FinQuiz - Curriculum Note, Study Session 2-3, Reading 5-12_Quant

8/17/2019 FinQuiz - Curriculum Note, Study Session 2-3, Reading 5-12_Quant

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Reading 5 The Time Value of Money

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• Money has a time value because a unit of money received today is worth more than a unit of money to be receivedtomorrow.

2. INTEREST RATES: INTERPRETATION

Interest rates can be interpreted in three ways.

1)

Required rates of return: It refers to the minimum rateof return that an investor must earn on his/herinvestment.

2) Discount rates: Interest rate can be interpreted as therate at which the future value is discounted toestimate its value today.

3) Opportunity cost: Interest rate can be interpreted asthe opportunity cost which represents the returnforgone by an investor by spending money todayrather than saving it. For example, an investor canearn 5% by investing $1000 today. If he/she decides tospend it today instead of investing it, he/she will forgo

earning 5%.

Interest rate = r = Real risk-free interest rate + Inflationpremium + Default risk premium +Liquidity premium + Maturitypremium

• Real risk-free interest rate: It reflects the single-periodinterest rate for a completely risk-free security when

no inflation is expected.• Inflation premium: It reflects the compensation forexpected inflation.Nominal risk-free rate = Real risk-free interest rate +

Inflation premium o E.g. interest rate on a 90-day U.S. Treasury bill (T-bill)

refers to the nominal interest rate.• Default risk premium: It reflects the compensation fordefault risk of the issuer.

• Liquidity premium: It reflects the compensation forthe risk of loss associated with selling a security at avalue less than its fair value due to high transactioncosts.

• Maturity premium: It reflects the compensation forthe high interest rate risk associated with long-termmaturity.

3. THE FUTURE VALUE OF A SINGLE CASH FLOW

The future value of cash flows can be computed usingthe following formula:

where,

PV = Present value of the investmentFVN = Future value of the investment N periods from

todayPmt = Per period payment amountN = Total number of cash flows or the number of a

specific periodr = Interest rate per period(1 + r)N = FV factor

Example:

Suppose,PV = $100, N = 1, r = 10%. Find FV.

• The interest rate earned each period on the originalinvestment (i.e. principal) is called simple interest e.g.$10 in this example.

Simple interest = Interest rate × Principal

If at the end of year 1, the investor decides to extendthe investment for a second year. Then the amountaccumulated at the end of year 2 will be:

or

•

Note that FV2> FV1 because the investor earnsinterest on the interest that was earned in previousyears (i.e. due to compounding of interest) inaddition to the interest earned on the originalprincipal amount.

• The effect of compounding increases with theincrease in interest rate i.e. for a given compoundingperiod (e.g. annually), the FV for an investment with10% interest rate will be > FV of investment with 5%interest rate.


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NOTE:

• For a given interest rate, the more frequently thecompounding occurs (i.e. the greater the N), thegreater will be the future value.

• For a given number of compounding periods, thehigher the interest rate, the greater will be the futurevalue.

Important to note:

Both the interest rate (r) and number of compoundingperiods (N) must be compatible i.e. if N is stated inmonths then r should be 1-month interest rate, un-annualized.

3.1 The Frequency of Compounding

With more than one compounding period per year,

where,

r s = stated annual interest rate

m = number of compounding periods per year

N = Number of years

Stated annual interest rate: It is the quoted interest ratethat does not take into account the compounding

within a year.

Stated annual interest rate = Periodic interest rate ×

Number of compounding

periods per year

Periodic interest rate = r s / m = Stated annual interest

rate / Number of

compounding periods

per year

where,

Number of compounding periods per year = Number of

compounding periods in one year × number of years =m×N

NOTE:

The more frequent the compounding, the greater will bethe future value.

Example:

Suppose,

A bank offers interest rate of 8% compounded quarterlyon a CD with 2-years maturity. An investor decides toinvest $100,000.

• PV = $100,000• N = 2• r s = 8% compounded quarterly• m = 4• r s / m = 8% / 4 = 2%• mN = 4 (2) = 8

FV = $100,000 (1.02)8 = $117,165.94

3.2 Continuous Compounding

When the number of compounding periods per yearbecomes infinite, interest rate is compoundedcontinuously. In this case, FV is estimated as follows:

where,

e = 2.7182818

• The continuous compounding generates themaximum future value amount.

Example:

Suppose, an investor invests $10,000 at 8% compoundedcontinuously for two years.

FV = $10,000 e 0.08 (2) = $11,735.11

3.3 Stated and Effective Rates

Periodic interest rate = Stated annual interest rate /Number of compounding periodsin one year (i.e. m)

E.g. m = 4 for quarterly, m = 2 for semi-annuallycompounding, and m = 12 for monthly compounding.

Effective (or equivalent) annual rate (EAR = EFF %): It isthe annual rate of interest that an investor actually earnson his/her investment. It is used to compare investmentswith different compounding intervals.

EAR (%) = (1 + Periodic interest rate) m– 1

• Given the EAR, periodic interest rate can becalculated by reversing this formula.

Periodic interest rate = [EAR(%) + 1]1/m –1

For example, EAR% for 10% semiannual investment willbe:

m = 2stated annual interest rate = 10%EAR = [1 + (0.10 / 2)] 2 – 1 = 10.25%

Practice: Example 4, 5 & 6,Volume 1, Reading 5.

Practice: Example 1, 2 & 3,Volume 1, Reading 5.


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• This implies that an investor should be indifferentbetween receiving 10.25% annual interest rate andreceiving 10% interest rate compounded

semiannually .

EAR with continuous compounding:

EAR = ers – 1

• Given the EAR, periodic interest rate can becalculated as follows:

EAR + 1 = ers

• Now taking the natural logarithm of both sides wehave:

ln (EAR + 1) = ln e rs (since ln e = 1)ln (EAR + 1) = rs

Now taking the natural logarithm of both sides we have:

EAR + 1 = lners (since ln e = 1)

EAR + 1 = r s

NOTE:

Annual percentage rate (APR): It is used to measure thecost of borrowing stated as a yearly rate.

APR = Periodic interest rate × Number of paymentsperiods per year

4. THE FUTURE VALUE OF A SERIES OF CASH FLOWS

Annuity:

Annuities are equal and finite set of periodic outflows/inflows at regular intervals e.g. rent, lease, mortgage,car loan, and retirement annuity payments.

• Ordinary Annuity: Annuities whose payments beginat the end of each period i.e. the 1st cash flowoccurs one period from now (t = 1) are referred to asordinary annuity e.g. mortgage and loan payments.

• Annuity Due: Annuities whose payments begin at the start of each period i.e. the 1st cash flow occursimmediately (t = 0) are referred to as annuity duee.g. rent, insurance payments.

Present value and future value of Ordinary Annuity:

The future value of an ordinary annuity stream iscalculated as follows:

FVOA = Pmt [(1+r)N–1 + (1+r)N–2 + … +(1+r)1+(1+r)0]

−

where,

Pmt = Equal periodic cash flows

r = Rate of interest

N = Number of payments, one at the end of each

period (ordinary annuity).

The future value of an ordinary annuity stream is

calculated as follows:

⋯

Or

−

−

Present value and future value of Annuity Due:

The present value of an annuity due stream is calculatedas follows (section 6).

( )( )0

1

11 1

=+

+

−

=−

t at Pmt r

r Pmt PV

N

AD

Or

( ))1(

111

r

r

r Pmt PV

N

AD +

+

−

=

PVAD = PVOA+ Pmtwhere,

Pmt = Equal periodic cash flows

r = Rate of interest

N = Number of payments, one at the beginning of

each period (annuity due).

• It is important to note that PV of annuity due > PV ofordinary annuity.


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NOTE:

PV of annuity due can be calculated by settingcalculator to “BEGIN” mode and then solve for the PV ofthe annuity.

The future value of an annuity due stream is calculatedas follows:

( ) ( )rr

r Pmt FV

N

AD +

−+= 111

Or

FVAD = FVOA × (1 + r)

• It is important to note that FV of annuity due >FV ofordinary annuity.

Example:

Suppose a 5-year, $100 annuity with a discount rate of10% annually.

Calculating Present Value for Ordinary Annuity:

Or

−

Using a Financial Calculator: N= 5; PMT = –100; I/Y= 10; FV=0; CPTPV= $379.08

Calculating Future Value for Ordinary Annuity:FVOA =100(1.10)4+100(1.10)3+100(1.10)2+100(1.10)1+100=610.51

Or

( )51.610

10.0

110.1100

5

=

−=

OA FV

Using a Financial Calculator: N= 5; PMT = -100; I/Y = 10;PV=0; CPTFV = $610.51

Annuity Due: An annuity due can be viewed as = $100lump sum today + Ordinary annuity of $100per period for four years.

