mathematics investing

Mathematics of Mathematics of InvestingInvestingFranklin Templeton Franklin Templeton Learning AcademyLearning Academy

What we'll coverWhat we'll cover

The Future Value EquationThe Future Value Equation Arithmetic of Accumulation StrategiesArithmetic of Accumulation Strategies Measuring RiskMeasuring Risk Asset Allocation Mathematics Asset Allocation Mathematics

The Future Value EquationThe Future Value Equation Arithmetic of Accumulation StrategiesArithmetic of Accumulation Strategies Measuring RiskMeasuring Risk Asset Allocation MathematicsAsset Allocation Mathematics

FV = PV(1 + r)FV = PV(1 + r)nn

FV = Future ValueFV = Future ValuePV = Present ValuePV = Present Valuer = Rate of Return/ Coupon Rater = Rate of Return/ Coupon Raten = No. of compounding periodsn = No. of compounding periods

Mr. Mr. BachchanBachchan plans to buy a studio after 5 plans to buy a studio after 5 years. years.

The current cost of such a studio is estimated The current cost of such a studio is estimated to be Rs.3.5 crore.to be Rs.3.5 crore.

Assuming prices rise @ 3% p.a., Assuming prices rise @ 3% p.a., how much how much will the studio be expected to cost 5 years will the studio be expected to cost 5 years down the line?down the line?FV = PV (1 + r)n

FV = 3.5 (1 + 3%)5

FV = Rs.4.057 crore

Ms. Ms. AishwaryaAishwarya invested Rs.1 crore in a noinvested Rs.1 crore in a no--load mutual fund scheme in their IPO, four load mutual fund scheme in their IPO, four years ago.years ago.

According to the latest fact sheet, the scheme According to the latest fact sheet, the scheme has shown a CAGR since inception of 10% has shown a CAGR since inception of 10% p.a.p.a.

How much is Ms. How much is Ms. Aishwarya'sAishwarya's investment investment worth today?worth today?FV = PV (1 + r)n

FV = 1 (1 + 10%)4

FV = Rs.1.464 crore

Mr. Mr. ShahrukhShahrukh plans to take his wife on a plans to take his wife on a cruise, after 4 years.cruise, after 4 years.

The cruise is expected to cost Rs.300,000 at The cruise is expected to cost Rs.300,000 at that time.that time.

Assuming the risk free rate of return to be Assuming the risk free rate of return to be 4% p.a., how much should he invest today, 4% p.a., how much should he invest today, to to realiserealise this dream, without taking any risk?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. Mr. TendulkarTendulkar dreams of sending his dreams of sending his daughter for a higher education after 4 years, daughter for a higher education after 4 years, for which he is ready to invest 350,000 today.for which he is ready to invest 350,000 today.

The education is expected to cost 500,000 at The education is expected to cost 500,000 at that time.that time.

How much should his money earn for him to How much should his money earn for him to realiserealise his dream?his dream?FV = PV (1 + r)n

r = (FV/ PV) 1/n -1r = (500000/ 350000) 1/4 -1r = 9.33% p.a.

Ms. Ms. KareenaKareena invested Rs.300,000 in different invested Rs.300,000 in different investment options. Her investments are investment options. Her investments are currently valued at Rs.400,000.currently valued at Rs.400,000.

She plans to She plans to encashencash her investments when her investments when the value crosses Rs.1,000,000the value crosses Rs.1,000,000

Assuming her investments grow @ 10% p.a., Assuming her investments grow @ 10% p.a., how soon can she expect to how soon can she expect to encashencash them?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)FV = PV(1 + r)nn

Applications aside, what do you think this Applications aside, what do you think this equation really signifies?equation really signifies?

The essence of how to create wealth!

Wealth creation is nothing but enhancement Wealth creation is nothing but enhancement of future valueof future value

FV = PV (1 + r)FV = PV (1 + r)nn

Enhancing Future ValueEnhancing Future Value

The more you The more you save, makes a save, makes a

differencedifferenceThe sooner The sooner you start, you start, makes a makes a

differencedifference

PV nr

The more The more you earn, you earn, makes a makes a

differencedifference

The more you save, makes a differenceThe more you save, makes a difference

Growth rate of 7% p.a.

Amount saved per month

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

Total Amount Saved

Value after 25 years

The sooner you start, makes a differenceThe sooner you start, makes a differenceRs. 1000 invested p.m. @ 7% p.a. till the age of 60

Starting Age

25 420,000 1,811,561

30 360,000 1,227,087

35 300,000 814,797

40 240,000 523,965

Total Amount Saved

Value at the age of 60

The more you earn, makes a differenceThe more you earn, makes a difference

Rs. 1000 invested p.m.

