4000
2000
0 1 2 3 4 5 6 7 8
INVESTMENT PLANS
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FUTURE VALUE
Future Value of a Single Amount
Future Value of an Annuity
Shorter period of time OR Shorter
compounding periods for different frequency
Frequency: Monthly, Semi-annually or Quarterly
Example
Suppose you invest INR 3,000 for three years
in a savings account that pays 5% interest per
year. What will be the growth of your
investment after 3 years if you let your interest
income be reinvested?
Solution:
PV= 3,000 r = 5% or 0.05 n = 3 FV=?
Compounding = Investing Money +
Reinvestment of interest earned on
PV = 3000, r=0.05 or 5%, n=3
PV = 3000, r=0.05 or 5%, n=3
years Then, FV=?
years. Then, FV=?
3000+ 3000*5% = 3000 + 150
3000 + 3000 * (5% + 5% + 5%)
= 3150
= 3000 + 3000* 15%
3150+ 3150*5%= 3150 + 157.5
= 3000 + 3000* (15/100)
= 3307.5
= 3000 + 3000 *0.15
3307.5 + 3307.5*5% = 3307.5 +
= 3000 + 450
165.375 = 3472.875
FV = 3450
FV =3472.875
Formula for Future value of single
amount
FVn =PV*(1+r)n
FVn =PV* FVIF (r,n)
Where as,
(1+r)n = Future Value Interest Factor
= FVIF (r,n)
= FVF (r,n)
OR
(1+r)n = Compounded Value Interest Factor
= CVIF (r,n)
=CVF (r,n)
WITHOUT THE USE OF WITH THE USE OF TABLE
TABLE VALUE VALUE
PV = 3000, r=0.05 or 5%, n=3
PV = 3000, r=0.05 or 5%, n=3 years Then, FV=?
years. Then, FV=?
FVn =PV(1+r)n
FVn =PV(1+r)n FV = PV*FVIF (3,5%)
FV3 = 3000(1+0.05)3 TABULATED VALUE from Present
= 3000(1.05)3 value and Future value tables
= 3000* (1.05*1.05*1.05)
= 3000* 1.157625 FV3 =3000*1.1576
= 3472.875 = 3472.8
Practice Exercise
Sum-1: If you deposit Rs.3,000 today in a bank
which pays 5% interest compounded annually,
How much will the deposit grow to after 7
years and 25 years?
Sum-2: Calculate the value of 5 years hence if
you deposit Rs. 4,000 today and the interest
rate is, (i) 8% or (ii) 10% or (iii) 12% or (iv)
15%
Calculate the value for all 4 options.
SUM-1:
Option-1: 4221.30
Option-2: 10,159.06
SUM-2:
(i) 5,877.2 (ii) 6442.04 (iii) 7049.36 (iv)
8045.44
Annuity
Annuity refers to a stream of constant cash flows may be
payments or receipts occurring at regular intervals of time.
Example: EMI of loans, premium of insurance policy, Interest
received on deposits, etc.
Ordinary/ •Cash Flows occur at the end of each period
Deferred Annuity
•Cash Flows occur at the beginning of each
Annuity Due period
FUTURE VALUE OF AN ANNUITY
Formula:
FVAn = A*(1+r)n-1 + A*(1+r)n-2+………
FVAn = A*[{(1+r)n-1} / r]
FVAn = A* FVIFA(r,n)
Whereas,
FVA = Future value of an annuity
A = Constant periodic flow
[(1+r)n-1] / r = Future Value Interest Factor of an Annuity
= FVIFA(r,n)
=FVFA(r,n)
Example
Suppose you have decided to deposit Rs. 4,000
per year in your child’s minority account for
10 years. What will be the accumulated
amount in that account at the end of 10 years if
interest rate is 7%?
Solution:
Annual deposit = A =Rs. 4,000
n=10 years,
r= 7% or 0.07
FVA =?
FVAn = A* [{(1+r)n-1} / r]
Solution:
Annual deposit = A =Rs. 4,000 n=10 years, r= 7% or 0.07
FVA =?
FVAn = A* [{(1+r)n-1} / r]
FVA = 4,000 * [{ (1+0.07) 10 -1)/ 0.07]
= 4,000* [{(1.07) 10 -1)}/0.07]
= 4,000*[{1.9672-1}/0.07]
= 4,000* (0.9672/0.07)
= 4,000* 13.82
= 55,265.7919
= 55,266
Practice Exercise
SUM-3: Twelve annual payments of 6,000 are
made into a deposit account that pays 17%
interest per year. What will be the future value
of this annuity at the end of 12 years?