Calculating Present Value for Annuity Due:

−

Calculating Future value for Annuity Due:

−

4.2 Unequal Cash Flows

Source: Table 2.

FV at t = 5 can be calculated by computing FV of eachpayment at t = 5 and then adding all the individual FVse.g. as shown in the table above:

FV of cash flow at t =1 is estimated asFV = $1,000 (1.05) 4 = $1,215.51

5.1 Finding the Present Value of a Single Cash Flow

The present value of cash flows can be computed usingthe following formula:

• The PV factor = 1 / (1 + r) N; It is the reciprocal of theFV factor.

Practice: Example 7, 11, 12 & 13,Volume 1, Reading 5.


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NOTE:

• For a given discount rate, the greater the number ofperiods (i.e. the greater the N), the smaller will be thepresent value.

• For a given number of periods, the higher thediscount rate, the smaller will be the present value.

5.1 The Frequency of Compounding

With more than one compounding period per year, where,

r s = stated annual interest rate

m = number of compounding periods per year

N = Number of years

6.2The Present Value of an Infinite Series of Equal

Cash Flows i.e. Perpetuity

Perpetuity: It is a set of infinite periodic outflows/ inflows

at regular intervals and the 1st

cash flow occurs oneperiod from now (t=1). It represents a perpetual annuitye.g. preferred stocks and certain government bondsmake equal (level) payments for an indefinite period oftime.

PV = Pmt / r

This formula is valid only for perpetuity with levelpayments.

Example:

Suppose, a stock pays constant dividend of $10 peryear, the required rate of return is 20%. Then the PV is

calculated as follows.

PV = $10 / 0.20 = $50

6.3 Present Values Indexed at Times Other than t =0

Suppose instead of t = 0, first cash flow of $6 begin at theend of year 4 (t = 4) and continues each year thereaftertill year 10. The discount rate is 5%.

• It represents a seven-year Ordinary Annuity.

a)

First of all, we would find PV of an annuity at t = 3 i.e.N = 7, I/Y = 5, Pmt = 6, FV = 0, CPTPV 3 = $34.72

b) Then, the PV at t = 3 is again discounted to t = 0.

N = 3, I/Y = 5, Pmt = 0, FV = 34.72, CPT PV 0 = $29.99

• An annuity can be viewed as the difference

between two perpetuities with equal, levelpayments but with different starting dates.

Example:

• Perpetuity 1: $100 per year starting in Year 1 (i.e. 1st payment is at t =1)

• Perpetuity 2: $100 per year starting in Year 5 (i.e. 1st payment is at t = 5)

• A 4-year Ordinary Annuity with $100 payments peryear and discount rate of 5%.

4-year Ordinary annuity = Perpetuity 1 – Perpetuity 2

PV of 4-year Ordinary annuity = PV of Perpetuity 1 – PVof Perpetuity 2

i. PV0 of Perpetuity 1 = $100 / 0.05 = $2000ii. PV4 of Perpetuity 2 = $100 / 0.05 = $2000iii. PV0 of Perpetuity 2 = $2000 / (1.05) 4 = $1,645.40iv. PV0 of Ordinary Annuity = PV 0 of Perpetuity 1 - PV 0

of Perpetuity 2= $2000 - $1,645.40= $354.60

6.4The Present Value of a Series of Unequal Cash

Flows

Suppose, cash flows for Year 1 = $1000, Year 2 = $2000,Year 3 = $4000, Year 4 = $5000, Year 5 = 6,000.

A. Using the calculator’s “CFLO” register , enter the cashflows

• CF0 = 0• CF1 = 1000• CF2 = 2000• CF3 = 4000

Practice: Example 15,Volume 1, Reading 5.



Practice: Example 8 & 9,Volume 1, Reading 5.


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• CF4 = 5000• CF5 = 6000

Enter I/YR = 5, press NPVNPV or PV = $15,036.46

Or

B. PV can be calculated by computing PV of eachpayment separately and then adding all theindividual PVs e.g. as shown in the table below:

Source: Table 3.

7.1 Solving for Interest Rates and Growth Rates

An interest rate can be viewed as a growth rate (g).

g = (FVN/PV)1/N –1

7.2 Solving for the Number of Periods

N = [ln (FV / PV)] / ln (1 + r)

Suppose, FV = $20 million, PV = $10 million, r = 7%.Number of years it will take $10 million to double to $20million is calculated as follows:

N = ln (20 million / 10 million) / ln (1.07) = 10.24 ≈ 10 years

7.3 Solving for the Size of Annuity Payments

Annuity Payment = Pmt =

Suppose, an investor plans to purchase a $120,000

house; he made a down payment of $20,000 andborrows the remaining amount with a 30-year fixed-ratemortgage with monthly payments.

• The amount borrowed = $100,000• 1st payment is due at t = 1• Mortgage interest rate = 8% compounding monthly.o PV = $100,000o r s = 8%o m = 12o Period interest rate = 8% / 12 = 0.67%o N = 30o mN = 12 × 30 = 360

−

−

Pmt = PV / Present value annuity factor

= $100,000 / 136.283494 = $733.76

• Thus, the $100,000 amount borrowed is equivalent to360 monthly payments of $733.76.

IMPORTANT Example:

Calculating the projected annuity amount required tofund a future-annuity inflow.

Suppose Mr. A is 22 years old. He plans to retire at age 63(i.e. at t = 41) and at that time he would like to have aretirement income of $100,000 per year for the next 20years. In addition, he would save $2,000 per year for thenext 15 years (i.e. t = 1 to t = 15) by investing in a bondmutual fund that will generate 8% return per year onaverage.

So, to meet his retirement goal, the total amount heneeds to save each year from t = 16 to t = 40 isestimated as follows:

Calculations:

It should be noted that:

PV of savings (outflows) must equal PV of retirementincome (inflows)

a) At t =15, Mr. A savings will grow to:

−

b) The total amount needed to fund retirement goal i.e.

PV of retirement income at t = 15 is estimated usingtwo steps:

i. We would first estimate PV of the annuity of$100,000 per year for the next 20 years at t = 40.

−

ii. Now discount PV 40 back to t = 15. From t = 40 to t

= 15 total number of periods (N) = 25.

N = 25, I/Y = 8, Pmt = 0, FV = $981,814.74, CPT PV

= $143,362.53

• Since, PV of savings (outflows) must equal PV ofretirement income (inflows)The total amount he needs to save each year (fromt = 16 to t = 40) i.e.

Practice: Example 17 & 18,Volume 1, Reading 5.


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Annuity = Amount needed to fund retirement goals- Amount already saved

= $143,362.53 - $54,304.23 = $89,058.30• The annuity payment per year from t = 16 to t = 40 isestimated as:Pmt = PV / Present value annuity factoro PV of annuity = $89,058.30o N = 25o r = 8%

Annuity payment = pmt = $89,058.30 / 10.674776

= $8,342.87

Source: Example 21, Volume 1, Reading 5.

7.4 Equivalence Principle

Principle 1: A lump sum is equivalent to an annuity i.e. if

a lump sum amount is put into an accountthat generates a stated interest rate for al lperiods, it will be equivalent to an annuity.

Examples include amortized loans i.e. mortgages, carloans etc.

Example:

Suppose, an investor invests $4,329.48 in a bank today at5% interest for 5 years.

$

$ • Thus, a lump sum initial investment of $4,329.48 cangenerate $1,000 withdrawals per year over the next5 years.

• $1,000 payment per year for 5 years represents a 5-year ordinary annuity.

Principle 2: An annuity is equivalent to the FV of thelump sum.

For example from the example above stated.FV of annuity at t = 5 is calculated as:

N = 5, I/Y = 5, Pmt = 1000, PV = 0,CPTFV = $5,525.64And the PV of annuity at t = 0 is:

N = 5, I/Y = 5, Pmt = 0, FV = 5,525.64,CPT PV =$4,329.48.

7.5 The Cash Flow Additivity Principle

The Cash Flow Additivity Principle: The amounts ofmoney indexed at the same point in time are additive.

Example:

Interest rate = 2%.Series A’s cash flows:

t = 0 0

t = 1 $100

t = 2 $100

Series B’s cash flows:

t = 0 0

t = 1 $200t = 2 $200

• Series A’s FV = $100 (1.02) + $100 = $202• Series B’s FV = $200 (1.02) + $200 = $404• FV of (A + B) = $202 + $404 = $606

FV of (A + B) can be calculated by adding the cashflows of each series and then calculating the FV of thecombined cash flow.

• At t = 1, combined cash flows = $100 + $200 = $300• At t = 2, combined cash flows = $100 + $200 = $300

Thus, FV of (A+ B) = $300 (1.02) + $300 = $606

Example:

Suppose,

Discount rate = 6%At t = 1→ Cash flow = $4At t = 2→ Cash flow = $24

It can be viewed as a $4 annuity for 2 years and a lumpsum of $20.

