3. risk and return

3. Risk & Return

• Risk and return concepts & measures

• Relating returns and risks

• Time value of money

Return Concepts • Return concepts

1. Actual return (historical return, realised return, ex post facto return)

2. Expected return (ex ante, predicted, probability-based) 3. Required return (compensation for time & risk)

• In theory – Expected return is equal to required return, if security price is

equal to its fair value – If expected returns is not equal to required return, security is

mispriced

• Investment decision – ‘Buy’ a security if expected return > required return* – A security was a good investment if actual return was greater

than or equal to the required return * greater than sign is due to a ‘margin of safety’ (relevant for future but not for past)

Risk Concepts

• Sources of total risk – Market risk vs company-specific (unique) risk

– Systematic risk vs unsystematic risk

• In theory – Unsystematic risks are diversifiable and there is no

compensation for these risks.

• Risk measures – Historical risk (ex post facto)

– Predicted risk (ex ante, probability-based)

Systematic & Unsystematic risks

• Systematic risks may be attributable to fluctuations in macro-economic factors like economic growth, government spending, money supply, interest rates, inflation, etc.

• Unsystematic risks usually arise from company-specific factors such as product failure, competition, production constraints etc. Unsystematic risks are diversifiable.

Measures for Return

• Measures for realised return – Holding period return: total return, annualised return – Return relative and cumulative wealth index – Average return over multiple periods (statistical measure)

• Arithmetic mean • Geometric mean (time-weighted return)

– Real world impact: inflation, taxes & exchange rates • Inflation: real vs nominal return • Taxes on income/dividends and capital gains • Impact of exchange rates for international investments

• Estimation of expected return – Expected holding period return based on a target price, or – Expected return based on probability distribution of returns

Holding Period Return • Holding period return HPR = [(P1-P0) + C]/P0

where P0 is price at the beginning of the holding period (or purchase price), P1 is price at the end of the holding period, C is the cashflow to the investor during the period (dividend paid on shares/ interest paid on bonds) Restating the formula, P0 = (P1 + C)/(1+R); (P1 + C) = P0 (1+R)

• Annualised return If r denotes annual return and t denotes holding period in years (1+HPR) = (1+r)t

HPR = (1+r)t - 1 r = (1+HPR)1/t - 1

• Real return = (1+Nominal return)/(1+Inflation) - 1

Statistical measures for Risk

• Historical risk

– Based upon variability of returns

• Variance & standard deviation (volatility)

• Criticism of above measures – Do not differentiate ‘down-side risk’ : semi-variance

– Ignore shape of probability distribution: Skewness and kurtosis

• Predicted risk

– Based on variability of expected returns

Measures of Security Return and Risk Historical (Ex post facto)

Expected (Ex ante)

Mean Return 𝑟 = Σri/n E(r) = Σpi ri

Variance (σ2) and Standard Deviation (σ)

σ2 = Σ(ri – 𝑟 )2/(n-1) σ2 = Σpi(ri – E(ri))2

1. 𝑟 is mean historical return & σ the standard deviation of ri, & ri denotes return for period i, where i = 1,2….n. 2. E(r) is expected return & σ is the standard deviation of expected ri, & i =1,2.. denote mutually exclusive states with expected return ri and probability pi

Relating Return & Risk

• Suppose that an investor is provided two investment alternatives as shown above. The two alternatives have the same expected return. – However, a risk averse investor would prefer the risk-free alternative in this case.

For the same level of expected return, the risk-free alternative is more valuable. – In fact, the risk averse investor may be ready to accept lower expected return, till a

point where the investor is indifferent between an alternative which provides a lower but certain expected return, and an alternative which provides a higher but uncertain expected return. The former is known as the certainty equivalent value of the latter.

– Hence required return for risky investment > required return for a risk free certainty equivalent investment

– The difference in required returns between a risky investment and its certainty equivalent risk-free investment is known as risk premium.