Growth Rate

6% 164,699 696,459

8% 184,166 957,367

10% 206,552 1,337,890

12% 232,339 1,897,635



Numbers to ponder overNumbers to ponder over

1,000 p.m.1,000 p.m. @ 15% p.a. over @ 15% p.a. over 33 years33 years ~ ~ 1 crore1 crore




Future Value, Multiple cash flowsFuture Value, Multiple cash flows

FV = CF1(1+r)FV = CF1(1+r)nn + + CF2(1+r)CF2(1+r)(n(n--1)1)+ .. + + .. + CFn(1+r)CFn(1+r)

CF=Cash flowCF=Cash flow

The Ease of ExcelThe Ease of Excel

Function DescriptionPV Present ValueNper No. of compounding periodsPmt Payment made/ received each periodRate Rate of return/ interest rate per periodFV Future Value

Points to rememberDenote outflows with a negative (-) signBe consistent about the units

Mr. Mr. DravidDravid invests Rs.200,000 in an equity fund.invests Rs.200,000 in an equity fund. He also opts for an SIP in the fund @ Rs.5000 per month.He also opts for an SIP in the fund @ Rs.5000 per month. Assuming his investment were to grow @ 11% p.a., how much Assuming his investment were to grow @ 11% p.a., how much

money can he expect to have after 10 years?money can he expect to have after 10 years?

Mr. Mr. LaxmanLaxman is planning a purchase after 3 years, the eventual is planning a purchase after 3 years, the eventual cost of which is expected to be Rs.400,000. For this he has cost of which is expected to be Rs.400,000. For this he has invested Rs.100,000 (invested Rs.100,000 (lumpsumlumpsum) in a bond fund. ) in a bond fund.

Assuming his investment grows @ 6.5% p.a., please advise him Assuming his investment grows @ 6.5% p.a., please advise him whether he will be able to achieve his goal or whether he needs 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 to do an SIP as well. If so, what should be the amount of a quarterly SIP?quarterly SIP?

Mr. Mr. GangulyGanguly has retired at the age of 60. His total investments has retired at the age of 60. His total investments as on that date are worth Rs.10 as on that date are worth Rs.10 lakhslakhs. .

He receives a pension of Rs.5000 p.m. and needs to draw He receives a pension of Rs.5000 p.m. and needs to draw another Rs.10000 p.m. from his investments.another Rs.10000 p.m. from his investments.

Assuming he lives till the age of 75 years, and is not keen on 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 leaving any money to his family, how much return should his investments earn to help him achieve his objectives?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.Ref the previous example. What if Mr. What if Mr. GangulyGanguly were to require a sum of Rs.20000 p.m. were to require a sum of Rs.20000 p.m.

from his investments for the first six months of his retirement from his investments for the first six months of his retirement and Rs.10000 p.m. thereafter?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. Mr. SehwagSehwag invested Rs.10,000 in the IPO of an equity fund on invested Rs.10,000 in the IPO of an equity fund on 29 Sep 1994 (NAV=10.00)29 Sep 1994 (NAV=10.00)

He again invested Rs.10,000 on 24 October 2000 in the same He again invested Rs.10,000 on 24 October 2000 in the same fund and plan at an NAV of 19.66.fund and plan at an NAV of 19.66.

He withdrew Rs.6000 from the fund on 8 May 2001 at an NAV He withdrew Rs.6000 from the fund on 8 May 2001 at an NAV of 20.64.of 20.64.

What would be the annualized return of Mr. What would be the annualized return of Mr. SehwagSehwag from the from the scheme as on March 31, 2003? The NAV on that date was scheme as on March 31, 2003? The NAV on that date was 22.50. Ignore loads in your calculations.22.50. Ignore loads in your calculations.

Rate, IRR & XIRRRate, IRR & XIRR

XIRRXIRRFixed/ Variable cash flows Fixed/ Variable cash flows across Irregular intervalsacross Irregular intervals

IRRIRRVariable cash flows across Variable cash flows across Regular intervalsRegular intervals

RateRateFixed cash flows across Regular Fixed cash flows across Regular intervalsintervals

Function best Function best suitedsuited

SituationSituation

Mr. Mr. HrithikHrithik has a choice between investing inhas a choice between investing in A. A 1 year bond with a coupon rate of 7% p.a., interest paid A. A 1 year bond with a coupon rate of 7% p.a., interest paid

monthlymonthly B. A 1 year bond with a coupon rate of 7.25% p.a., interest paidB. A 1 year bond with a coupon rate of 7.25% p.a., interest paid

halfhalf--yearlyyearly Which of the two would you recommend?Which of the two would you recommend?

The Future Value EquationThe Future Value Equation Arithmetic of Accumulation Arithmetic of Accumulation

StrategiesStrategies Measuring RiskMeasuring Risk Asset Allocation MathematicsAsset Allocation Mathematics

Arithmetic of Rupee Cost AveragingArithmetic of Rupee Cost Averaging

Month Amount Invested Rs.