N = 2, I/Y = 6, Pmt = 4, FV = 0,

CPT PV of $4 annuity = $7.33

N = 2, I/Y = 6, Pmt = 0, FV = 20,

CPTPV of lump sum = $17.80

Total = $7.33 + $17.80 = $25.13

Practice: End of Chapter Practice

Problems for Reading 5.


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Reading 6 Discounted Cash Flow Applications


2. NET PRESENT VALUE AND INTERNAL RATE OF RETURN

Capital Budgeting refers to an investment decision-making process used by an organization to evaluateand select long-term investment projects.

Capital Structure is the mix of debt and equity used tofinance investments and projects.

Working capital management refers to themanagement of the company’s short-term assets (i.e.inventory) and short-term liabilities (i.e. accountspayable).

Capital budgeting usually uses the followingassumptions:

1. Decisions are based on cash flows; not on accounting

profits (i.e. net income):

• In addition, intangible costs and benefits are often

ignored because it is assumed that if these benefitsor costs are real, they will eventually be reflected incash flows.

• The relevant cash flows need to be considered areincremental cash flows. Sunk costs should be ignoredin the analysis.

2.

Timing of cash flows is critical i.e. cash flows that arereceived earlier are more valuable than cash flowsthat are received later.

3. Cash flows are based on opportunity costs:

Opportunity costs should be included in project costs.These costs refer to the cash flows that could be

generated from an asset if it was not used in theproject.

4. Cash flows are analyzed on an after-tax basis. Cashflows on after-tax basis should be incorporated in theanalysis.

5. Financing costs are ignored. Financing costs arereflected in the required rate of return which is used todiscount after-tax cash flows and investment outlaysto estimate net present value (NPV) i.e. only projectswith expected return > cost of the capital (requiredreturn) will increase the value of the firm.

• Financing costs are not included in the cash flows;because when financing costs are included in bothcash flows and in the discount rate, it results indouble-counting the financing costs.

6.

Capital budgeting cash flows are not accounting net

income.

For details, refer to Reading 35, Capital Budgeting.

Independent projects are projects whose cash flows areindependent of each other. Since projects areunrelated, each project is evaluated on the basis of itsown profitability.

Mutually exclusive projects compete directly with eachother e.g. if Projects A and B are mutually exclusive, youcan choose A or B, but you cannot choose both.

2.1 Net Present Value and the Net Present Value Rule

NPV = Present value of cash inflows - initial investment

−

where,

CFt = After-tax cash flow at timet

r = required rate of return for the investment

CF0 = investment cash outflow at time zero

Decision Rule:

• Accept a project if NPV ≥ 0• Do not Accept a project if NPV< 0

Independent projects: All projects with positive NPV areaccepted.

Mutually exclusive projects: A project with the highestNPV is accepted.

• Positive NPV investments increase shareholderswealth.

• NPV is inversely related to opportunity cost of capitali.e. the higher the opportunity cost of capital, thesmaller the NPV.

Advantages:

1) NPV directly measures the increase in value to thefirm.

2) NPV assumes that cash flows are reinvested at r(opportunity cost of capital).

Practice: Example 1,

Volume 1, Reading 6.


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2.2The Internal Rate of Return and the Internal Rate of

Return Rule

IRR is the discount rate that makesPresent value offuture cash inflows = initial investment

• In simple words, IRR is the discount rate where NPV =0.

• IRR is calculated using trial and error method or by

using a financial calculator.

As the name implies, internal rate of return (IRR) dependsonly on the cash flows of the investment i.e. no externaldata is needed to calculate it.

Example:

IRR is found by solving the following:

Solution:

IRR = 13.45%

Important to Note:

In the equation of calculating IRR, the IRR must becompatible with the timing of cash flows i.e. if cash flowsare semi-annual (quarterly), the IRR will be semi-annual(quarterly).

When project’s cash flows are a perpetuity, IRR can be

estimated as follows:

−

Decision Rule:

• Accept a project if IRR ≥ Cost of Capital.• Do not Accept a project if IRR< Cost of Capital.

NOTE:

• When IRR = opportunity cost of capitalNPV = 0.• When IRR> opportunity cost of capitalNPV> 0.• When IRR< opportunity cost of capitalNPV< 0.•

If projects are independent, accept both if IRR of

both projects ≥ Cost of Capital.• If projects are mutually exclusive and project A IRR>

project B IRR and both IRR≥ Cost of Capital, acceptProject A because IRRA>IRRB .

Advantages of IRR:

1) IRR considers time value of money.2) IRR considers all cash flows.3) IRR involves less subjectivity.4) It is easy to understand.5) It is widely accepted.

Limitations of IRR:

1) IRR is based on the assumption that cash flows arereinvested at the IRR; however, this may not alwaysbe realistic.

2) IRR provides result in percentages; however,percentages can be misleading and involves difficultyin ranking projects i.e. a firm rather earn 100% on a$100 investment, or 10% on a $10,000 investment.

3) In case of non-conventional cash flow pattern, there

are or can be multiple IRRs or No IRR at all.

2.3 Problems with the IRR Rule

No conflict exists between the decision rules for NPV andIRR when:

1)

Projects are independent.2) Projects have conventional cash flow pattern.

Conflict exists between the decision rules for NPV and IRRwhen:

1) Projects are mutually exclusive.2) Projects have non-conventional cash flow pattern.

NPV and IRR rank projects differently due to following

reasons:

1) Differences in cash flow patterns.

2) Size (scale) differences: Required rate of return favors

small projects because the higher the opportunitycost, the more valuable these funds are. Sometimes,the larger, low-rate-of-return project has the betterNPV.

3) Timing differences: Project with shorter paybackperiod provides more CF in early years forreinvestment. Therefore, when required rate of returnis high, it favors project with early CFs.

NPV versus IRR:

• NPV rule is based on external market-determineddiscount rate because it assumes reinvestment at r(opportunity cost of capital).

• IRR assumes that cash flows are reinvested at IRR;thus, IRR and IRR rankings are not affected by anyexternal interest rate or discount rate.

• It is more realistic to assume reinvestment atopportunity cost ‘r’; thus, NPV method is best.

It implies that whenever there is a conflict between NPVand IRR decision rule and to choose between mutuallyexclusive projects, we should always use NPV rule.

Practice: Example 2 & 3,



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3. PORTFOLIO RETURN MEASUREMENT

Holding Period Return (HPR): A holding period returnrefers to the return earned by an investor from holdingan asset for a specified period of time e.g. 1 day, 1week, 1 month, 5 years etc.

Total return = Capital gain (or loss) yield + Dividend yield

− −

−

where,

P = price

D = dividend

t-1 = beginning of the period

t = end of period

3.1 Money-Weighted Rate of Return

The money-weighted rate of return (MWR) measures thecompound growth rate in the value of all funds investedin the account over the entire evaluation period. In U.S.,it is known as “dollar-weighted return”. It represents aninternal rate of return (IRR) of an investment. Like IRR,

• Amounts invested (initial market value of theportfolio) are cash outflows for the investor.

• All additions to the portfolio are cash outflows for theinvestor.

• Amounts returned (receipts) or withdrawn by theinvestor are cash inflows for the investor.

• The ending market value of the portfolio is a cashinflow for the investor.

It is computed as follows:

where,

IRR represents the MWR.

T = number of periods

CFt = cash flow at time t

•

MWR is preferred to use to evaluate theperformance of the portfolio manager when themanager has discretion over the deposits and

withdrawals made by clients.

Advantages of MWR: MWR requires an account to bevalued only at the beginning and end of the evaluationperiod.

Disadvantages of MWR:

• MWR is highly affected by the size and timing of

external cash flows to an account.• It is not appropriate to use when investment

manager has little or no control over the externalcash flows to an account.

Example:

Assume,

• Amount invested in a mutual fund at the beginningof 1st year = $100

• Amount invested in a mutual fund at the beginningof 2nd year = $950

• Amount withdrawn at the end of 2nd year = $350• Value of investments at the end of 3rd year = $1,270

CF0 = –100CF1 = –950CF2 = +350

CF3 = +1,270

−

−

Solve for IRR, we have →IRR = 26.11%

3.2 Time-Weighted Rate of Return

The time-weighted rate of return (TWR) measures thecompound rate of growth over a stated evaluationperiod of one unit of money initially invested in theaccount.

• In TWR, the account needs to be valued wheneveran external cash flow occurs.

• TWR measures the actual rate of return earned bythe portfolio manager.

• TWR is preferred to use to evaluate the performanceof the portfolio manager when the manager has nocontrol over the deposits and withdrawals made by

clients.

When there are no external cash flows, TWR is computedas follows:

−

In order to calculate time weighted return, first of all,holding period return for each sub-period is computedand then these sub-period returns must be linkedtogether (known as chain-linking process) to computethe TWR for the entire evaluation period.


