Chapter 12
Risk Analysis and Returns
Babita Goyal
Key words:
Risk, sensitivity analysis, probability distribution of risk, correlated cash flows,
investment decision trees, expected monetary value, risk profile, certainty equivalent, risk adjusted
discount method, Portfolio, expected return, portfolio risk, correlation between assets,
diversification, systematic error, unsystematic error, market portfolio, risk free asset, CML, SML,
CAPM.
Suggested readings:
1.
Chandra P. (1970), Appraisal Implementation, Tata-McGraw Hill Publishing Company
Limited, New Delhi.
2.
Gupta P.K. and Mohan M. (1987), Operations Research and Statistical Analysis, Sultan
Chand and Sons, Delhi.
3.
Hampton J.J. (1992), Financial Decision Making (4th edition), Prentice hall of India
Private Limited
4.
Khan M.Y. and Jain P.K. (2004), Financial Management (4th edition), Tata-McGraw Hill
Publishing Company Limited.
5.
Swarup K., Gupta P.K. and Mohan M. (2001), Operations Research, Sultan Chand and
Sons, Delhi.
449
12.1
Introduction
Till now, we have assumed that all investments or cash flows have equal risk and the average cost of
capital for the firm could be used to evaluate investment proposals. However, this is not the case and
investment proposals differ in the risk involved. These investments or the cash flows are related to
the future and the future is uncertain. There is an uncertainty in anticipating the future due to
changing economic, technological, and political conditions. Thus the prediction of future events is
difficult which results in variability in the expected stream of returns. This variability has been
termed as 'risk'.
All the projects do not possess the same type of variability. For example, a proposal to manufacture
a new product is riskier than a proposal to renovate an existing plant. Similarly the investments
made in the Government bonds at fixed yield are likely to give almost 'certain' returns and hence are
less risky than the investment made in the stocks of a private company. In view of such variations,
while appraising an investment, risk associated with that proposal should be taken into consideration.
Consider the two proposals A and B
Table 12.1
Proposal
A
B
Possible outcome
Probability
Rs. 8,000
0.5
0
0.5
Rs. 4,000
1
The expected gain from both the proposals is Rs. 4,000. However proposal A is riskier than proposal
B. In such situations B is a preferred option.
450
12.2
Measures of risk- of a single asset
Risk refers to variability, which has several measures. For a single investment/asset the associated
risk can be expressed from these measures. There are two different viewpoints from which risk can
be measured.
(i)
Behavioral point of view
When judging the risk from behavioral point of view, we
can use either of the following methods:
(a)
Sensitivity analysis; and
(b)
Probability distribution associated with the risk.
(ii)
Quantities point of view
When judging the risk from quantitative point of view, we
can use
(a)
Expected monetary value;
(b)
Range;
(c)
Mean absolute deviation;
(d)
Standard deviation;
(e)
Coefficient of variation; and
(f)
Semi variance.
Now, we discuss these techniques in detail.
(i)
Behavioral point of view
(a)
Sensitivity analysis
Sometimes called 'what-if analysis', this method answers questions regarding changes in the key
input processes. For example,
(i)
What will happen to the NPV of the project if the sales are 20% below the expected point?
(ii)
What will be the rate of return if the project winds up in fifth year of its life instead of
anticipated eight years?
451
This method is based on the statistic range to measure the variability, which in this case is the
difference between the most optimistic and the most pessimistic outcome/ return estimate of the
security or asset. Since the level of the outcome is directly related to the state of the economy, viz.,
recession, normal or boom, sensitivity analysis provides a measure of the risk under these different
conditions of the economy. Higher is the range, greater is the variability, and riskier is the project.
Consider the following investment proposal:
Table 12.2
Project details
Investment A
Investment B
Investment C
100
100
100
Pessimistic
14
12
8
Normal (most likely)
16
16
18
Optimistic
18
20
22
4
8
14
Initial outlay (Rs.)
Annual returns (Rs.)
Range
On the basis of the sensitivity analysis, the investments in increasing order of risk are A, B and C.
The procedure of sensitivity analysis consists of the following steps:
(i)
Set up the relationship between the basic variables of the process e.g., the quantity sold,
unit-selling price etc. and the criterion to study the performance e.g., NPV or IRR.