Sale Price Rs. No. of Units Purchased

1 1000 12 83.333

2 1000 15 66.667

3 1000 9 111.111

4 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 AveragingArithmetic of Value Averaging

1 1000 12 83.33 0 83.33 10002 2000 15 133.33 83.33 50.00 7503 3000 9 333.33 133.33 200.00 18004 4000 12 333.33 333.33 0.00 0

TOTAL 48 3550

Month Amount Rs.

NAVTotal Value Rs.

Units to own Units to buy

Existing Units

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)


Which of these funds would you select? Which of these funds would you select?

Which of these funds would you select? Which of these funds would you select?

Standard DeviationStandard Deviation

It is a statistical measure of historic volatility It is a statistical measure of historic volatility of a fund/ portfolio. of a fund/ portfolio.

It measures the dispersion of a fund's It measures the dispersion of a fund's periodic returns (often based on 36 months periodic returns (often based on 36 months of monthly returns). of monthly returns).

The wider the dispersions, the larger the The wider the dispersions, the larger the standard deviation and the higher the risk.standard deviation and the higher the risk.

BetaBeta

A measure of how volatile a fundA measure of how volatile a funds past s past returns have been compared with an returns have been compared with an appropriate benchmarkappropriate benchmark

By definition, the benchmarkBy definition, the benchmarks beta is 1s beta is 1 If Beta > 1, the returns of the Fund are If Beta > 1, the returns of the Fund are

expected to rise and fall more than those of expected to rise and fall more than those of the benchmarkthe benchmark

RR--squaredsquared

A measure of how much of a fundA measure of how much of a funds past s past returns can be explained by the returns returns can be explained by the returns from the overall market (or its benchmark)from the overall market (or its benchmark)

If a fundIf a funds total return were synchronised s total return were synchronised with the overall marketwith the overall markets return, its Rs return, its R--squared would be 1.00 (100%)squared would be 1.00 (100%)

If a fund bore no relationship to the If a fund bore no relationship to the marketmarkets returns, its Rs returns, its R--squared would be 0squared would be 0

Sharpe RatioSharpe Ratio

A measure of a fundA measure of a funds risks risk--adjusted returns adjusted returns per unit of risk assumedper unit of risk assumed

The higher the ratio, the better the FundThe higher the ratio, the better the Funds s historical riskhistorical risk--adjusted performanceadjusted performance

Return Return Risk free ReturnRisk free Return

Standard DeviationStandard Deviation

DurationDuration

It is the weighted average of the It is the weighted average of the maturities of a bond's maturities of a bond's cashflowscashflows

It measures the sensitivity of the bond It measures the sensitivity of the bond price to changes in interest ratesprice to changes in interest rates


Markowitz: Markowitz: Portfolio Portfolio Selection, Selection, 1952: 1952: Key conclusion: Key conclusion: Dividing a portfolio Dividing a portfolio over asset classes over asset classes that do not move that do not move up/ down at the up/ down at the same time helps same time helps bring down the risk bring down the risk of the portfolio.of the portfolio.



CorrelationCorrelation

Correlation mCorrelation measures the extent to which the easures the extent to which the returns of a group of investment options have returns of a group of investment options have moved together over time. It ranges from moved together over time. It ranges from 1 1 to +1to +199+1 = the movement of two funds has been +1 = the movement of two funds has been

exactly the same i.e. perfect positive correlation.exactly the same i.e. perfect positive correlation.99 --1 = the two funds have moved in diametrically 1 = the two funds have moved in diametrically

opposite directions i.e. perfect negative opposite directions i.e. perfect negative correlation.correlation.

Rate of Return & Asset AllocationRate of Return & Asset Allocation

Return Derived from Asset Allocation

Asset Allocation Derived from Return

i

i

Making Asset Allocation workMaking Asset Allocation work

60%60%40%40%Switch from Growth Funds to Switch from Growth Funds to Income Funds to rebalanceIncome Funds to rebalance

55%55%45%45%Market fluctuationsMarket fluctuations60%60%40%40%Frozen AllocationFrozen Allocation

Income Income FundsFunds

Growth Growth FundsFunds

REBALANCING REBALANCING AN EXAMPLEAN EXAMPLE

REBALANCING HELPS INVESTORS ENTER REBALANCING HELPS INVESTORS ENTER EQUITIES AT EQUITIES AT LOWSLOWS AND EXIT AT AND EXIT AT HIGHSHIGHSWITHOUT HAVING TO WITHOUT HAVING TO TIMETIME THE MARKETTHE MARKET

Bull Market conditionsBull Market conditions

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