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r twr = (1+r t,1 )×(1+r t,2 ) × … (1+r t,n ) –1

• Note that unless the sub-periods represent a year,the time-weighted rate of return will not beexpressed as an annual rate.

• Each subperiod return within the full evaluationperiod has a weight = (length of the subperiod /length of the full evaluation period).

If the investment is for more than one year , time-weighted return can be annualized by calculatinggeometric mean of n annual returns:

Time – weighted return = [(1+R1)(1+R2)…(1+Rn)]1/n – 1

Where,Rit = return for year i

n = total number of annual returns

Method of computing Time-weighted Return for the Year:

i. Calculate holding period return for each day (i.e. 365days daily returns) using the following formula:

where,r i = r 1, r 2, …r 365

ii. Calculate annual return for the year by linking thedaily holding period returns as follows:

Time – weighted return = [(1+R1)(1+R2)…(1+R365)] – 1

This annual return represents the precise time-weightedreturn for the year IF withdrawals and additions to theportfolio occur only at the end of day. Otherwise, it

represents the approximate time-weighted return for theyear.

Time-weighted return can be annualized by calculatinggeometric mean of n annual returns:

Time-weighted return = [(1+R1)(1+R2)…(1+Rn)]1/n –1

where,

Rit = return in period t

n = total number of periods

Advantage of TWR: TWR is not sensitive to any external

cash flows to the account i.e. additions and withdrawalsof funds.

Disadvantage of TWR:

• TWR requires determining a value for the accounteach time any cash flow occurs.

• Marking to market an account on daily basis isadministratively more cumbersome, expensive andpotentially more error-prone.

Example:

• Beginning portfolio value for period 1 = $10,000• Ending portfolio value for period 1 = $10,050• Dividends received before additional investment in

period 1 = $100

• Beginning portfolio value for period 2 = $10,350• Ending portfolio value for period 2 = $10,850• Dividends received in period 2 = $100

ℎ ℎ

−

ℎ ℎ

−

The annual return (based on the geometric average)over the entire period is

r = [(1.0150)(1.05850)] –1=0.0739 or 7.39%

TWR versus MWR:

• When funds are contributed to an account prior to aperiod of strong (positive) performance, MWR>TWR.

• When funds are withdrawn from an account prior toa period of strong (positive) performance,MWR 10% ofaccount) and during that evaluation period,account’s performance is highly volatile, then MWRand TWR will provide significantly different results.




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4. MONEY MARKET YIELDS

Money market instruments are short-term debtinstruments i.e. having maturities of one year or less.These instruments pay par value (face value) at maturityand are usually discount instruments i.e. they do not paycoupons, but instead are sold below (at discount from)their par (face) value. For example, T-bills are discountinstruments where,

• Investor buys the T-bill at (Face value – discount) andreceives face value at maturity.

• Investor earns a dollar return equal to the discountwhen he/she holds the T-bill to maturity.

Other types of money-market instruments includecommercial paper and bankers’ acceptances whichare discount instruments and negotiable certificates ofdeposit which are interest bearing instruments (that paycoupons).

1)

Bank Discount Basis: T-bills are quoted on a 360-daydiscount basis rather than price basis using the bankdiscount rate(a 360-day year is commonly used inpricing money market instruments). The bank discountrate is defined as:

−

−

where,

r BD = Annualized yield on a bank discount basisn = Actual number of days remaining to maturityLimitations of Yield on a bank discount basis: Bankdiscount yield is not a meaningful measure of investors’return because:

1. It is based on the FV (par value) of the bond insteadof its purchase price; but returns should beevaluated relative to the amount invested (i.e.purchase price).

2. It is annualized based on a 360-day year rather thana 365-day year.

3. It is annualized based on simple interest; thus, itignores the compound interest.

•

The discount rate for the T-bill can be used to find PVof other cash flows with risk characteristics similar tothose of the T-bill.

• However, when risk of cash flows is higher than thatof T-bill, the T-bill's yield can be used as a base rate and a risk premium is added to it to represent higherrisk of cash flows.

2) Holding period yield (HPY): HPY reflects the returnearned by an investor by holding the instrument tomaturity.

−

where,

P0 = initial purchase price of the instrument

P1 = Price received for the instrument at its maturity

D1 = Cash distribution paid by the instrument at its

maturity (i.e. interest)

For interest-bearing instruments: The purchase and saleprices must include any accrued interest* when thebond is purchased/sold between interest paymentdates.

*Coupon interest earned by the seller from the lastcoupon date but not received by the seller as the nextcoupon date occurs after the date of sale.

NOTE:

• When the price is quoted including accrued interest,it is called Full price.

• When the price is quoted without accrued interest, itis called Clean price.

3) Effective annual yield (EAY):

EAY = (I + HPY) 365/t - 1

Rule: The bank discount yield < effective annual yield.

4) Money market yield (or CD equivalent yield): Moneymarket yield can be used to compare the quotedyield on a T-bill to quoted yield on interest-bearingmoney-market instruments that pay interest on a 360-day basis.

• Generally, the money market yield is equal to theannualized holding period yield (assuming a 360-dayyear) i.e.

Money market yield = rMM = (HPY) × (360/ t)

• Unlike bank discount yield, the money market yield isbased on purchase price.

• Thus, money market yield > bank discount yield.

Or




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360360

5)

Bond-equivalent yield: When a semi-annual yield isannualized by multiplying it by 2, it is referred to as thebond-equivalent yield. It ignores compounding ofinterest. The bond equivalent yield is calculated asfollows:

Practice: Example 7, Volume 1,

Reading 6 & End of Chapter

Practice Problems for Reading 6.


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Reading 7 Statistical Concepts and Market Returns


2.1 The Nature of Statistics

Statistics refer to the methods used to collect andanalyze data. Statistical methods include descriptivestatistics and statistical inference (inferential statistics).

• Descriptive statistics: It describes the properties of alarge data set by summarizing it in an effectivemanner.

•

Statistical inference: It involves use of a sample tomake forecasts, estimates, or judgments about thecharacteristics of a population

2.2 Populations and Samples

• A population is a complete set of outcomes or allmembers of a specified group.

• A parameter describes a characteristic of apopulation e.g. mean value, the range ofinvestment returns, and the variance.

Since analyzing the entire population involves high costs,it is preferred to use a sample.

• A sample is a subset of a population.• A sample statistic or statistic describes acharacteristic of a sample.

2.3 Measurement Scales

Measurement scales are the specific set of rules used toassign a symbol to the event in question. There are fourtypes of measurement scales.

a)

Nominal Scale: It is a simple classification systemunder which the data is categorized into varioustypes.

• It does not rank the data.• It is the weakest level of measurement.

Example:

Mutual funds can be categorized according to theirinvestment strategies i.e.

• Mutual Fund 1 refers to a small-cap value fund.•

Mutual Fund 2 refers to a large-cap value fund.

b) Ordinal Scale: This scale categorizes data into variouscategories and also rank them into an order based onsome characteristic.

• It is a stronger level of measurement relative tonominal scale.

• However, the intervals separating the ranks in ordinal

scale cannot be compared with each other.

Example:

Under Morningstar and Standard & Poor's star ratings formutual funds,

• A fund that is assigned 1 star represents a fund withrelatively the worst performance.

•

A fund that is assigned 5 stars represents a fund withrelatively the best performance.

c) Interval Scale: This scale rank the data into an orderbased on some characteristic and the differencesbetween scale values are equal e.g. Celsius andFahrenheit scales.

• The zero point of an interval scale does not reflect atrue zero point or natural zero e.g. 0°C does notrepresent absence of temperature; rather, it reflectsa freezing point of water.

• As a result, it cannot be used to compute ratios e.g.40°C is two times larger than 20°C; however, it doesnot represent two times as much temperature.

• Since difference between scale values are equal,scale values can be added and subtractedmeaningfully.

Example:

The difference in temperature between 15°C and 20°C isthe same amount as the difference between 40°C and45°C. Also, 10°C + 5°C = 15°C

d) Ratio Scale: It is the strongest level of measurement.Under this scale,

• The data is ranked order based on somecharacteristic.

• The differences between scale values are equal;therefore, scale values can be added andsubtracted meaningfully.

• A true zero point as the origin exists. E.g. zero moneymeans no money.o Thus, it can be used to compute ratios and to add

and subtract amounts within the scale.

Example:

Money is measured on a ratio scale i.e. the purchasingpower of $100 is twice as much as that of $50.




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3. SUMMARIZING DATA USING FREQUENCY DISTRIBUTIONS

Data can be summarized using a frequency distribution.In a Frequency distribution, data is grouped intomutually exclusive categories and shows the number ofobservations in each class.

•

It is also useful to identify the shape of thedistribution.

Construction of a Frequency Distribution table:

Step 1: Arrange the data in ascending order. Step 2: Calculate the range of the data.