(ii)
Estimate the range of the variation and the most likely value of the basic variables.
(iii)
Study the effect on the measure of performance of the variations in the basic variables.
Typically, one variable is chosen at a time.
Example 1:
The NPV relationship for an investment proposal is as follows
NPV =
n
∑
( Q( P − V ) − F − D ) (1 − T ) + D
(1 + k )
t =1
452
t
+
S
− I
(1 + k ) n
where
Q = Number of units sold annually;
V = Variable unit cost;
F = Total fixed cost excluding depreciation and interest;
D = Annual depreciation charge;
T = Income tax rate;
k = Cost of capital;
n = Life of the project in years;
S = Net salvage value; and
I = Initial cost of the project.
The range and the most likely value of each of the basic variables is given below:
Table 12.3
Variables
Range
Most likely value
Q
1000-2000
1600
P
Rs. 600-1000
Rs. 750
V
Rs. 300-500
Rs. 400
F
Rs. 1,20,000-1,20,000
Rs. 1,20,000
D
Rs. 1,50,000-1,50,000
Rs. 1,50,000
T
0.6-0.6
0.6
k
0.08-0.11
0.10
n
3-5
5
S
Rs. 4,00,000-4,00,000
Rs. 4,00,000
I
Rs. 12,00,000-12,00,000
Rs. 12,00,000
Holding the other variables constant, we study the relationship between (i) k and NPV; and (ii) P and
NPV.
453
(i)
k and NPV
Putting the most likely values of the other variables in the given relation, we have the relationship
between k and NPV as follows
NPV =
5
∑
(1600(750 − 400) − 1, 20, 000 − 1, 60, 000 ) (1 − 0.6) + 1, 60, 000
(1 + k )
t =1
=
5
∑
t =1
+
t
400000
− 1, 20, 000
(1 + k )5
2, 72, 000
4, 00, 000
+
− 1, 20, 000
(1 + k )t
(1 + k )5
Giving different values to k in its range, we have corresponding NPV as follows
Table 12.4
k (%)
0.08
0.09
0.10
0.11
NPV (Rs.)
158080
118080
79552
42700
Graphically the situation may be represented as follows
NPV (Rs.'000)
150
.
100
.
50
.
.
.
0.08
0.09
.
.
0.10
0.11
k (%)
Fig. 12.1
(ii)
P and NPV
Putting the most likely values of the other variables in the given relation, we have the relationship
between P and NPV as follows
454
100
NPV =
5
∑
(1600( P − 400) − 1, 20, 000 − 1, 60, 000 ) (1 − 0.6) + 1, 60, 000
(1 + 0.10)
t =1
=
5
640 P
∑ (1.10)
t =1
t
−
5
∑
t =1
t
+
4, 00, 000
− 1, 20, 000
(1 + 0.10)5
2, 08, 000 4, 00, 000
+
− 1, 20, 000
(1.10)t
(1.10)5
Giving different values to P in its range, we have corresponding NPV as follows
Table 12.5
P (Rs.)
600
700
750
800
900
1000
NPV (Rs.)
-284384
-41760
79552
200864
443488
686112
Graphically the situation may be represented as follows
NPV('000)
700
600
.
.
400
.
.
300
.
200
.
0
.
-100
.
-200
.
-300
.
500
.
.
.
.
.
.
.
600
700
750
800
900
1000
Fig. 12.2
455
P
The graphs show that NPV may be very sensitive to even small changes in the parameters. For
example, changing P from Rs. 700 to Rs. 750 has transition from negative value to positive value.
A useful way of presenting the results of sensitivity analysis is to show the behavior of the measure
of performance for different values of unfavorable changes in the basic variables from their most
likely values.
Consider the following tables
Table 12.6:
k vs NPV
%age unfavorable variation in k
k (%)
NPV (Rs.)
5
10.5
60,418
10
11.0
42,700
15
11.5
24,681
20
12.0
7,575
Table 12.7:
P vs NPV
%age unfavorable variation in P
P (Rs.)
NPV (Rs.)
5
713
(10,308)
10
675
(1,02,500)
15
638
(1,92,267)
20
600
(2,84,384)
456
NPV (Rs. ‘000)
.
40 .
20 .
0 .
60
-25
.
k and NPV
.
.
