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NPTEL Course

Course Title: Security Analysis and Portfolio Management

Instructor: Dr. Chandra Sekhar Mishra

Module-2

Session-4

Risk and Return – II

Outline

● Concept of Required Rate of Return ● Portfolio Risk and Return

In this session we will discuss the concepts of required rate of return, different types of risk and risk & return in the context of a portfolio of investments. Required rate of Return: The investors invest in financial instruments expecting some rate of return. This depends on the return from the next best investment option – known as opportunity cost – and is affected by:

● Time value of money – today’s one rupee is better than tomorrow’s. ● Expected rate of inflation ● Risk involved because of uncertainty in the cash flows from the investment

The Real Risk Free Rate (RFR): This is the return that has no risk the following features:

● Assumes no inflation. ● Assumes no uncertainty about future cash flows. ● Influenced by time preference for consumption of income and investment opportunities

in the economy. By adjusting for inflation, nominal risk free rate can be converted into real RFR.

Real RFR = 1 + Nominal RFR

1 + Inflation rate - 1

Nominal Risk-Free Rate is dependent upon the conditions in the Capital Markets and expected rate of inflation Facets of Fundamental Risk: Risk is fundamentally inherent to investment. All types of investments do not fit to the pattern of ‘no risk, hence risk free return’. Since the investors bear the risk i.e. the uncertainty in future cash flows, they will be demanding a premium for the risk borne. Although, mathematically risk can involve upward or downward swing in the return, practically it is the downside that bothers the investor. The different types of risk that can drive the risk premium are:

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● Business risk ● Financial risk ● Liquidity risk ● Exchange rate risk ● Country risk

For more details about these risks one may refer session # 3. These risks are also non diversifiable or systematic in nature.

Thus, Risk premium:

f(Business Risk, Financial Risk, Liquidity Risk, Exchange Rate Risk, Country Risk); or

f(Systematic Market Risk)

Risk Premiumand Portfolio Theory: The relevant risk measure for an individual asset is its co-movement with the market portfolio.Systematic risk relates the variance of the investment to the variance of the market.Beta measures this systematic risk of an asset. Beta is the regression coefficient of returns of security on returns of market portfolio. Fundamental Risk versus Systematic Risk: Fundamental risk comprises of business risk, financial risk, liquidity risk, exchange rate risk, and country risk.Systematic risk refers to the portion of an individual asset’s total variance attributable to the variability of the total market portfolio. Relationship BetweenRisk and Return: The following figure depicts the relationship between risk and return. The line that combines return and risk is known as Security Market Line (SML). As one perceives more risk, one raises the expected rate of return. The return over and above RFR is known as risk premium.

Figure 4.1

Source: Reilly and Brown, 2006

The required rate of return can change its position on SML based on the movement in the risk. Figure 4.2 reflects that.

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Figure 4.2

Source: Reilly and Brown, 2006 The risk premium can change because of change in the slope of the SML. Slope indicates the return per unit of risk premium.

RPi = E(Ri) - NRFR Where: RPi = risk premium for asset i E(Ri) = the expected return for asset i NRFR = the nominal return on a risk-free asset

Similarly the market risk premium for the market portfolio (contains all the risky assets in the market) can be computed as below:

RPm = E(Rm)- NRFR Where: RPm = risk premium on the market portfolio E(Rm) = expected return on the market portfolio NRFR = expected return on a risk-free asset

Figure 4.3 shows the new SML because of change in slope of SML.

Figure 4.3

Source: Reilly and Brown, 2006

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The security market line can shift because of change in macroeconomic conditions. For example if the inflation rate goes up, other things remaining constant, the SML will shift upward (figure 4.4).

Figure 4.4

Measuring Portfolio Return and Risk: Portfolio return: This is the weighted average return of assets that are part of a portfolio and is measured as below:

Rp = w1 x R1 + w2 x R2 +……. wn x Rn Otherwise, Where,

Rp = Expected return of portfolio; Rj = Expected rate of return from jth asset and wj is the weight of investment in jth asset.

Example: the following table provides the proportion of investment and expected return on four different stocks.

Asset A B C D Proportion in Portfolio (%) 20 30 40 10 Expected Return from Asset .18 .16 .20 .24

Expected Return from Portfolio = 0.20 x .18 + 0.30 x 0.16 + 0.40 x 0.20 + 0.10 x 0.24

n

j

jjp RwR1

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= 0.036 + 0.048 + 0.080 + 0.024 = 0.158 = 15.8% Portfolio Risk Where,

σp = Standard deviation of portfolio return Ri = Expected return on portfolio in a given condition i Pi = Probability of happening of condition i n = number of conditions

Concept of Portfolio Risk: Risk of a portfolio depends on the risk of individual assets of portfolio and the covariance of returns of assets of the portfolio.Risk of portfolio is expected to be reduced with inclusion of more assets in the portfolio.In a two asset portfolio, if return on one asset is negatively correlated the portfolio risk will be lower than when the returns of assets are positively correlated.