Range = Maximum Value - Minimum valueStep 3: Choose the appropriate number of classes (k):

Determining the number of classes involves judgment.

NOTE:

A large value of k is useful to obtain detailed information

regarding the extreme values of a distribution.

Step 4: Determine the class interval or width using thefollowing formula i.e.

i≥ (H-L)/k

where,

i= Class intervalH = Highest observed valueL = Lowest observed valuek= Number of classes

Interval: An interval represents a set of values within

which an observation lies.

• If too few intervals are used, then the data is over-summarized and may ignore importantcharacteristics.

• If too many intervals are used, then the data isunder-summarized.

• The smaller (greater) the value of k, the larger(smaller) the interval.

Example:

Suppose,

H = $35,925L = $15,546k = 7

Class interval = ($35,925 - $15,546)/7 = $2,911≈ $3,000.

It is important to note that:

• We will always round up (not down), to ensure thatthe final class interval includes the maximum valueof the data.

• The class intervals (also known as ranges or bins) do

not overlap.

Step 5: Set the individual class limits i.e.

• Ending points of intervals are determined by

successively adding the interval width to theminimum value.

• The last interval would be the one which includes themaximum value.

NOTE:

The notation [20,000 to 25,000) means 20,000 ≤observation < 25,000A square bracket shows that theendpoint is included in the interval.

Step 6: Count the number of observations in each classinterval.

Absolute Frequency: The actual number of observationsin a given class interval is called the absolute frequencyor simply frequency; as shown in the table below i.e.there are 8 observations that fall under the price interval15 up to 18.

Relative frequency:

Relative frequency = Absolute frequency / Total numberof observations

Cumulative Absolute Frequency: The cumulativeabsolute frequency is found by adding up the absolutefrequencies. It reflects the number of observations thatare less than the upper limit of each interval.

Cumulative Relative Frequency: The cumulative relative frequency is found by adding up the relativefrequencies. It reflects the percentage of observationsthat are less than the upper limit of each interval.


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E.g. in the table above after the “ relative frequency ”,the cumulative relative frequency for the

• 2nd class interval would be 0.10 + 0.2875 = 0.3875 itindicates that 38.75% of the observations lie belowthe selling price of 21.

• 3rd class interval would be 0.3875 + 0.2125 = 0.60 itindicates that 60% of the observations lie below theselling price of 24.

E.g. in the table below cumulative relative frequency forthe 2nd class interval would be 0.10 + 0.2875 = 0.3875 andfor the 3rd class interval would be 0.3875 + 0.2125 = 0.60

NOTE:

The frequency distributions of annual returns cannot becompared directly with the frequency distributions ofmonthly returns.

For details, refer to discussion before table 4,Volume 1, Reading 7.

4.1 The Histogram

A histogram is the graphical representation of afrequency distribution.

• The classes are plotted on the horizontal axis.• The class frequencies are plotted on the vertical axis.

•

The heights of the bars of histogram represent theabsolute class frequencies.

• Since the classes have no gaps between them,there would be no gaps between the bars of thehistogram as well.

4.2The Frequency Polygon and the Cumulative

Frequency Distribution

Frequency polygon: It also graphically represents thefrequency distribution.

• The mid-point of each class interval is plotted on thehorizontal axis.

• The corresponding absolute frequency of the class

interval is plotted on the vertical axis.• The points representing the intersections of the classmidpoints and class frequencies are connected by aline.

Cumulative frequency distribution: This graph can beused to determine the number or the percentage of theobservations lying between a certain values. In thisgraph,

• Cumulative absolute or cumulative relativefrequency is plotted on the vertical axis.

• The upper interval limit of the corresponding classinterval is plotted on the horizontal axis.o For extreme values (both negative and positive),

the cumulative distribution tends to flatten out.o Steeper (flatter) slope of the curve indicates large

(small) frequencies (# of observations).

NOTE:

Change in the cumulative relative frequency = Relativefrequency of the next interval.




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5. MEASURES OF CENTRAL TENDENCY

A measure of central tendency indicates the center ofthe data. The most commonly used measures of centraltendency are:

1. Arithmetic mean or mean: It is the sum of theobservations in the dataset divided by the number ofobservations in the dataset.

2.

Median: It is the middle number when theobservations are arranged in ascending ordescending order. A given frequency distribution hasonly one median.

3. Mode: It is the observation that occurs most frequentlyin the distribution. Unlike median, a mode is notunique which implies that a distribution may havemore than one mode or even no mode at all.

4. Weighted mean: It is the arithmetic mean in which

observations are assigned different weights. It iscomputed as:

⋯ where,

X1 , X 2 ,…,Xn = observed valuesw1 , w 2 ,…,w3 = Corresponding weights, sum to 1.

• An arithmetic mean is a special case of weightedmean where all observations are equally weightedby the factor 1/ n (or l/N).

•

A positive weight represents a long position and anegative weight represents a short position.

•

Expected value: When a weighted mean iscomputed for a forward-looking data, it is referred toas the expected value.

Example:

Weight of stocks in a portfolio = 0.60Weight of bonds in a portfolio = 0.40Return on stocks = –1.6%Return on bonds = 9.1%

A portfolio's return is the weighted average of the returns

on the assets in the portfolio i.e.

Portfolio return = (w stock × R stock) + (w bonds × R bonds)= 0.60(-1.6%) + 0.40 (9.1%) = 2.7%.

5.

Geometric mean (GM): The geometric mean can beused to compute the mean value over time tocompute the growth rate of a variable.

with Xi ≥ 0for i= 1, 2, …, n.

Or

or as

∑ G = elnG

• It should be noted that the geometric mean can be

computed only when the product under the radicalsign is non-negative.

The geometric mean return over the time period can becomputed as:

− • Geometric mean returns are also known ascompound returns.

Advantages of Measures of central tendency:

•

Widely recognized.• Easy to compute.• Easy to apply.

5.1.1) The Population Mean

It is the arithmetic mean of the total population and iscomputed as follows:

∑ where,

µ = Population meanN = Number of observations in the entire population

Xi = ith observation.

• The population mean is a population parameter.• A given population has only one mean.




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5.1.2) The Sample Mean

The sample mean is the arithmetic mean value of asample; it is computed as:

∑ where,

= sample mean

Xi = ith observationn = number of observations in the sample

• The sample mean is a statistic.• It is not unique i.e. for a given population; differentsamples may have different means.

Cross-sectional mean: The mean of the cross-sectionaldata i.e. observations at a specific point in time is calledcross-sectional mean.

Time-series mean: The mean of the time-series data e.g.monthly returns for the past 10 years is called time-series

mean.

5.1.3) Properties of the Arithmetic Mean

Property 1: The sum of the deviations* around the meanis always equal to 0.

*The difference between each outcome and the mean

is called a deviation.

Property 2: The arithmetic mean is sensitive to extremevalues i.e. it can be biased upward ordownward by extremely large or smallobservations, respectively.

Advantages of Arithmetic Mean:

• The mean uses all the information regarding the sizeand magnitude of the observations.

• The mean is also easy to calculate.• Easy to work with algebraically

Limitation: The arithmetic mean is highly affected byoutliers (extreme values).

• Trimmed Mean: It is the arithmetic mean of thedistribution computed after excluding a stated small% of the lowest and highest values.

• Winsorized mean: In a winsorized mean, a stated %of the lowest values is assigned a specified low valueand a stated % of the highest values is assigned aspecified high value and then a mean is computedfrom the restated data. E.g. in a 95% winsorizedmean,

o The bottom 2.5 % of values are set = 2.5th percentile value.

o The upper 2.5% of values are set = 97.5th percentilevalue.

5.2 The Median

Population median: A population median divides a

population in half.

Sample median: A sample median divides a sample inhalf.

Steps to compute the Median:

1. Arrange all observations in ascending order i.e. fromsmallest to largest.

2. When the number of observations (n) is odd, themedian is the center observation in the ordered list i.e.

Median will be located at =

position

• (n+1)/2 only identifies the location of the median,

not the median itself.

3. When the number of observations (n) is even, thenmedian is the mean of the two center observations inthe ordered list i.e.

Median will be located at mean of

.

Advantage: Median is not affected by extremeobservations (outliers).

Limitations:

•

It is time consuming to calculate median.• The median is difficult to compute.• It does not use all the information about the size andmagnitude of the observations.

• It only focuses on the relative position of the rankedobservations.

Example:

Suppose, current P/Es of three firms are 16.73, 22.02, and29.30.

n = 3→ (n + 1) / 2 = 4/ 2 = 2nd position.

Thus, the median P/E is 22.02.






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5.3 The Mode

Population mode: A population mode is the mostfrequently occurring value in the population.

Sample mode: A sample mode is the most frequentlyoccurring value in the sample.

Unimodal Distribution: A distribution which has only one

mode is called a unimodal distribution.