ANSWER: 1,96,944
SHORTER PERIOD OF TIME
FV = PV (1 + (r/m))m*n
Where as,
m= Frequency of compounding per year
r= Annual rate of Interest
n= Number of years
Example
Suppose you deposit Rs. 10,000 with an investment company which pays
16% interest with quarterly compounding. How much will this deposit grow
to in 5 years?
Solution:
PV = 10,000 , r =0.16 or 16%, n=5, m=4 times per year
FV = ?
FV = PV*(1 + (r/m))m*n
FV = PV*(1 + (r/m))m*n
=
10,000(1+ (0.16/4)) 4*5
= 10,000(1+0.04) 20
=10,000(1.04) 20
= 10,000*2.1911
= 21,911.23
=Rs. 21,911
Practice Exercise
SUM-4: How much would a deposit of
5,000 at the end of 6 years be if the
interest rate is 12% and if the
compounding is done Quarterly?
Answer: 10,164
PRESENT VALUE
Present Value of a Single Amount
Present Value of An Annuity
Present Value of an Uneven Cash Flow
Retirement Plan
Shorter period of time OR Shorter
Discounting periods
PRESENT VALUE OF A SINGLE
AMOUNT
FORMULA:
PV= FVn *[1/ (1+r)n]
PV = FV * PVIF (r,n)
Where as,
1/ (1+r)n = Present Value Interest Factor
= PVIF (r,n)
= PVF (r,n)
OR
1/(1+r)n = Discounted Value Interest Factor
= DVIF (r,n)
=DVF (r,n)
Example
Find the present value of Rs. 12,000 receivable after 8 years if the rate of
discount is,
(a)
10% (b) 12% (c) 15%
Solution: FV=12,000 n=8 PV=?
PV= FVn [1/ (1+r)n]
(a)
If r =0.10 or 10%
PV = 12,000*[1/ (1+0.10) 8]
= 12,000*[1/(1.08) 8]
=12,000* (0.9090) 8
= 12,000*0.4665
= 5,598
Practice Exercise
(b) 4846.8
(c) 3922.8
SUM-5:Suppose someone promises to give you
Rs. 50,000 four years hence. What is the
present value of this amount if the interest rate
is 12%?
PRESENT VALUE OF AN ANNUITY
FORMULA:
PVAn = A/(1+r) + A/(1+r)2 +……...+ A/(1+r)n-1+ A/(1+r)n
PVAn = A [1/(1+r) + 1/(1+r)2 +…………..+ 1/(1+r)n-1+ 1/(1+r)n
PVAn = A [{1- (1/(1+r)n )}/r]
PVAn = A* PVIFA(r,n)
Whereas,
1- (1+r)n/r =PVIFA(r,n)
PVAn = Present Value of an annuity
PVIFA(r,n) = Present Value Interest factor for an annuity
Example
What is the present value of a 5-year annuity
of Rs. 2,000 at 10%?
Solution:
A=2,000 r=0.10 or 10% n=5 PVA=?
PVAn = A*[{1- (1/(1+r)n )}/r]
PVAn = A*[{1- (1/(1+r)n )}/r]
= 2,000*[{1- (1/(1+0.10) 5)}/0.10]
= 2,000*[{1-(1/(1.10) 5 )}/0.10]
=2,000*[{1-(1/1.61051)}/0.10]
= 2,000*[(1-0.6209)/0.10]
= 2,000*[0.3791/0.10]
= Rs. 7,582
Rate = 10% PVCF
(1/1+0.10) n Present
Year Cash Flow r = 10%
(1/1.10) n Value of
Cash Flow
(1/1.10) 1
1 2,000 0.9091 1818.2
2 2,000 (1/1.10) 2 0.8264 1652.8
3 2,000 (1/1.10) 3 0.7513 1502.6
4 2,000 (1/1.10) 4 0.6830 1366
5 2,000 (1/1.10) 5 0.6209 1241.8
PVA 7581.4
Practice Exercise
SUM-6:Suppose one expects to receive
4,000 Rs. annually for 6 years at the end
of each year. What is the present value of
this stream of benefits if the discount rate
is 7%?
