Mathematics of Investing

8/13/2019 Mathematics of Investing

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Mathematics of Investing


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February 12, 2014 2

What we'll cover

The Future Value EquationAsset Allocation Mathematics

Arithmetic of Accumulation Strategies

Arithmetic of Financing Life


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February 12, 2014 3

The Future Value EquationAsset Allocation MathematicsArithmetic of Accumulation Strategies



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February 12, 2014 4

FV = PV(1 + r)nFV = Future Value

PV = Present Value

r = Rate of Return/ Coupon Rate

n = No. of compounding periods


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February 12, 2014 5

Mr. A Bachchan plans to buy a house after 5 years.The current cost of such a house is estimated to beRs. 35 lakhs.

Assuming property prices rise @ 3% p.a., how much

will the house be expected to cost 5 years down theline?

FV = PV (1 + r)nFV = 35 (1 + 3%)5

FV = 40.57 lakhs


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February 12, 2014 6

Ms M Dixit invested Rs. 10 lakh in a no-load mutualfund scheme in their IPO, four years ago.

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

How much is Ms Dixit's investment worth today?

FV = PV (1 + r)nFV = 10 (1 + 10%)4

FV = 14.64 lakhs


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February 12, 2014 7

Mr A Devgan has a dream to take his wife on a luxurycruise in the Caribbean, after 4 years.

The cruise is expected to cost Rs.3 lakhs at that time.

Assuming the risk free rate of return to be 7% p.a.,

how much should he invest today, to realise thisdream, without taking any risk?

FV = PV (1 + r)nPV = FV/ (1 + r)n

PV = 3/ (1 + 7%)4

PV = 2.29 lakhs


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February 12, 2014 8

Mr J Shroff dreams of sending his daughter toHarvard after 4 years, for which he is ready to invest35 lakhs today.

The education is expected to cost Rs.50 lakhs at thattime.

How much should his money earn for him to realisehis dream?

FV = PV (1 + r)nr = (FV/ PV) 1/n-1

r = (50/ 35) 1/4-1

PV = 9.33% p.a.


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February 12, 2014 9

Ms A Rai invested Rs.30 lakhs in different investmentoptions. Her investments are currently valued atRs.40 lakhs.

She plans to encash her investments and retire whenthe value crosses Rs.1 crore.

Assuming her investments grow @ 10% p.a., howsoon can she expect to retire?

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

n = log(100/ 40)/ log(1+10%)

PV = 9.6 years


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February 12, 2014 10

FV = PV(1 + r)n

Applications aside, what do you think this equationreally signifies?

The essence of how to create wealth!


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Wealth creation is nothing but enhancement of future value

FV = PV (1 + r)n

Enhancing Future Value

The more yousave, makes a

difference

The sooneryou start,makes a

difference

PV nr

The moreyou earn,

makes adifference


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Themore you save,makesadifference

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


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Thesooneryoustart,makesadifference

Rs. 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


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Themoreyouearn,makesadifference

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

Value after 10

years

Value after

25 years


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Future Value, Multiple cash flows

FV = CF1(1+r)n+

CF2(1+r)

(n-1)

+ .. + CFn(1+r)CF=Cash flow


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The Ease of Excel

Function Description

PV Present Value

Nper No. of compounding periods

Pmt Payment made/ received each period

Rate Rate of return/ interest rate per period

FV Future Value

Points to rememberDenote outflows with a negative (-) sign

Be consistent about the units


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Mr S Tendulkar invests Rs. 2 lakhs 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 muchmoney can he expect to have after 10 years?


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Mr R Dravid is planning a holiday in Switzerland after 3 years,

the eventual cost of which is expected to be Rs 4 lakhs. For thishe has invested Rs 1 lakh (lumpsum) in an income fund.

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

quarterly SIP?


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Mr S Ganguly has retired at the age of 60. His total investments

as on that date are Rs 10 lakhs.

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

Assuming he lives till the age of 75 years, and is not keen onleaving any money to his family, how much return should hisinvestments earn to help him achieve his objectives?

Simple Annualized Return:

0.73% X 12 = 8.76%

Compounded AnnualizedReturn:

(1 + 0.73%)12- 1 = 9.12%


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Ref the previous example.

What if Mr Ganguly were to require a sum of Rs 20000 p.m.from his investments only for the first six months of hisretirement?

Simple Annualized Return:

0.82% X 12 = 9.84%

Compounded Annualized Return:

(1 + 0.82%)12- 1 = 10.30%


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Mr V Sehwag invested Rs 10000 in the IPO of Prima Plus

(Growth Option).

He again invested Rs 10000 on 24 October 2000 in the samescheme and plan at an NAV of 19.66.

He withdrew Rs.6000 from the scheme on 8 May 2001 at anNAV of 20.64.

What would be the annualized return of Mr Sehwag from thescheme as on March 31, 2003? The NAV on that date was22.50. Ignore loads in your calculations.


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Rate, IRR & XIRR

Situation Function bestsuited

Fixed cash flows across Regularintervals

Rate

Variable cash flows acrossRegular intervals

IRR

Fixed/ Variable cash flowsacross Irregular intervals

XIRR


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Mr H Roshan has a choice between investing in

A. A 1 year bond with a coupon rate of 7% p.a., interest paidmonthly

B. A 1 year bond with a coupon rate of 7.25% p.a., interest paidhalf-yearly

Which of the two would you recommend?


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The Future Value Equation

Asset Allocation Mathematics




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Markowitz: PortfolioSelection, 1952:Dividing aportfolio over asset classes

that do not move up/ downat the same time helpsbring down the risk of theportfolio.

Significance Relative to Risk


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Rate of Return & Asset Allocation

Return Derived from Asset Allocation

Asset AllocationDerived fromReturn


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Periodic Rebalancing

EXAMPLE Growth Funds Income Funds

Frozen Allocation 40% 60%

45% 55%

Switch from Growth Funds to IncomeFunds to rebalance

40% 60%

REBALANCING HELPS INVESTORS ENTER

EQUITIES AT LOWS AND EXIT AT HIGHSWITHOUT HAVING TO TIME THE MARKET

Making Asset Allocation Work

Bull Market skew


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Arithmetic of Rupee Cost Averaging

Month Amount Invested(Rs.)

Sale Price(Rs.)

No. of UnitsPurchased

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. 4000/344.444units)


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Arithmetic of Value Averaging

1 12 1000 83.33 83.33 1000

2 15 2000 133.33 50.00 750

3 9 3000 333.33 200.00 1800

4 12 4000 333.33 0.00 0

TOTAL 48 3550

Month Amount

Invested

(Rs.)

Sale

Price

(Rs.)

Total

Value

Units to own Units to

buy

Average Sales Price of Units : Rs. 12 ( i.e. Rs. 48/4 months)Average Purchase Cost of Units : Rs 10.65 ( i.e. Rs. 3550/333.33units)


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Context of Investing

PROFESSION INVESTMENTS

INCOME

CURRENT

EXPENDITURE

SAVINGS

FUTUREEXPENDITURE


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Thank you

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Mathematics of Investing