Bimodal Distribution: A distribution which has two modesis called a bimodal distribution.

Trimodal Distribution: A distribution which has threemodes is called a Trimodal distribution.

A distribution would have no mode when all the values ina data set are different.

Modal Interval: Data with continuous distribution (e.g.stock returns) may not have a modal outcome. In suchcases, a modal interval is found i.e. an interval with thelargest number of observations (highest frequency). Themodal interval always has the highest bar in thehistogram.

Important to note: The mode is the only measure ofcentral tendency that can be used with nominal data.

5.4.2) The Geometric MeanGeometric mean v/s Arithmetic mean:

• The geometric mean return represents the growthrate or compound rate of return on an investment.

• The arithmetic mean return represents an averagesingle-period return on an investment.

• The geometric mean is always ≤ arithmetic mean.• When there is no variability in the observations (i.e.

when all the observations in the series are the same),geometric mean = arithmetic mean

• The greater the variability of returns over time, themore the geometric mean will be lower than thearithmetic mean.

• The geometric mean return decreases with anincrease in standard deviation (holding thearithmetic mean return constant).

• In addition, the geometric mean ranks the two funds

differently from that of an arithmetic mean.

5.4.3) The Harmonic Mean

with Xi> 0 for i = 1,2, …, n.

• It is a special case of the weighted mean in whicheach observation's weight is inversely proportional toits magnitude.

Important to note:

• Harmonic mean formula cannot be used to

compute average price paid when differentamounts of money are invested at each date.

• When all the observations in the data set are thesame, geometric mean = arithmetic mean =harmonic mean.

• When there is variability in the observations,harmonic mean < geometric mean < arithmeticmean.

6. OTHER MEASURES OF LOCATION: QUANTILE

Measures of location: Measures of location indicate boththe center of the data and location or distribution of thedata. Measures of location include measures of centraltendency and the following four measures of location:

• Quartiles• Quintiles• Deciles• Percentiles

Collectively these are called “Quantiles”.

6.1 Quartiles, Quintiles, Deciles, and Percentiles

1) Quartiles divide the distribution into four differentparts.

• First Quartile = Q1 = 25th percentile i.e. 25% of theobservations lie at or below it.

• Second Quartile = Q2 = 50th percentile i.e. 50% of the

Practice: Example on 5.4.3,







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observations lie at or below it.• Third Quartile = Q3 = 75th percentile i.e. 75% of the

observations lie at or below it.

2) Quintiles divide the distribution into five different parts.In terms of percentiles, they can be specified as P20,P40, P60, & P80.

3) Deciles divide the distribution into ten different parts.

4) Percentiles divide the distribution into hundreddifferent parts. The position of a percentile in an arraywith n entries arranged in ascending order isdetermined as follows:

where,

y = % point at which the distribution is being divided.Ly = location (L) of the percentile (Py).n = number of observations.

•

The larger the sample size, the more accurate thecalculation of percentile location.

Example:

Dividend Yields on the components of the DJ

Euros STOXX 50

No. CompanyDividend

Yield(%)

1 AstraZeneca 0.00

2 BP 0.00

3 Deutsche Telekom 0.00

4 HSBC Holdings 0.00

5 Credit Suisse Group 0.26

6 L’Oreal 1.09

7 SwissRe 1.27

8 Roche Holding 1.33

9 Munich Re Group 1.36

10 General Assicurazioni 1.39

11 Vodafone Group 1.41

12 Carrefour 1.5113 Nokia 1.75

14 Novartis 1.81

15 Allianz 1.92

16 Koninklije Philips Electronics 2.01

17 Siemens 2.16

18 Deutsche Bank 2.27

19 Telecom Italia 2.27

No. CompanyDividend

Yield(%)

20 AXA 2.39

21 Telefonica 2.49

22 Nestle 2.55

23 Royal Bank of Scotland Group 2.60

24 ABN-AMRO Holding 2.65

25 BNP Paribas 2.65

26 UBS 2.65

27 Tesco 2.95

28 Total 3.11

29 GlaxoSmithKline 3.31

30 BT Group 3.34

31 Unilever 3.53

32 BASF 3.59

33 Santander Central Hispano 3.6634 Banco Bilbao VizcayaArgentaria 3.67

35 Diageo 3.68

36 HBOS 3.78

37 E.ON 3.87

38 Shell Transport and Co. 3.88

39 Barclays 4.06

40 Royal Dutch Petroleum Co. 4.27

41 Fortus 4.28

42 Bayer 4.45

43 DiamlerChrysler 4.68

44 Suez 5.13

45 Aviva 5.15

46 Eni 5.66

47 ING Group 6.16

48 Prudential 6.43

49 Lloyds TSB 7.68

50 AEGON 8.14

Source: Example 9,Table 17, Volume 1, Reading 7.

Calculating 10th percentile (P10):Total number ofobservations in the table above = n = 50

L10 = (50 + 1) × (10 / 100) = 5.1

• It implies that 10th percentile lies between 5th observation (X5 = 0.26) and 6th observation (X6 =1.09).


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Thus,P10 = X5 + (5.1 – 5) (X6 – X5) = 0.26 + 0.1 (1.09 – 0.26)

= 0.34%

Calculating 90th percentile (P90):

L90 = (50 + 1) × (90 / 100) = 45.9

• It implies that 90th percentile lies between the 45th observation (X45 = 5.15) and 46th observation (X46 =

5.66).

Thus,P90 = X45 + (45.9 – 45) (X46 – X45) = 5.15 + 0.90 (5.66 – 5.15)

= 5.61%

Calculating 1stQuartile (i.e.P25):

L25 = (50 + 1) × (25 / 100) = 12.75

• It implies that 25th percentile lies between the 12th observation (X12 = 1.51) and 13th observation (X13 =1.75).

Thus,

P25 = Q1 = X12 + (12.75 – 12) (X13 – X12) = 1.51 + 0.75 (1.75 –1.51) = 1.69%

Calculating 2nd Quartile (i.e.P50):

L50 = (50 + 1) × (50 / 100) = 25.5

• It implies that P50 lies between the 25th observation(X25 = 2.65) and 26th observation (X26 = 2.65).

• Since, X25 = X26 = 2.65, no interpolation is needed.

Thus,

P50 = Q2 = 2.65% = Median

Calculating 3rd Quartile (i.e.P75):

L75 = (50 + 1) × (75 / 100) = 38.25


Thus,

P75 = Q3 = X38 + (38.25 – 38) (X39 – X38)= 3.88 + 0.25 (4.06 – 3.88)= 3.93%

Calculating 20th percentile (P20) = 1st Quintile:

L20 = (50 +1) × (20 /100) = 10.2


Thus,

1st quintile = P20 = X10 + (10.2 – 10) (X11 – X10) = 1.39 + 0.20(1.41 – 1.39) = 1.394% or 1.39%

Source: Example 9, Volume 1, Reading 7, P. 356.

6.2 Quantiles in Investment Practice

Quantiles are frequently used by investment analysts torank performance i.e. portfolio performance. Forexample, an analyst may rank the portfolio ofcompanies based on their market values to compareperformance of small companies with large ones i.e.

• 1st decile contains the portfolio of companies withthe smallest market values.

• 10th decile contains the portfolio of companies withthe largest market values.

Quantiles are also used for investment researchpurposes.

7. MEASURES OF DISPERSION

The variability around the central mean is calledDispersion. The measures of dispersion provideinformation regarding the spread or variability of thedata values.

Relative dispersion: It refers to the amount ofdispersion/variation relative to a reference value orbenchmark e.g. coefficient of variation. (It is discussedbelow).

Absolute Dispersion: It refers to the variation around themean value without comparison to any reference pointor benchmark. Measures of absolute dispersion include:1) Range:

Range = Maximum value - Minimum value

Advantage: It is easy to compute.

Disadvantages:

• It does not provide information regarding the shapeof the distribution of data.

• It only reflects extremely large or small outcomesthat may not be representative of the distribution.

NOTE:

Interquartile range (IQR) = Third quartile - First quartile= Q3 – Q1

• It reflects the length of the interval that contains the


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middle 50% of the data.• The larger the interquartile range, the greater the

dispersion, all else constant.

2) Mean absolute deviation (MAD):It is the average ofthe absolute values of deviations from the mean.

∑ −

where, = Sample meann = Number of observations in the sample

• The greater the MAD, the riskier the asset.

Example:

Suppose, there are 4 observations i.e. 15, -5, 12, 22.

Mean = (15 – 5 + 12 + 22)/4 = 11%MAD = (|15 – 11| + |–5 – 11| + |12 – 11| + |22 – 11|)/4

= 32/4 = 8%

Advantage:

MAD is superior relative to range because it is based onall the observations in the sample.

Drawback:

MAD is difficult to compute relative to range.

3)

Variance: Variance is the average of the squareddeviations around the mean.

