3 - 1

Risk and Returnin Capital Budgeting

Risk And Return of A Single Asset

• Risk refers to the variability of expected returns associated with a given security or asset.

• Return- Periodic cash receipts & Appreciation

( Depreciation in the price of the asset

3 - 3

Return of a Single Asset

price)security eginning(opening/b 1- t period, time at pricesecurity P

price)security nding(closing/e t period, time at pricesecurity P

t period, time of end the at dividend hincome/cas annual whereD

(1)P

PPDR

1-t

t

t

1t

1ttt

If the price of a share on April 1 (current year) is Rs 25, the annual dividend received at the end of the year is Re 1 and the year-end price on March 31 is Rs 30, the rate of return = [Re 1 + (Rs 30 – Rs 25)]/Rs 25 = 0.24 = 24 per cent. The rate of return of 24 per cent has two components:

(i) Current yield, i.e. annual income ÷ beginning price = Re 1/Rs 25 = 0.04 or 4 per cent and

(ii) Capital gains/loss = (ending price – beginning price) ÷ beginning price = (Rs 30 – Rs 25)/ 25 = 0.20 = 20 per cent.

Measurement of Risk

The two major concerns of an investor, while choosing a security (asset) as an investment, are the expected return from holding the security and the risk that the realised return may fall short of the expected return. To obtain a more concrete measure of risk, two statistical measures of variability of return, namely, standard deviation and coefficient of variation, can be used.

Probability (Distribution) Probability distribution is a model that relates probabilities to the associated outcome. Probability is the chance that a given outcome will occur.

Table 2: Table 2: Expected Rates of Returns (Probability Distribution)

Possible outcomes

(1)

Probability

(2)

Returns(per cent)

(3)

Expected returns[(2) × (3)]

(4)

Asset X

Pessimistic (recession)Most likely (normal)Optimistic (boom)

0.200.600.201.00

141618

2.89.63.6

16.0

Asset Y

Pessimistic (recession)Most likely (normal)Optimistic (boom)

0.200.600.201.00

81624

1.69.64.8

16.0

Based on the probabilities assigned (probability distribution of) to the rate of return, the expected value of the return can be computed. The expected rate of return is the weighted average of all possible returns multiplied by their respective probabilities. Thus, probabilities of the various outcomes are used as weights. The expected return,

considered outcomes of number n

return its withassociatedy probabilit iPr

outcome possible ith the for return iR where

(2)n

1i ixPriRR

Table 3: Standard Deviation of Returns

Asset X

i

123

14%16  18  

16%16  16  

(–2)%02

4%0  4  

0.200.600.20

0.80%0

0.80

1.6

Asset Y

123

81624

161616

(–8)08

640

64

0.200.600.20

12.80

12.825.6

Standard Deviation Standard deviation measures the dispersion around the expected value. Expected value of a return is the most likely return on a given asset/security.

n

t ixRtR1Pr

2

iR

centper 06.56.25 yr

R RRi 2RRi iPr ii RR Pr

2

centPer 26.16.1Pr3

1

2

iix RRr

Risk And Return of Portfolio

Risk and Return of Portfolio

A portfolio means a combination of two or more securities (assets). A large number of portfolios can be formed from a given set of assets. Each portfolio has risk-return characteristics of its own.

Portfolio Expected Return

The expected rate of return on a portfolio is the weighted average of the expected rates of return on assets comprising the portfolio. Symbolically, the expected return for a n-asset portfolio is defined by Equation 5.

E (rp) = Σwi E (ri)

where E (rp) = Expected return from portfolio

wi = Proportion invested in asset i

E (ri) = Expected return for asset i

n = Number of assets in portfolio

Example 2  

Suppose the expected return on two assets, L (low-risk low-return) and H (high-risk high-return), are 12 and 16 per cents respectively. If the corresponding weights are 0.65 and 0.35, the expected portfolio return is = [0.65 × 0.12 + 0.35 × 0.16] = 0.134 or 13.4 per cent.

Portfolio Risk (Two-Asset Portfolio)

Total risk is measured in terms of variance (σ2, pronounced sigma square) or standard deviation (σ, pronounced sigma) of returns. The overall risk of the portfolio includes the interactive risk of an asset relative to the others, measured by the covariance of returns. The covariance, in turn, depends on the correlation between returns on assets in the portfolio. The total risk of a portfolio made up of two assets is defined by the Equation 6.

σ2p = w2

1 σ 21 + w2

2 σ 22 + 2w1 w2 (σ12)

Alternatively,

σ 2p= (w1 σ1)2 + (w2 σ2)2 + 2w1w2 (ρ12 σ1 σ2)

where σ2p = Var (rp) or variance of returns of the portfolio

w1 = Fraction of total portfolio invested in asset 1

w2 = Fraction of total portfolio invested in asset 2

σ21 = Variance of asset 1

σ1 = Standard deviation of asset 1

σ22 = Variance of asset 2

σ2 = Standard deviation of asset 2

σ12 = Covariance between returns of two assets (= ρ12 σ1 σ2)

ρ12 = Coefficient of correlation (pronounced Rho) between the returns of two assets.

Let us assume that standard deviations of assets L and H, of our Example 2 are 16 and 20 per cents respectively. If the coefficient of correlation between their returns is 0.6 and the two assets are combined in the ratio of 3:1, the expected return of the portfolio is determined as follows:

E (rportfolio) = wLE (rL) + wH E (rH)

= (0.75 × 12%) + (0.25 × 16%) = 9.0% + 4.0% = 13 per cent

The variance of the portfolio is given by:

σ2p = (w1 σ1)2 + (w2 σ2)2 + 2 w1 w2 (ρ12 σ1 σ2)

= (0.75 × 16)2 + (0.25 × 20)2 + 2 (0.75) (0.25) [(0.6) (16 × 20)]= 144 + 25 + (0.375)(192) = 144 + 25 + 72 = 241

Thus, σp = 15.52 per cent

The above discussion shows that the portfolio risk depends on three factors: (a) Variance (or standard deviation) of each asset in the portfolio; (b) Relative importance or weight of each asset in the portfolio; (c) Interplay between returns on two assets or interactive risk of an asset relative to other, measured by the covariance of returns.

241