Mathematics of
Investing
Franklin Templeton
Learning Academy
What we'll cover
The Future Value Equation
Arithmetic of Accumulation Strategies
Measuring Risk
Asset Allocation Mathematics
The Future Value Equation
Arithmetic of Accumulation Strategies
Measuring Risk
Asset Allocation Mathematics
FV = PV(1 + r)n
FV = Future Value
PV = Present Value
r = Rate of Return/ Coupon Rate
n = No. of compounding periods
Mr. Bachchan plans to buy a studio after 5
years.
The current cost of such a studio is estimated
to be Rs.3.5 crore.
Assuming prices rise @ 3% p.a., how much
will the studio be expected to cost 5 years
down the line?
FV = PV (1 + r)n
FV = 3.5 (1 + 3%)5
FV = Rs.4.057 crore
Ms. Aishwarya invested Rs.1 crore in a noload mutual fund scheme in their IPO, four
years ago.
According to the latest fact sheet, the scheme
has shown a CAGR since inception of 10%
p.a.
How much is Ms. Aishwarya's investment
worth today?
FV = PV (1 + r)n
FV = 1 (1 + 10%)4
FV = Rs.1.464 crore
Mr. Shahrukh plans to take his wife on a
cruise, after 4 years.
The cruise is expected to cost Rs.300,000 at
that time.
Assuming the risk free rate of return to be
4% p.a., how much should he invest today,
to realise this dream, without taking any risk?
FV = PV (1 + r)n
PV = FV/ (1 + r)n
PV = 300000/ (1 + 4%)4
PV = Rs.256,441
Mr. Tendulkar dreams of sending his
daughter for a higher education after 4 years,
for which he is ready to invest 350,000 today.
The education is expected to cost 500,000 at
that time.
How much should his money earn for him to
realise his dream?
FV = PV (1 + r)n
r = (FV/ PV) 1/n -1
r = (500000/ 350000)
r = 9.33% p.a.
1/4
-1
Ms. Kareena invested Rs.300,000 in different
investment options. Her investments are
currently valued at Rs.400,000.
She plans to encash her investments when
the value crosses Rs.1,000,000
Assuming her investments grow @ 10% p.a.,
how soon can she expect to encash them?
FV = PV (1 + r)n
n = log(FV/ PV)/ log(1+r)
n = log(1000000/ 400000)/ log(1+10%)
n = 9.6 years
FV = PV(1 + r)n
Applications aside, what do you think this
equation really signifies?
The essence of how to create wealth!
Enhancing Future Value
Wealth creation is nothing but enhancement
of future value
n
PV
r
FV = PV (1 + r)
The more you
save, makes a
difference
The more
you earn,
makes a
difference
The sooner
you start,
makes a
difference
The more you save, makes a difference
Growth rate of 7% p.a.
Total Amount
Saved
Value after
25 years
5,000
1,500,000
4,073,986
3,000
900,000
2,444,391
1,500
450,000
1,222,196
1,000
300,000
814,797
Amount saved per month
The sooner you start, makes a difference
Rs. 1000 invested p.m. @
7% p.a. till the age of 60
Total Amount
Saved
Value at the
age of 60
25
420,000
1,811,561
30
360,000
1,227,087
35
300,000
814,797
40
240,000
523,965
Starting Age
The more you earn, makes a difference
Rs. 1000 invested p.m.
Value after 10
years
Value after
25 years
6%
164,699
696,459
8%
184,166
957,367
10%
206,552
1,337,890
12%
232,339
1,897,635
Growth Rate
Numbers to ponder over
1,000 p.m. @ 15% p.a. over 33 years ~ 1 crore
5,000 p.m. @ 15% p.a. over 22 years ~ 1 crore
10,000 p.m. @ 15% p.a. over 18 years ~ 1 crore
15,000 p.m. @ 15% p.a. over 15 years ~ 1 crore
Future Value, Multiple cash flows
FV = CF1(1+r)n +
CF2(1+r)(n-1)+ .. +
CFn(1+r)
CF=Cash flow
The Ease of Excel
Function
PV
Nper
Pmt
Rate
FV
Description
Present Value
No. of compounding periods
Payment made/ received each period
Rate of return/ interest rate per period
Future Value
Points to remember
Denote outflows with a negative (-) sign
Be consistent about the units
Mr. Dravid invests Rs.200,000 in an equity fund.
He also opts for an SIP in the fund @ Rs.5000 per month.
Assuming his investment were to grow @ 11% p.a., how much
money can he expect to have after 10 years?
Mr. Laxman is planning a purchase after 3 years, the eventual
cost of which is expected to be Rs.400,000. For this he has
invested Rs.100,000 (lumpsum
(lumpsum)) in a bond fund.
Assuming his investment grows @ 6.5% p.a., please advise him
whether he will be able to achieve his goal or whether he needs
to do an SIP as well. If so, what should be the amount of a
quarterly SIP?
Mr. Ganguly has retired at the age of 60. His total investments
as on that date are worth Rs.10 lakhs.
lakhs.
He receives a pension of Rs.5000 p.m. and needs to draw
another Rs.10000 p.m. from his investments.
Assuming he lives till the age of 75 years, and is not keen on
leaving any money to his family, how much return should his
investments earn to help him achieve his objectives?
Simple Annualized Return:
0.73% X 12 = 8.76%
Compounded Annualized
Return:
(1 + 0.73%)12 - 1 = 9.12%
Ref the previous example.