.
5
10
15
.
-75 .
-100 .
-125 .
%age unfavorable variation
.
20
-50
P and NPV
.
-175 .
-200 .
-150
-225
.
.
-275 .
-300 .
-250
Fig. 12.3
Such a presentation is helpful in identifying the variables, which are crucial to the success of the
project.
This method thus, indicates the need for further analysis if the measures of performance are sensitive
to the small changes in the values of basic variables. Thus the method is an indicator of the
robustness of the project.
Example 2:
The initial investment outlay of a capital investment project consists of Rs. 100
lakh for plant and machinery. The working capital requirement is Rs. 560 lakh. The following
information is available regarding the project
457
Table 12.8
Description
Value
1
Sales (lakh units per annum for five years)
Selling price per unit
Rs. 120
Variable cost per unit
Rs. 60
Rs. 15,00,000
Fixed costs (excluding depreciation)
Rate of depreciation (WDV) (%)
25
Tax
40
Post-tax cu off rate (%)
12
Rs. 23,73,000
Salvage value
Find the financial viability of the project by NPV method. What will be the effect of reduction in SP
by 10%?
Sol:
The cash flows associated with the project are as follows
Table 12.9
Cash out flow
Rs. 150 lakh
Plant and machinery
Rs. 100 lakh
Working capital
Rs. 50 lakh
458
We now determine the cash flows after tax of the proposal.
Table 12.10
Particulars
Cash flows in years (Rs. lakh)
1
2
3
4
5
120
120
120
120
120
VC
60
60
60
60
60
FC
15
15
15
15
15
45
45
45
45
45
25
18.75
14.06
10.55
7.91
20
26.25
30.94
34.45
37.09
8
10.5
12.38
13.78
14.84
EAT
12
15.75
18.56
20.67
22.25
CFAT
37
34.5
32.62
31.22
30.16
Sales
Less:
EBDT
Less:
Depreciation
EBT
Less:
Tax (@ 40%)
(EAT + depreciation)
Salvage value
23.73
Recovery of WC
50
103.89
PV factor (12%)
0.893
0.797
0.712
0.636
0.567
PV of cash flows
33.04
27.50
23.22
19.86
58.91
162.53
Less:
150
Cash inflows
NPV
12.53
Since NPV is positive soothe project should be accepted.
449
Table 12.11:
When SP decreases by 10%
Particulars
Cash flows in years (Rs. lakh)
1
2
3
4
5
108
108
108
108
108
VC
60
60
60
60
60
FC
15
15
15
15
15
33
33
33
33
33
25
18.75
14.06
10.55
7.91
8
14.25
18.94
22.45
25.09
3.2
5.7
7.58
8.98
10.04
EAT
4.8
8.55
11.36
13.47
15.05
CFAT
(EAT + depreciation)
29.8
27.3
25.42
24.02
22.96
Sales
Less:
EBDT
Less:
Depreciation
EBT
Less:
Tax (@ 40%)
Salvage value
23.73
Recovery of WC
50
96.69
PV factor (12%)
0.893
0.797
0.712
0.636
0.567
PV of cash flows
26.61
21.75
18.10
15.27
54.82
136.55
Less:
150
Cash inflows
NPV
Now,
(13.45)
The original NPV = Rs. 12.53 lakh
New NPV = -Rs. 13.45 lakh
Change in NPV = -Rs. 13.45 lakh - Rs. 12.53 lakh = - Rs.25.98 (-207.34%)
The project is sensitive w.r.to SP and a decrease of 10% in NPV has rendered the project unviable.
449
This method gives a crude estimate of the risk since it does not involve the chances of any particular
outcome to turn up. Also, the assumption that only one variable is changing, while the others are
held at a constant level, may be somewhat unrealistic.
Hertz simulation model
This model, developed by David B. Hertz, is similar to sensitivity analysis. The model consists of
the following steps:
(i)
Estimate the range of the values for each of the factors included in the model, e.g., selling
price, sale in units, costs, revenue etc. This is equivalent to obtaining the probability
distribution of the factors included in the model.
(ii)
From the probability distribution of the factors, choose values at random with the
respective probabilities to determine the expected values of the factors. The combined
values of all the factors would yield the rate of return from that combination.