Risk-free alternative Risky alternative

P0 Rs 40,000 Rs 40,000

P1 Rs 50,000 (p=100%) Rs 100,000 (p=50%), Rs 0 (p=50%)

E (P1) Rs 50,000 Rs 50,000

E (ri) 25% 25%

σ 0 125%

Investor’s Risk Preference: Indifference Curves

Highly Risk Averse

Moderately Risk Averse

Less Risk Averse

Risk Neutral

Risk Seeker

Speculator

Gambler

Exp

ecte

d R

etu

rn

Standard Deviation

The total satisfaction derived by an investor from a given return and risk trade-off is the ‘utility’ of that investor for that combination. Investors may derive the same utility from different combinations of return and risk. A curve joining all the return and risk combinations for which the utility to the investor is the same is known as an indifference curve. The risk aversion of an investor determines the slope of an indifference curve, while the level of utility determines its position. In the first graph, investor A is more risk averse than investor B. For A, curve A1 represents higher utility than curve A2, because every point on A1 provides more return for the same level of risk than A1. Similarly curve B1 represents higher utility than B2.

Exp

ecte

d R

etu

rn

Standard Deviation

A1 A2 B1

B2

Measuring relationship between Return and Risk

• The models quantifying the relationship between required return and risk are called return generating models or return models (also known as asset pricing theories)

• The most common models/theories* are – The market model

– The capital asset pricing model

– The arbitrage pricing theory

– The multifactor models

*These will be discussed in subsequent sessions

Time value of Money

• Relationship between Future Value (FV) & Present Value (PV) of money – FV = PV (1+r)n

• The power of compounding means that small differences in rates of return can result in significant changes in cumulative wealth

• Calculating FV & PV – of one-time cashflow (formula above) – of uneven cash flows – of annuity – of perpetuity

Time Value Calculations*

• Uneven Cash Flows: C1, C2….Cn – PV = ΣCt/(1+r)t

– FV = PV(1+r) n

• Annuity: Cash Flow C x n periods – PV = C [(1-(1/(1+r)n)/r] – FV = C [(1+r)n-1]/r

• Perpetuity: Cash Flow C – PV = C/r

• Growing Perpetuity: Cashflow C, Growth rate g – PV = C/(r-g) *Will be relevant during the sessions on Valuation

Relationship between Security Price and Returns

• Present value of an investment is inversely related to the required return – PV = ΣCi/(1+r)i ; C1…Cn are inflows, r is required return

• According to valuation theories, fair price of a security is equal to the PV of future inflows

• Assuming no mispricing, security price = fair price (PV); and expected returns = required returns

• Hence security price is inversely related to return

• Ex. bond prices fall, when interest rates rise

Relationship between Stock Returns and Return on Equity

1. Return on equity (RoE or Return on Networth) is an accounting concept, based upon historical value of equity and not on market stock prices

2. Hence RoE is not the same as stock return, which is based on the market price paid by the stock owner

3. According to valuation models, fair value of a stock is related to its RoE. 4. But since RoE does not impact risks, it is not related to required returns. 5. If stocks are not mispriced, expected returns are equal to required

returns. Hence, expected returns should not be related to RoE. 6. Statements 3 & 5 are not contradictory. If fundamentals of a company

increase (as reflected for ex. by an increase in RoE), its stock prices will also increase such that the expected return tends to become equal to the required return.

7. Conversely, if fundamentals of a company worsen, resulting in lower expected cash flows and RoE, its stock prices will decline such that the expected stock return tends to become equal to the required return.

Not for syllabus

Building blocks of Security Analysis

• Risk-return concepts: How to calculate historical and expected returns and risks?

• Time value of money: Relationship between present value and future cash flows

• Portfolio theory: How is return related to risk in a portfolio context

• CAPM (& other return models): How to calculate required return based on risk?

• EIC analysis: How to project future cash flows of a firm? • Valuation theory: How to estimate fair value of a security

based on future cash flows or dividends (projected based on EIC) and required returns (estimated based on a return model)?

• Efficient market hypothesis: Whether security analysis is a value-adding activity?

Reading references & Home work

• Reading Ref: PC Ch 4, BKMM Ch 5 & 6 – IBS Case Study: Comparing Sensex returns & Fixed deposits

• Home Work

– For your assigned company: • Take monthly stock prices for 5 years (60 data points) from BSE/NSE

websites and BSE 30 index/Nifty 50 • Adjust the prices for any bonus/splits/consolidations • Calculate the monthly returns from the monthly price series for the

stock and for the index • Calculate arithmetic and geometric mean returns for the stock and for

the index • Calculate the variance and standard deviation for the stock and for the

index • Comment upon the average return and risk for your stock compared

with the index