4)

Standard deviation (S.D.): Standard deviation is thepositive square root of the variance. It is easy to

interpret relative to variance because standarddeviation is expressed in the same unit ofmeasurement as the observations.

7.3.1) Population Variance

The population variance is computed as:

∑ − where,

µ= Population meanN = Size of the population

Example:

Returns on 4 stocks: 15%, –5%, 12%, 22%Population Mean ( ) = 11%

− − − − −

7.3.2) Population Standard Deviation

It is computed as:

∑ −

√ 7.4.1) Sample Variance

It is computed as:

− − where,

=Sample meann=Number of observations in the sample• The sample mean is defined as an unbiased

estimator of the population mean.• (n – 1) is known as the number of degrees offreedom in estimating the population variance.

7.4.2) Sample Standard Deviation

It is computed as:

−

−

Important to note:

• The MAD will always be ≤ S.D. because the S.D. givesmore weight to large deviations than to small ones.

• When a constant amount is added to eachobservation, S.D. and variance remain unchanged.

7.5Semivariance, Semideviation, and Related

Concepts

Semivariance is the average squared deviation below the mean.

− −

Semi-deviation (or semi-standard deviation) is thepositive square root of semivariance.

•

Semi-deviation will be < Standard deviation becausestandard deviation overstates risk.

Practice: Example 10, 11 & 12,



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Example:

Returns (in %): 16.2, 20.3,9.3, -11.1, and -17.0.

Thus, n = 5

Mean return = 3.54%

Two returns, -11.1 and -17.0, are < 3.54%.

Semi-variance =[(-11.1 - 3.54)2 + (-17.0- 3.54)2] / 5 – 1=636.2212/4 = 159.0553

Semi-deviation= √ = 12.6%.Target semi-variance is the average squared deviationbelow a stated target .

− −

where,

B = target value,n = number of observations.

Target semi-deviation is the positive square root of thetarget semi-variance.

NOTE:

• Semivariance (or Semideviation) and targetSemivariance (or target Semideviation) are difficultto compute compared to variance.

• For symmetric distributions, semi-variance =variance.

Example:

Stock returns = 16.2, 20.3, 9.3%, –11.1% and –17.0%.Target return = B = 10%

Target semi-variance = [(9.3 –10.0)2 + (–11.1 – 10.0)2 + (–17.0 – 10.0)2]/(5 – 1)

= 293.675

Target semi-deviation = √ = 17.14%7.6 Chebyshev's Inequality

Chebyshev's inequality can be used to determine the

minimum % of observations that must fall within a giveninterval around the mean; however, it does not give anyinformation regarding the maximum % of observations.

According to Chebyshev's inequality:

The proportion of any set of data lying within k standarddeviations of the mean is always at least [1 – 1/ (K 2 )] for all k >1.

Regardless of the shape of the distribution and forsamples and populations and for discrete andcontinuous data:

• Two S.D. interval around the mean must contain atleast 75% of the observations.

• Three S.D. interval around the mean must contain atleast 89% of the observations.

Example:

When k = 1.25, then according to Chebyshev'sinequality,

• The minimum proportion of the observations that liewithin + 1.25s is [1 - 1/ (1.25)2] = 1 - 0.64 = 0.36 or 36%.

7.7 Coefficient of Variation

Coefficient of Variation (CV) measures the amount of

risk (S.D.) per unit of mean value.

When stated in %, CV is:

where,

s = sample S.D. = sample mean.• CV is a scale-free measure (i.e. has no units of

measurement); therefore, it can be used to directlycompare dispersion across different data sets.• Interpretation of CV: The greater the value of CV, thehigher the risk.

• An inverse CV

=

S

XIt indicates unit of mean

value (e.g. % of return) per unit of S.D.

7.8 The Sharpe Ratio

The Sharpe ratio for a portfolio p, based on historicalreturns is:

ℎ −






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− • Excess return on Portfolio = Mean portfolio return-Mean Risk free return it reflects the extra returnrequired by investors to assume additional risk.

• The larger the Sharpe ratio, the better the risk-adjusted portfolio performance.

• When Sharpe ratio is positive, it decreases with an

increase in risk, all else equal.• When Sharpe ratio is negative, it increases with anincrease in risk; thus, in case of negative Sharperatio, larger Sharpe ratio cannot be interpreted asbetter risk-adjusted performance.

• When two portfolios have same S.Ds, then theportfolio with the negative Sharpe ratio closer to 0 issuperior to other portfolio.

• However, when two portfolios have different S.Ds,then the portfolio with the negative Sharpe ratio

closer to 0 cannot be interpreted as superior to otherportfolio.

Ex-ante Sharpe Ratio: It is the forward-looking sharp ratiofor a portfolio based on expected mean return, the risk-free return and the S.D. of return.

Limitation of Sharpe Ratio: It uses standard deviation as ameasure of risk; however, Standard deviation is

appropriate to use as a risk measure for symmetricdistributions. Thus, it overstates risk-adjustedperformance.

8. SYMMETRY AND SKEWNESS IN RETURN DISTRIBUTIONS

Symmetrical return distribution or Normal distribution: It isa return distribution that is symmetrical about its meani.e. equal loss and gain intervals have same frequencies.It is referred to as normal distribution.

• A symmetrical distribution has skewness = 0

Characteristics of the normal distribution:

1) In a normal distribution, mean = median.2) A normal distribution is completely described by two

parameters i.e. its mean and variance.3) Approximately:

• 68% of the observations lie between ± one standarddeviation from the mean.

• 95% of the observations lie between ± two standarddeviations.

• 99% of the observations lie between ± threestandard deviations.

Skewed distribution: The distribution that is notsymmetrical around the mean is called skewed.

a) Positively skewed or right-skewed Distribution: It is areturn distribution that reflects frequent small losses

and a few extreme gains i.e. limited but frequentdownside.

• It has a long tail on its right side.• It has skewness> 0.• In a positively skewed unimodal distribution mode< median < mean.

• Generally, investors prefer positive skewness (all elseequal).

b) Negatively skewed or left-skewed Distribution: It is areturn distribution that reflects frequent small gainsand a few extreme losses i.e. unlimited but lessfrequent upside.

• It has a long tail on its left side.• It has skewness< 0.• In a negatively skewed unimodal distribution mean < median < mode.

Sample skewness (or sample relative skewness) iscomputed as follows:

− − ∑ −

where,

n = number of observations in the sample s = sample S.D.n / (n-1)(n – 2) = It is used to correct for downward bias

in small samples.




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For larger values of n, sample skewness is computed as:

≈ ∑ −

• For n ≥ 100 a skewness coefficient of +/- 0.5 isconsidered unusually large.

9. KURTOSIS IN RETURN DISTRIBUTIONS

Kurtosis is used to identify how peaked or flat thedistribution is relative to a normal distribution.

Leptokurtic: It is a distribution that is more peaked (i.e.greater number of observations closely clustered aroundthe mean value) and has fatter tails (i.e. greater numberof observations with large deviations from the meanvalue) than the normal distribution.

• It has more frequent extremely large deviations fromthe mean than a normal distribution.

•

Ignoring fatter tails in analysis results inunderestimation of the probability of extremeoutcomes.

• The more leptokurtic the distribution is, the higher therisk.

Platykurtic: It is a distribution that is less peaked thannormal.

Mesokurtic: It is a distribution that is identical to thenormal distribution.

The Sample excess kurtosis is computed as:

− − − ∑ −

− −

− − • For a normal distribution (mesokurtic), kurtosis = 3.0.• For a leptokurtic distribution, kurtosis> 3.• For a platykurtic distribution, kurtosis < 3.

NOTE:

Kurtosis is free of scale (i.e. it has no units ofmeasurement).

It is always positive number because the deviations areraised to the 4th power.

Excess kurtosis = Kurtosis – 3

• A normal or mesokurtic distribution has excesskurtosis = 0.

• A leptokurtic distribution has excess kurtosis > 0.• A platykurtic distribution has excess kurtosis < 0.

For larger sample size(n), Excess Kurtosis is computedusing the following formula:

∑ −

−

∑ −

− • For n ≥ 100 (taken from a normal distribution), a

sample excess kurtosis of ≥ 1.0 would be consideredunusually large.

10. USING GEOMETRIC AND ARITHMETIC MEANS

• For estimating single-period average return,arithmetic mean should be used.

• In contrast, for estimating average returns for morethan one period, geometric mean should be used.

≈






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Important to Note:

To plot past performance on a graph, it is moreappropriate to use semi-logarithm scale rather thanusing arithmetic scale.

Semi-logarithm graph: In this graph,

• There is an arithmetic scale on the horizontal axis fortime.

•

There is a logarithmic scale on the vertical axis forthe value of the investment.