(iii)
Repeat the steps (i) and (ii) a number of times to achieve a range of values for rate of return.
The basic objective of financial decision-making is to select a proposal having a criterion with
minimum risk. When the expected return and standard deviation of a series of investment proposals
have been obtained, the same procedure may be applied to study the effectiveness of various
combinations of them in meeting the objective of the firm. Hertz has suggested nine factors to be
considered while evaluating an investment proposal:
(i)
Market size;
(ii)
Selling price;
(iii)
Market growth rate;
(iv)
Eventual market share;
(v)
Total investment required;
(vi)
Useful life of the facilities;
(vii)
Residual value of investment;
(viii)
Operating costs; and
450
(ix)
Fixed costs.
These factors have a bearing on the NPV of the project. The DCF model considers the expected
values of the input variables, which are subject to chance fluctuations. Hertz suggested considering
range of each variable in determining the NPV.
An illustrative range of the factors can be as follows
Table 12.12
Factor
Range
Expected value
Market size
1,00,000-4,00,000
2,25,000
Selling price
250-350
300
Market growth rate
0-8%
5%
Eventual market share
5-20%
10%
Total investment required
Rs. 10,00,000-15,00,000
Rs. 12,00,000
Useful life of the facilities
5-10 years
8 years
Residual value of investment
Rs. 2,00,000-3,00,000
Rs. 2,50,000
Operating costs
Rs. 5,00,000-7,50,000
Rs. 5,50,000
Different levels of these factors can be considered to begin with. For example, in the first run a very
high operating cost can be combined with a low market share (starting of a product). In the second
run, the operating costs are decreased with a large market share. The process is continued in this
manner a large number of times. The outputs obtained form the distribution of the NPV. Then the
expected NPV along with the standard deviation can be obtained.
(iii)
Probability distribution of the risk
A better measure of the risk is the expected value of the return given by
E ( R) =
n
∑R
i =1
i
P( Ri )
where Ri = Return of the i th possible outcome;
P ( Ri ) = Probability of the i th outcome; and
n
= Total number of possible outcomes.
451
An investment with the higher expected return is considered to be less risky.
In the following example, the probability distribution associated with various possible outcomes is
given.
Table 12.13
Project details
Investment A
Investment B
Investment C
100
100
100
Pessimistic
14
12
8
Normal (most likely)
16
16
18
Optimistic
18
20
22
4
8
14
Initial outlay (Rs.)
Annual returns (Rs.)
Range
Then we want to find out least risky investment.
Table 12.14
Event (i)
A
Return (Ri)
Outcome probability P (Ri)
Ri P (Ri)
Pessimistic
14
0.20
2.8
Most likely
16
0.60
9.6
Optimistic
18
0.20
3.6
16.0
B
Pessimistic
12
0.10
1.2
Most likely
16
0.60
9.6
Optimistic
20
0.30
6.0
16.8
C
Pessimistic
8
0.05
0.4
Most likely
18
0.70
12.6
Optimistic
20
0.25
5.0
18.0
The investments in increasing order of risk are C, B and A.
452
However, in general, the state of the economy can assume more values than the number of listed
discrete values. In such situations, a continuous probability distribution can be used to explain the
stream of cash flows associated with the project. One commonly used distribution in such situations
is the normal distribution.
Consider the following data
Table 12.15
Rate of return (%) from an investment
Probability
-5 – 0
0.10
0–5
0.15
5 – 10
0.25
10 – 15
0.25
15 – 20
0.15
20 – 25
0.10
The mean of this data is 10% and the standard deviation is 7.15%
The distribution of the rates of return can be approximated by a normal distribution as follows:
.
.
.
-5%
0%
5%
x =10%
.
.
.
15%
20%
25%
Return
Fig. 12.4
Mathematical analysis
Now, we carry out mathematical analysis of the cash flows involving risk. The cash flows at
different times may be uncorrelated, moderately correlated, or perfectly correlated. We explain
below each of these types of cash flows.
