Risk and Return
in Capital Budgeting
3-1
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
Return of a Single Asset
R
D t  Pt  Pt 1 
Pt 1
(1)
whereD t  annual income/cas h dividend at the end of time period, t
Pt  security price at time period, t (closing/e nding security price)
Pt -1  security price at time period, t - 1 (opening/b eginning security price)
3-3
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.
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,
n
R   R xPr
(2)
i
i
i1
where R  return for the ith possible outcome
i
Pr  probability associated with its return
i
n  number of outcomes considered
Table 2: Expected Rates of Returns (Probability Distribution)
Possible outcomes
(1)
Probability
Returns
(per cent)
(3)
Expected returns
[(2) × (3)]
(4)
0.20
0.60
0.20
1.00
14
16
18
2.8
9.6
3.6
16.0
0.20
0.60
0.20
1.00
8
16
24
1.6
9.6
4.8
16.0
(2)
Asset X
Pessimistic (recession)
Most likely (normal)
Optimistic (boom)
Asset Y
Pessimistic (recession)
Most likely (normal)
Optimistic (boom)
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
2
 Rt  R x Pri
t 1
Table 3: Standard Deviation of Returns
Asset X
i
Ri
1
2
3
rx 
14%
16
18
 R
3
i
i 1
Ri  R
R
16%
16
16
(–2)%
0
2
R  R 
2
i
R  R   Pr
2
Pri
i
i
4%
0
4
0.20
0.60
0.20
0.80%
0
0.80
1.6
64
0
64
0.20
0.60
0.20
12.8
0
12.8
25.6

2
 R Pr  1.6  1.26 Per cent
Asset Y
1
2
3
ry  25.6  5.06 per cent
8
16
24
16
16
16
(–8)
0
8
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 = w21 σ 21 + w22 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:
= (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  241
σ2p
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.