• The values plotted on the vertical axis are gapedaccording to the differences between theirlogarithms.o Suppose, values of investment are $1, $10, $100

and $1,000. Each value are equally spaced on alogarithm scale because the difference in theirlogarithms is equal i.e. ln10 – ln1 = ln100 – ln10 =ln1000 – ln100 = 2.30.

• On the vertical axis, equal changes between valuesrepresent equal % changes.

• The growth at a constant compound rate is plottedas a straight line i.e. upward (downward) slopingcurve reflects increasing (decreasing) growth ratesover time.

Important to Note:

• The arithmetic mean is appropriate to use foranalyzing future (or expected) performance.

• In contrast, the geometric mean is appropriate touse for analyzing past performance.

Example:

Suppose,

• Total amount invested = $100,000• Probability of earning 100% return = 50%.• Probability of earning -50% return = 50%.o With 100% return, return in one period = 100% ×

$100,000 = $200,000.o With –50% return in the other period, return = –50%

× $100,000 = $50,000

Geometric mean return = % % –1 = 0With 50/50 chances of 100% or –50% returns, considerfour equally likely outcomes i.e. $400,000, $100,000,

$100,000, and $25,000.

Arithmetic mean ending wealth=($400,000 + $100,000 +$100,000 + $25,000) / 4

= $156,250.

• Actual returns are calculated as follows

o

$$

$ 100 300%

o

$$

$ 100 0%

o

$$

$ 100 0%

o

$$

$ 100 75%

Arithmetic mean return for two-period = (300% + 0% + 0% – 75%) / 4

= 56.25%.

Arithmetic mean return for single-period = [(1+56.25 %)1/2 –1]× 100 = 25%

≈ 25%

• According to this arithmetic mean return, arithmeticmean ending wealth = $100,000 × 1.5625 = $156,250.

Conclusion: In order to reflect the uncertainty in the cashflows, the expected terminal wealth of $156,250 shouldbe discounted at 25% arithmetic mean rate not thegeometric mean rate.

Source: Volume 1, Reading 7.

Practice: End of Chapter Practice

Problems for Reading 7.


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Reading 8 Probability Concepts


2. PROBABILITY, EXPECTED VALUE, AND VARIANCE

Random variable: A variable that has uncertainoutcomes is referred to as random variable e.g. thereturn on a risky asset.

Event: An event is an outcome or a set of outcomes of arandom process e.g. 10% return earned by the portfolioor tossing a coin three times.

• When an event is certain or impossible to occur, it isnot a random outcome.

Probability: Probability is a measure of the likelihood orchance that an event will occur in the future.

• If an event is possible to occur, it has a probabilitybetween 0 and 1.

• If an event is impossible to occur, it has a probabilityof 0.

• If an event is certain to occur, it has a probability of 1.

Properties of a Probability:

1) The probability of any event ‘E’ is a number that liesbetween 0 and 1 i.e.

0 ≤ P(E) ≤ 1

Where, P(E) = Probability of event E.

2) The sum of the probabilities of any set of mutuallyexclusive and exhaustive events always equals 1 e.g.if there are three events A, B & C, then theirprobabilities i.e. P(A) + P(B) + P(C) = 1.

Mutually exclusive events: When events are mutuallyexclusive, events cannot occur at the same time e.g.when a coin is tossed, the event of occurrence of ahead and the event of occurrence of a tail are mutuallyexclusive events. The following events are mutuallyexclusive.

• Event A: The portfolio earns a return = 8%.• Event B: The portfolio earns a return < 8%.

Exhaustive events: When events are exhaustive, it meansthat all possible outcomes are covered by the eventse.g. following events are exhaustive.

• Event A: The portfolio earns a return = 8%.• Event B: The portfolio earns a return < 8%.• Event C: The portfolio earns a return > 8%.

In the probability distribution of the random variable,each random outcome is assigned a probability.

Empirical (or statistical) probability: It is a probabilitybased on observations obtained from probabilityexperiments (historical data). The empirical frequencyof an event E is the relative frequency of event E i.e.

P(E) =

• Empirical probability of an event cannot becomputed for an event with no historical record orfor an event that occurs infrequently.

Example:

Total sample of dividend changes = 16,189.

• Frequency of observations that ‘change individends’ is increase = 14,911.

• Frequency of observations that ‘change individends’ is decrease = 1,278.

Probability that a dividend change is a dividend

increase =

≈ 0.92

Subjective probability: It is a probability based onpersonal assessment, educated guesses, and estimates.

Priori probability: It is a probability based on logicalanalysis, reasoning & inspection rather than onobservation or personal judgment.

• Priori and empirical probabilities are referred to as

objective probabilities.

Odds for Event E can be stated as:

E=

=

For example, given odds for E = "a to b," it implies thatthe

• For ‘a’ occurrences of E, we expect ‘b’ cases ofnon-occurrence.

Probability of E =

Odds against Event E can be stated as:

E =

For example, given odds against E =“a to b," it impliesthat the

Probability of E =


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Example:

Suppose odds for E = “1 to 7." Thus, total cases = 1 + 7 =8.It means that out of 8 cases there is 1 case ofoccurrence and 7 cases of non-occurrence.

The probability of E = 1/ (1 + 7) = 1/ 8.

Example:

Suppose,

• Winning probability = 1 / 16• Losing probability = 15 / 16• Profit when a person wins = $15• Loss when a person losses = $ -1

Expected profit = (1 / 16)($15) + (15/ 16)(-$1) = $0

Types of Probability:

1) Unconditional Probability: An unconditionalprobability is the probability of an event occurringregardless of other events e.g. the probability of thisevent A denoted as P(A). It may be viewed as stand-alone probability. It is also called marginalprobabilities.

2) Conditional Probability: A conditional probability is theprobability of an event occurring, given that anotherevent has already occurred.

P(A|B) Probability of A, given B.

NOTE:

The conditional probability of an event can be greaterthan, equal to, or less than the unconditional probability,depending on the facts.

Example:

Unconditional Probability: The probability that the stockearns a return above the risk-free rate (event A).

( )

− ( )

Conditional Probability: The probability that the stockearns a return above the risk-free rate (event A), giventhat the stock earns a positive return (event B).

P(A|B) =

%

Joint Probability: The probability of occurrence of allevents is referred to as joint probability. For example, the

joint probability of A and B denoted as P(AB) read as the

probability of A and B is the sum of the probabilities oftheir common outcomes.

• P(AB) = P(BA).

The conditional probability of A given that B hasoccurred:

( )() → () ≠

Multiplication Rule for Probability: For two events, A and

B, the joint probability that both events will happen isfound as follows:

P(A and B) = P(AB) = P(A|B) × P(B)P(B and A) = P(BA) = P(B|A) × P(A)

Addition Rule for Probabilities: The probability that eventA or B will occur (i.e. at least one of the two events

occurs) is found as follows:

P(A or B) = P(A) + P(B) – P *(A and B)

*To avoid double counting of probabilities of shared outcomes

When events A and B are mutually exclusive, P(AB) = 0;thus, the addition rule can be simplified as:

P(A or B) = P(A) + P(B)

Independent Events: Two events are independent if theoccurrence of one of the events does not affect theprobability of the other event. Two events A and B areindependent if

P(B |A) = P(B)Or if

P(A |B) = P(A)





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Dependent Events: Two events are dependent when theprobability of occurrence of one event depends on theoccurrence of the other.

Multiplication Rule for Independent Events:

P(A and B) = P(AB) = P(A) × P(B)P(A and B and C) = P(ABC) = P(A) × P(B) × P(C)

Example:

Suppose the unconditional probability that a fund is aloser in either period 1 or 2 = 0.50 i.e.

• P(fund is a period 1 loser) = 0.50• P(fund is a period 2 loser) = 0.50

Calculating the probability that fund is a Period 2 loser

and fund is a Period 1 loser i.e. P(fund is a Period 2 loserand fund is a Period 1 loser).

Using the multiplication rule for independent events:

P(Fund is a period 2 loser and fund is a period 1 loser) =P(fund is a period 2 loser) × P(fund is a period 1 loser) =0.50 × 0.50 = 0.25

Source: Example 6, CFA® Curriculum, Volume 1, Reading 8.

Complement Rule: For an event or scenario S, the eventnot-S is called the complement of S and is denoted asSC. Since either S or not-S must occur,

P(S) + P(SC) = 1

The Total Probability Rule: According to the totalprobability rule, the probability of any event P(A) can bestated as a weighted average* of the probabilities of theevent, given scenarios i.e. P(A│S1).

*where, weights = P(S1) × P(A│S1)

It is expressed as follows:

P(A) = P(AS) + P(ASC) = P(A│S) P(S) + P(A│SC) P(SC)

P(A) = P(AS1) + P(AS2) +… P(ASn)

= P(A│S1) P(S1) + P(A│S2) P(S2)+…P(A│Sn) P(Sn)

Where, S1, S2…,Sn are mutually exclusive and exhaustivescenarios or events.

•

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