(i)
Uncorrelated cash flows
453
When there is no correlation between the cash flows occurring at various time periods, such cash
flows are termed as uncorrelated cash flows. In such cases, the expected NPV of the project is given
as
NPV =
n
At
∑ (1 + i)
t =1
t
−I
1
σ ( NPV )
⎛ n σ t2 ⎞ 2
= ⎜∑
2t ⎟
⎝ t =1 (1 + i ) ⎠
where
NPV = Expected NPV ;
At
= Expected cash flow at time t ;
i
= Risk-free interest rate;
I
= Initial outlay; and
σt
= Standard deviation of the cash flow at time t.
It may be noted that in this case, the discount rate is the risk free interest rate because the time value
of money and the risk factor are being treated separately. The risk factor is being reflected in σ ( NPV )
of the project along with the computation of NPV.
Example 3:
The benefits associated with a project involving an initial outlay of Rs. 10,00,000
are as follows:
Table 12.16
Year 1
Cash flow (Rs.
Year 2
Probability
lakh)
Cash flow (Rs.
Year 3
Probability
lakh)
Cash flow (Rs.
Probability
lakh)
4
0.4
5
0.4
3
0.3
5
0.5
6
0.4
4
0.5
6
0.1
7
0.2
5
0.2
Assume that the cash flows are independent. Calculate the expected NPV and the standard deviation
of the NPV if the risk free rate of interest is 10%.
454
Sol:
A1 = 4(0.4)+5(0.5)+6(0.1) = 4.7 lacs
A2 = 5.8 lacs
A3 = 3.9 lacs
∴ NPV =
n
At
∑ (1 + i)
t =1
=
t
−I
⎛ 4.7
⎞
5.8
3.9
+
+
− 10 ⎟ lacs
⎜
2
3
(1.10)
⎝ 1.10 (1.10)
⎠
= 1.99 lacs
Again,
σ 12 = 0.4(4-4.7) 2 + 0.5(5 − 4.7) 2 + 0.1(6 − 4.7)2
= 0.41
σ 22 = 0.56
σ 32 = 0.49
1
∴ σ ( NPV )
⎛ n σ t2 ⎞ 2
= ⎜∑
2t ⎟
⎝ t =1 (1 + i ) ⎠
1
⎛ 0.41
0.56
0.44 ⎞ 2
= ⎜
+
+
⎟
2
4
(1.10)
(1.10)6 ⎠
⎝ (1.10)
= 0.999 lacs
Example 4:
The benefits associated with a project involving an initial outlay of Rs. 10,00,000
are as follows:
Table 12.17
Year 1
Cash flow (Rs.
Year 2
Probability
lakh)
Cash flow (Rs.
Year 3
Probability
lakh)
Cash flow (Rs.
Probability
lakh)
0
0.1
1
0.15
0
0.15
2
0.2
4
0.2
1.5
0.2
4
0.4
7
0.3
3
0.3
6
0.2
10
0.2
4.5
0.2
8
0.1
13
0.15
8
0.15
455
Assume that the cash flows are independent. Find
(i)
The probability that the NPV of the project is less than or equal to 0.
(ii)
The probability that the NPV of the project is equal to or more than Rs. 1,00,000.
(iii)
The probability that the NPV of the project is equal to or more than Rs. 3,00,000
Sol:
∴ NPV =
⎛ 4.0
⎞
7.0
3.0
+
+
− 10 ⎟ lakh
⎜
2
3
1.10
(1.10)
(1.10)
⎝
⎠
= 1.67 lakh
1
∴ σ ( NPV )
⎛ 219081 378473 189737 ⎞ 2
= ⎜
+
+
⎟
2
(1.10) 4 (1.10)6 ⎠
⎝ (1.10)
= Rs. 3,97,921
(i)
P ( the NPV of the project is less than or equal to 0)
= P ( NPV ≤ 0)
0 − 167000 ⎞
⎛ NPV − 167000
= P⎜
≤
⎟
397921
397921 ⎠
⎝
= P ( Z ≤ -0.4197)
= 0.1628
(ii)
P ( the NPV of the project is equal to or greater than 1,00,000)
= P ( NPV ≥ 100000)
100000 − 167000 ⎞
⎛ NPV − 167000
= P⎜
≥
⎟
397921
397921
⎝
⎠
= P ( Z ≥ -0.1683)
= 0.5675
456
(iii)
P ( the NPV of the project is equal to or greater than 3,00,000)
= P ( NPV ≥ 300000)
300000 − 167000 ⎞
⎛ NPV − 167000
= P⎜
≥
⎟
397921
397921
⎝
⎠
= P ( Z ≥ 0.3342)
= 0.3707
(ii)
Moderately correlated cash flows
In this type of cash flows, a moderate correlation is observed. To evaluate such cash flows, a series
of probability distributions is used. Consider the following example.