Answer: 19,066
PRESENT VALUE OF UNEVEN
CASHFLOW
What is the Present Value of following Cash flow
streams if the discount rate is 12%
End of the Stream A Stream B Stream C
year
1 100 1000 500
2 200 900 500
3 300 800 500
4 400 700 500
5 500 600 500
6 600 500 500
7 700 400 500
8 800 300 500
Solution for stream-A:
PV at 12%
CF = Cash r= 0.12
YEAR PVCF
Flow (1/1.12) n
1 100 0.8929 89.29
2 200 0.7972 159.44
3 300 0.7118 213.54
4 400 0.6355 254.2
5 500 0.5674 283.7
6 600 0.5066 303.96
7 700 0.4523 316.61
8 800 0.4039 323.12
PVA 1943.86
Stream – B : 3520.5
Stream- C: 2483.75
Practice Exercise
SUM-7: What is the present value of following
cash stream if discounted rate is 10%
Year 0 1 2 3 4 5 6 7
Cash 2000 4000 1000 0 2500 2000 3500 4800
flow
RETIREMENT PLAN
At the time of Retirement, Mr. X is given a
choice between two alternatives:
(a) An annual pension of Rs. 8,000 as he long
lives.
(b)A lump sum amount of Rs. 40,000.
If Mr. X expects to live for 15 years and the
interest rate is 15%, which option appears
more attractive?
Solution:
Option-(a): Option-(b):
A= 8,000 Rs. 40,000
r= 15% or 0.15
n= 15
PVA=?
Option-(a) Calculation
PVAn = A*[{1- (1/(1+r)n )}/r]
= 8,000[{1-(1/(1+0.15) 15 )}/0.15]
= 8,000 [{1-(1/(1.15) 15 )}/0.15]
= 8,000[{1-(1/8.137)}/0.15]
= 8,000 [{1- 0.1229}/0.15]
= 8,000{0.8771/0.15}
= 46,778.67
= Rs. 46,779
Shorter period of time or Shorter
Discounting Period
Formula :
PV = FV [1/ {1+(r/m)} m*n ]
Where as,
m= Frequency of discounting per year
r= Annual rate of Interest
n= Number of years
Example
Consider a cash flow of Rs. 15,000 to be
received at the end of 2 years. Determine the
Present value of this cash flow when the
discount rate is 12% and
(a) Discounting is done monthly.
(b)Discounting is done semi-annually
(c) Discounting is done quarterly
Solution:
FV = 15,000 r=0.12 or 12% n=2
(a) m=12 times per year
PV = FV [1/ {1+(r/m)}] m*n
= 15,000*[1/ {1+(0.12/12)}] 12*2
= 15,000* [1/ {1+0.01}] 24
= 15,000* [1/ 1.01] 24
= 15,000* 0.78761
= 11,813.49
= 11,813
(b) 11,881
(c) 11,841
PRESENT VALUE OF A PERPETUITY
PERPETUITY = AN ANNUITY OF INFINITE DURATION
Formula:
P∞ = A*PVIFA (r, ∞)
P∞ = A* (1/r)
Whereas,
P∞ = Present value of a perpetuity
A= Constant annual Payment
PVIFA (r, ∞) =
PVIFA (r, ∞) = 1/r
Example
Find Present Value of perpetuity if constant
yearly cash flow will be Rs.12,000 at a interest
rate of 16%.
Solution:
A= Rs.12,000, r=0.16 or 16%, P∞= ?
P∞ = A* (1/r)
= 12,000 (1/0.16)
= 75,000
RULE OF 72 and RULE of 69
• How long would it take to double the amount
at a given interest rate?
Rule of 69 = More
Rule of 72
accurate
Doubling period Doubling Period
= 72 / Interest Rate = 0.35 + (69 / Interest Rate)
Example:
Example:
If interest rate is 10%, the
If interest rate is 10%, Doubling
doubling period is 7.2 years
period is 7.25 years (0.35 +
(72/10).
(69/10))
If interest rate is 18%, the
doubling period is 4 years
(72/18).
Practice Exercise
SUM-8:If you deposit Rs. 8,000 today at 12%
rate of interest in how many years (roughly)
will this amount grow to Rs. 1,28,000?
Work this problem using the Rule of 72 and
Rule of 69
Effective Rate of Interest
Impact of Compounding Frequency on
Nominal Rate of Interest
Effective Interest Rate = {1+(r/m)} m] – 1
m= Frequency of discounting or compounding
per year
r= Nominal or Stated rate of Interest
Example
A borrower offers 13% nominal rate of interest with
quarterly compounding. What is the Effective rate of
Interest?
Solution:
r = 0.13 m= 4 times per year
Effective Interest Rate = {1+(r/m)} m] – 1
= [{1+ (0.13/4)} 4 ] – 1
= (1+0.0325) 4 - 1
Effective Interest Rate = 0.1364
Effective Interest Rate= 13.64%
THANK YOU
