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Investment ReturnsTo make a meaningful judgment about the

return we need to know the scale (size) and the timing of the return.

The solution to scale and timing problem is to express investment results as rates of return, or percentage return.

Important concepts:1. k = (FV/PV)1/n – 12. Total return % = (Capital gain + Dividend) / Initial Investment3. Internal rate of return (IRR)4. Return conversion

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What is Risk?One definition: Uncertainty of future outcomesAlternative definition: The probability of an

adverse outcomeWe will discuss several measures of risk that

are used in developing portfolio theory

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Overview of RiskAll financial assets are expected to produce cash flows and

the risk of an asset is judged in terms of the riskiness of its cash flows

The risk of an asset can be considerer in two ways: (1) on a stand-alone basis, where the asset’s cash flows are

analyzed by themselves or (2) in a portfolio context, where the cash flows from a

number of assets are combined and then the consolidated cash flows are analyzed

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Overview of RiskAn asset with a high degree of risk must

provide a relatively high expected rate of return to attract investors.

Investors in general are averse to risk, so they will not buy risky assets unless those assets have high expected returns.

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Estimating Return StatisticsYear Stock X (rx) Stock Y (ry)

1 110% 23%2 22% 16%3 -60% 10%4 -20% 8%5 30% 25%

Calculate expected return for stock X and Y.

Expected return (μ) = ∑ri / T

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Estimating Risk Standard deviation (σi) measures total,

or stand-alone, risk.

Calculate standard deviations of Stock X and Y.

Formula: Variance = (σi)2 = [∑ri

2 –(∑ri)2/T ] ÷ (T -1) and

Standard Deviation (σi) = √Variance

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Estimating Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return.Calculate the CV of Stock X and Y.

Formula: CV = Standard Deviation (σi) / Expected

return (μ)

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Estimating Return and Risk StatisticsYear Stock X (rx) rx

2 Stock Y (ry) ry

2

1 110% 12100% 23% 529%2 22% 484% 16% 256%3 -60% 3600% 10% 100%4 -20% 400% 8% 64%5 30% 900% 25% 625%

T = 5 ∑rx = 82% ∑rx2 =

17484%∑ry = 82% ∑ry

2 = 1574%

μ 16.4% 16.4%σi 63.52% 7.57%CV 3.87 0.46

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Expected Rate of ReturnExpected rate of return: If we multiply each possible

outcome by its probability of occurrence and then sum these products, we have a weighted average of outcomes. The weights are the probabilities , and the weighted average is the expected rate of return,

n

1iii

^

nn 22 11^

r P r

rP .... rP rP r

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Calculating the Expected Return:^

n^

i ii 1

^

X

^

Y

r expected rate of return

r r P

r = (110%) (0.2) (22%)(0.5) + (-60%)(0.3) = 15%

r = (20%) (0.2) (16%)(0.5) + (10%)(0.3) = 15%

Scenario Probability Company X Company Y

Good 0.2 110% 20%

Average 0.5 22% 16%

Bad 0.3 -60% 10%

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Calculating the Standard Deviation :

deviation Standard

2Variance

2

1

( )ˆn

i ii

r r P

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Standard Deviation Calculation1

2 2 2

X 2

X

12 2 2

Y 2

Y

(110% - 15%) (0.2) (22% - 15%) (0.5)

(-60% - 15%) (0.3) 0.5930 or 59.3%

(20% - 15%) (0.2) (16% - 15%) (0.5)

(10% - 15%) (0.3) 0.0361 or 3.61%

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Comments on Standard Deviation

Standard deviation (σi) measures total, or stand-alone, risk.

The larger σi is, the lower the probability that actual returns will be closer to expected returns.

Larger σi is associated with a wider probability distribution of returns.

Difficult to compare standard deviations, because return has not been accounted for.

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Coefficient of Variation (CV)A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

^

X

Y

Std devCV Mean return r

59.3%CV 3.9515%

3.61%CV 0.2415%

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Scenario Probability Company X Company Y

Good 0.2 110% 20%

Average 0.5 22% 16%

Bad 0.3 -60% 10%

μ 15% 15%

σi59.3% 3.61%

CV 3.95 0.24

Estimating Return and Risk Statistics

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Investor attitude towards riskRisk aversion – assumes investors dislike

risk and require higher rates of return to encourage them to hold riskier securities.

Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

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Investment Decision Making Under RiskRisk = Probability of loosing money or

probability of earning less then certain amount/percentage.

If returns have a normal distribution, we can make decision based on expected return and standard deviation.Investments by Ravi Shukla, page 38-40.

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Normal Distribution and Z value

Z Formula:

Z = [X - Expected return] / Standard DeviationWhere, X = Percentage return defined as risk.

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Z Table

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