Example 5:
Initial outlay of a project is Rs. 1,50,000. The following table summarizes the cash
flows associated with the project along with their probabilities.
Table 12.18
Year 1
Cash flow (Rs.)
Year 2
Probability
Cash flow (Rs.)
60000
50000
Year 3
Probability
0.6
Cash flow (Rs.)
Probability
40000
0.7
50000
0.3
0.4
80000
0.6
80000
0.4
90000
0.8
1,00,000
0.2
60000
0.6
70000
0.4
80000
0.3
90000
0.7
100000
0.8
110000
0.2
The risk free arte of interest is 5%. What is the expected NPV of the project and what is the standard
deviation of the NPV?
Sol:
There are 8 possible cash flow streams and we calculate the NPV of each of the stream.
457
Table 12.19
Cash flow stream
Probability
NPV (Rs.)
1.
0.168 (= 0.4 × 0.6×
0.7)
50, 000 60, 000 40, 000
+
+
− 1,50, 000 = − 13, 406
1.05
(1.05) 2 (1.05)3
2.
0.072
50, 000 60, 000 50, 000
+
+
− 1,50, 000 = − 4, 767
1.05
(1.05) 2 (1.05)3
3.
0.096
50, 000 80, 000 60, 000
+
+
− 1,50, 000 = 22, 012
1.05
(1.05) 2 (1.05)3
4.
0.064
50, 000 60, 000 70, 000
+
+
− 1,50, 000 = 30, 650
1.05
(1.05) 2 (1.05)3
5.
0.144
80, 000 90, 000 80, 000
+
+
− 1,50, 000 = 39, 288
1.05
(1.05) 2 (1.05)3
6.
0.336
80, 000 90, 000 90, 000
+
+
− 1,50, 000 = 47,928
1.05
(1.05) 2 (1.05)3
7.
0.096
80, 000 1, 00, 000 1, 00, 000
+
+
− 1,50, 000 = 56,565
1.05
(1.05) 2
(1.05)3
8,
0.024
80, 000 1, 00, 000 1,10, 000
+
+
− 1,50, 000 = 64,961
1.05
(1.05) 2
(1.05)3
Expected NPV =
8
∑ p NPV
i
i =1
i
= Rs. 30916
σ NPV =
(iii)
8
∑ p ( NPV − Expected NPV )
i =1
i
i
2
= Rs. 25052.44
Perfectly correlated cash flows
In this type of cash flows, there is perfect linear relationship between cash flows spread over
different time periods. In such cases, the expected NPV of the project is given as
NPV =
n
At
∑ (1 + i)
t =1
σ ( NPV ) =
458
n
σt
∑ (1 + i)
t =1
−I
t
t
Example 6:
An investment project involves a current outlay of Rs. 1,00,000. The cash flows
spread over a horizon of four years are perfectly correlated and the mean and standard deviation of
the cash flows over different years are as follows:
Table 12.20
Year
At
σt
1
Rs. 50000
15000
2
Rs. 30000
10000
3
Rs. 40000
20000
4
Rs. 30000
12000
If the risk free rate of interest is 6%, find the expected NPV of the project. What is the standard
deviation of the expected NPV?
Sol:
NPV =
n
At
∑ (1 + i)
t =1
−I
t
=
50000 30000 40000 30000
+
+
+
− 100000
1.06 (1.06) 2 (1.06)3 (1.06) 4
=
Rs.31217.29
σ ( NPV ) =
n
σt
∑ (1 + i)
t =1
t
= Rs.4935
Investment decision trees
We have seen that decision trees are pictorial representation of the decision-making path of a
problem. Any decision taken with the help of tables can also be made with the help of a decision
tree. We now present the decision tree of the investment problem that we considered in example 5.
The probabilities of the cash flows at any time point have been shown. The probability distribution
of the NPV has been reached. Now it is easy to calculate the expected NPV of the project and its
standard deviation.