What if Mr. Ganguly were to require a sum of Rs.20000 p.m.
from his investments for the first six months of his retirement
and Rs.10000 p.m. thereafter?
Simple Annualized Return:
0.82% X 12 = 9.84%
Compounded Annualized Return:
(1 + 0.82%)12 - 1 = 10.30%
Mr. Sehwag invested Rs.10,000 in the IPO of an equity fund on
29 Sep 1994 (NAV=10.00)
He again invested Rs.10,000 on 24 October 2000 in the same
fund and plan at an NAV of 19.66.
He withdrew Rs.6000 from the fund on 8 May 2001 at an NAV
of 20.64.
What would be the annualized return of Mr. Sehwag from the
scheme as on March 31, 2003? The NAV on that date was
22.50. Ignore loads in your calculations.
Situation
Rate, IRR & XIRR
Function best
suited
Fixed cash flows across Regular Rate
intervals
Variable cash flows across
Regular intervals
IRR
Fixed/ Variable cash flows
across Irregular intervals
XIRR
Mr. Hrithik has a choice between investing in
A. A 1 year bond with a coupon rate of 7% p.a., interest paid
monthly
B. A 1 year bond with a coupon rate of 7.25% p.a., interest paid
halfhalf-yearly
Which of the two would you recommend?
The Future Value Equation
Arithmetic of Accumulation
Strategies
Measuring Risk
Asset Allocation Mathematics
Arithmetic of Rupee Cost Averaging
Month
Amount Invested Sale Price Rs. No. of Units
Rs.
Purchased
1000
12
83.333
1000
15
66.667
1000
111.111
1000
12
83.333
TOTAL
4000
48
344.444
Average Sales Price of Units : Rs. 12 ( i.e. Rs. 48/4 months)
Average Purchase Cost of Units : Rs. 11.61 ( i.e. Rs. 4000/344.444
units)
Arithmetic of Value Averaging
Month
1
2
3
4
TOTAL
Total
Value
Rs.
1000
2000
3000
4000
NAV
Units to own
Existing
Units
Units to
buy
Amount
Rs.
12
15
9
12
48
83.33
133.33
333.33
333.33
0
83.33
133.33
333.33
83.33
50.00
200.00
0.00
1000
750
1800
0
3550
Average Sales Price of Units : Rs. 12 ( i.e. Rs. 48/4 months)
Average Purchase Cost of Units : Rs. 10.65 ( i.e. Rs. 3550/333.33
units)
The Future Value Equation
Arithmetic of Accumulation Strategies
Measuring Risk
Asset Allocation Mathematics
Which of these funds would you select?
Which of these funds would you select?
Standard Deviation
It is a statistical measure of historic volatility
of a fund/ portfolio.
It measures the dispersion of a fund's
periodic returns (often based on 36 months
of monthly returns).
The wider the dispersions, the larger the
standard deviation and the higher the risk.
Beta
measure of how volatile a funds past
returns have been compared with an
appropriate benchmark
By definition, the benchmarks beta is 1
If Beta > 1, the returns of the Fund are
expected to rise and fall more than those of
the benchmark
R-squared
A measure of how much of a funds past
returns can be explained by the returns
from the overall market (or its benchmark)
If a funds total return were synchronised
with the overall markets return, its Rsquared would be 1.00 (100%)
If a fund bore no relationship to the
markets returns, its R-squared would be 0
Sharpe Ratio
Return Risk free Return
Standard Deviation
A measure of a funds risk-adjusted returns
per unit of risk assumed
The higher the ratio, the better the Funds
historical risk-adjusted performance
Duration
It is the weighted average of the
maturities of a bond's cashflows
It measures the sensitivity of the bond
price to changes in interest rates
The Future Value Equation
Arithmetic of Accumulation Strategies
Measuring Risk
Asset Allocation Mathematics
Markowitz: Portfolio
Selection, 1952:
Key conclusion:
Dividing a portfolio
over asset classes
that do not move
up/ down at the
same time helps
bring down the risk
of the portfolio.
Markowitz: Portfolio
Selection, 1952:
Key conclusion:
Dividing a portfolio
over asset classes
that do not move
up/ down at the
same time helps
bring down the risk
of the portfolio.
Markowitz: Portfolio
Selection, 1952:
Key conclusion:
Dividing a portfolio
over asset classes
that do not move
up/ down at the
same time helps
bring down the risk
of the portfolio.
Correlation
Correlation measures the extent to which the
returns of a group of investment options have
moved together over time. It ranges from 1
to +1
9 +1 = the movement of two funds has been
exactly the same i.e. perfect positive correlation.
9 -1 = the two funds have moved in diametrically
opposite directions i.e. perfect negative
correlation.
Rate of Return & Asset Allocation
Return Derived from Asset Allocation
Asset Allocation
Derived from Return
Making Asset Allocation work
REBALANCING AN EXAMPLE
Frozen Allocation
Market
fluctuations
Bull
Market
conditions
Switch from Growth Funds to
Income Funds to rebalance
Growth
Funds
40%
45%
40%
Income
Funds
60%
55%
60%
REBALANCING HELPS INVESTORS ENTER
EQUITIES AT LOWS
LOWS AND EXIT AT HIGHS
HIGHS
WITHOUT HAVING TO TIME
TIME THE MARKET