● Alternate formula to measure of portfolio risk Where,

σport= the standard deviation of the portfolio Wi= the weights of an individual asset in the portfolio, where weights are determined by the proportion of value in the portfolio The variance of rates of return for asset I Covij = the covariance between the rates of return for assets i and j where, Covij = ρijσiσj

Portfolio Risk: Example Asset E(Ri) Wi i

2 i A 0.10 0.50 0.0049 0.07 B 0.20 0.50 0.0100 0.10 If correlation coefficient between returns of A and B is 0.6, σport= (0.5 x 0.5 x 0.07 x 0.07 + 0.5 x 0.5 x 0.10 x 0.10 + 0.5 x 0.5 x 0.6 x 0.07 x 0.10 + 0.5 x 0.5 x 0.6 x 0.10 x 0.07)½=(0.005825)½=0.076322=7.632%

n

i

n

jij

ijjii

n

iiport Covwww

1 1

2

1

2

.

Variance

1

2

P2

n

ii

P

PRRi

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References

● Reilly and Brown (2006), Investment Analysis and Portfolio Management, 8e, Thomson (Cengage) Learning, New Delhi

● Bodieet al (2009), Investments, 8e, Tata McGraw Hill, New Delhi ● Prasanna Chandra (2008), Investment Analysis and Portfolio Management, 3e, Tata

McGraw Hill, New Delhi

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Questions and Answers

Q.1:

a) If nominal rate of return and inflation rate are 12.4% and 5.6% respectively, what is the real rate of return?

b) If real rate of return and nominal rate of return are 8.5% and 15.4% respectively, what is the inflation rate?

Ans.:

a) Real rate of return = (1+.124)/(1+.056) – 1 = 0.0644 i.e. 6.44% b) Inflation rate = (1+.154)/(1+.085) – 1 = 0.0636 = 6.36%

Q.2: What is systematic risk? What are the different types of risk that make systematic market risk?

Ans.: Systematic risk relates the variance of the investment to the variance of the market. Systematic risk also refers to the portion of an individual asset’s total variance attributable to the variability of the total market portfolio. The different types of risks that make the systematic risk are: Business Risk, Financial Risk, Liquidity Risk, Exchange Rate Risk and Country Risk among others.

Q.3: The following table provides the market values of stocks in one’s portfolio and their expected rates of return. What is the expected rate of return for the portfolio?

Stock: Infosys RIL ONGC SBI DRL TATA Steel

Market Value (Rs.’000)

30,000 20,000 40,000 50,000 20,000 40,000

E(Ri) 0.18 .016 0.12 0.20 -0.10 0.10

Ans:

Stock: Infosys RIL ONGC SBI DRL TATA Steel

Portfolio

Market Value (Rs.’000)

30,000 20,000 40,000 50,000 20,000 40,000 200,000

Weight 0.15 0.10 0.20 0.25 0.10 0.20 1.00

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E(Ri) 0.18 .016 0.12 0.20 -0.10 0.10

Wi*Ri 0.027 0.0016 0.024 0.05 -0.01 0.02 0.1126

Hence the expected return for the portfolio = 11.26%

Q.4: The weights, returns and variance of returns of two stocks that constitute an investment portfolio are given below. Find the portfolio return and risk.

Stock E(Ri) Wi Variance (i2) Standard Deviation

(i)

A 0.12 0.40 0.0064 0.08

B 0.18 0.60 0.0100 0.10

Correlation coefficient between returns of Stocks A and B = 0.8

Portfolio Return = 0.40*0.12 + 0.60*0.18 = 0.156 = 15.6%

n

i

n

j

ijjii

n

iiport Covwww

1

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Portfolio Standard Deviation = (0.4 x 0.4 x 0.08 x 0.08 + 0.6 x 0.6 x 0.10 x 0.10 + 0.4 x 0.6 x 0.8 x 0.08 x 0.10 + 0.6 x 0.4 x 0.8 x 0.10 x 0.08)½ = (0.0049312) ½ = 0.070223 = 7.02%

Q.5: The weights, returns, standard deviation of returns of three stocks (P, Q and R) along with the correlation matrix of returns of such stocks are as below. Find the portfolio return and risk.

Stock E(Ri) Wi Standard Deviation (i)

P 0.16 0.30 0.12

Q 0.20 0.40 0.14

R 0.24 0.30 0.16

Correlation Coefficients:

P and Q Q and R P and R

0.6 0.7 0.5

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Ans.:

Portfolio Return: 0.30*0.16 + 0.40*0.20 +0.30*0.24 = 0.048+0.080+0.072 = 0.20 = 20%

Portfolio Risk (p) = (0.302*0.122 + 0.402*0.142 + 0.302*0.162 + 2*0.6*0.30*0.40*0.12*0.14 + 2*0.7*0.40*0.30*0.14*0.16 + 2*0.5*0.30*0.30*0.16*0.12) ½

= (0.0217984) ½ =0.1476 = 14.76%