459
0.6
35,000
0.8
30,000
NPV
-15612
0.24
40,000
0.5
30,000
0.4
-11414
0.16
0.5
0.05
45,000
1684
0.5
50,000
0.05
5882
0.7
60,000
42046
0.2
40,000
Invest Rs. 1,00,000
Expected NPV = Rs. 30,916
S.D. = Rs. 25,052.44
0.6
50,000
0.22
0.3
70,000
0.5
50,000
50442
0.09
0.4
60,000
Fig 12.5
460
75,000
63540
0.8
0.16
90,000
67134
0.2
0.04
(ii)
Quantitative techniques of risk management
(a)
Expected monetary value
In this technique, the expected monetary value of the project over different possible likelihood is
calculated once the cash flows have been specified along with their probability distribution.
n
EMVt =
∑ CF
i =1
it
pit ;
where
EMVt = The expected monetary value at time t ;
CFit = i th cash flow at time t ; and
Probability of the i th cash flow at time t.
pit =
Example 7:
The initial outlay of a project is Rs. 10,00,000. The expected returns from the
project are as follows:
Table 12.21
Event
State of
nature
Year 1
Cash flow
Year 2
Probability
(Rs.)
A
Cash flow
Year 3
Probability
(Rs.)
Cash flow
Probability
(Rs.)
Deep
1,00,000
0.10
2,00,000
0.10
1,00,000
0.10
recession
2,00,000
0.15
4,00,000
0.15
2,00,000
0.20
C
Normal
3,00,000
0.50
6,00,000
0.40
3,00,000
0.30
D
Good
4,00,000
0.20
8,00,000
0.20
4,00,000
0.30
E
Excellent
5,00,000
0.05
10,00,000
0.15
5,00,000
0.10
recession
B
Mild
If the cost of capital is 10%, using monetary value method, find whether the project is viable or not.
461
Sol:
To find the viability of the project, we calculate the expected monetary value for each of the
period.
Table 12.22
Event
Year 1
Cash
Prob.
Year 2
Expecte
Cash flow
flow
d value
(Rs.)
(Rs.)
(Rs.)
Prob.
Year 3
Expected
Cash
value
flow
value
(Rs.)
(Rs.)
(Rs.)
Prob.
Expected
A
1,00,000
0.10
10,000
2,00,000
0.10
20,000
1,00,000
0.10
10,000
B
2,00,000
0.15
30,000
4,00,000
0.15
60,000
2,00,000
0.20
40,000
C
3,00,000
0.50
15,000
6,00,000
0.40
2,40,000
3,00,000
0.30
90,000
D
4,00,000
0.20
80,000
8,00,000
0.20
1,60,000
4,00,000
0.30
1,20,000
E
5,00,000
0.05
25,000
10,00,000
0.15
1,50,000
5,00,000
0.10
50,000
Total
2,95,000
6,30,000
Thus, we have
Table 12.23
Year
Expected cash flow (Rs.)
1
2,95,000
2
6,30,000
3
3,10,000
Then the NPV of the project is
NPV =
n
At
∑ (1 + i)
t =1
t
−I
=
2,95, 000 6,30, 000 310000
+
+
− 1000000
1.10
(1.10) 2
(1.10)3
=
Rs.21,345
Hence the project should be accepted.
462
3,10,000
(b)
Range
Range refers to the difference between the largest possible outcome and the smallest possible
outcome, i.e.
Rg = Rh − Rl
where
Rg = Range of the distribution;
Rh = Highest value of the distribution; and
Rl = Smallest value of the distribution.
( c)
Mean absolute deviation
Mean absolute deviation refers to the quantitative distance of the possible outcomes from their
expected value, i.e.,
n
∑ P( R ) |
MAD =
i
i =1
(d)
Ri − E ( R) |
Standard deviation of risk
Standard deviation of risk measures the dispersion of the possible outcomes from their expected
value, i.e.,
σ =
n
∑ P( R ) ( R − E ( R) )
i
i =1
2
i
Higher value of standard deviation (σ) corresponds to higher risk.
In the following table
Table 12.24
Project details
Investment A
Investment B
100
100
Pessimistic
14
12
Normal (most likely)
16
16
Optimistic
18
20
4
8
Initial outlay (Rs.)
Annual returns (Rs.)
Range
463
For investment A
0.20 (14 − 16 ) + 0.60 (16 − 16 ) + 0.20 (18 − 16 )
σA =
=
2
2
= 1.26%
1.6
For investment B
0.10 ( 4.8 ) + 0.60 ( 0.8 ) + 0.30 ( 3.2 )
σB =
=
2
2
= 2.4%
5.76
For investment C
0.05 (10 ) + 0.25 ( 2 )
σc =
=
2
6
2
= 2.45%
The investments in increasing order of risk are A, B and C.
(e)
Coefficient of variation
The coefficient of variation measures the risk per unit of expected return, i.e.
CV =
σ
E ( R)
where E ( R) is the expected value and σ is the standard deviation of the return.
In the above example, for investment A
CV ( A) =
σ
E ( R)
=
1.26
= 0.079
16
For investment B
CV ( B ) =
2.4
= 0.143
16.8
CV (C ) =
2.45
= 0.136
18
For investment C
Thus B is the riskiest investment followed by C and A.
464
2
2
12.3
Selection of a project
Once we know the measures of performance, e.g. NPV, IRR or some other measure and the risk or
variability associated with the measure, then the key question is whether or not to accept the project.
There are three methods that can be used in decision-making.
(i)
Risk-profile method;
(ii)
Risk adjusted discount rate method; and
(iii)
Certainty equivalent technique.
Now, we will discuss these methods.
(i)
Risk-profile method
In this method we make use of probability distribution of some absolute measure, say NPV, which is
then, converted into probability distribution of some relative measure, say, profitability index.
For example, consider the probability distribution of a NPV of a project that has an initial outlay of
Rs. 1,00,000 is normal represented by.
Probability
.
.
.
0
20000
40000
Fig.12.6
Then the profitability index of the project can be calculated as
Profitabilty index =
465
NPV + Investment
Investment
NPV (Rs.)
Since profitability index is a linear function of NPV, it will follow the same distribution, i.e.,
Probability
.
1.0
.
.
1.2
1.4
PI
Fig.12.7
Then we compare the dispersion of the profitability index (or any other relative measure) of the
project with the maximum risk profile acceptable for the expected profitability index of the firm.
Suppose that corresponding to expected profitability index of 1.2, the acceptable risk profile can be
displayed as follows:
Probability
PI
Expected PI
.
Expected PI
1.2
Fig.12.8
Then we see that the dispersion of the profitability index is less than the maximum risk acceptable to
the firm for the given level of expected profitability index. Hence the project is acceptable to the
firm.
466
It may be noted that the risk dispersion that may be acceptable to the firm increases with the high
value of profitability index. This is so because as PI increases, the value of NPV is less likely to
become negative even if dispersion is high
Probability
Area where NPV is negative
.
.
5
1
.
Expected PI
9
Fig.12.9
Another aspect of this method is the willingness of the firm to take risk. If the management is averse
to risk, the risk dispersion curve will be more compact than the curve of the firm, which is willing to
take more risk.
(ii)
Risk adjusted rate discount method
In this method the discount rate associated with the project is adjusted to reflect the risk attached
with the project. The discount rate of the project is compared with the cost of capital to the firm. If
the risk of the project is equal to the risk of the existing investments to the firm, the discount rate to
the project is the average cost of capital to the firm; if the risk of the project is greater than the risk of
the existing investments to the firm, the discount rate to the project is higher than the average cost of
capital to the firm; and if the risk of the project is lower than the risk of the existing investments to
the firm, the discount rate to the project is lower than the average cost of capital to the firm. The
risk-adjusted discount rate is given by
rk = i + n + d k where
rk = Risk - adjusted discount rate for project k ;
467
i = Risk-free rate of interest;
n = Adjustment for the firm's normal risk; and
d k = Adjustment for the differential risk of project k .
Here (i + n) measures the firm's cost of capital. Depending upon the extent of the risk of the project
under consideration as compared to the existing risk of the firm, dk may be positive or negative. The
adjustment for the differential risk is a function of the management's perception and attitude towards
risk.
Once rk, the risk adjusted discount rate of the firm is specified, NPV of the firm can be calculated and
the project is acceptable if the NPV is positive, i.e.
n
At
∑ (1 + r )
NPV =
i =1
t
− I;
k
where
At
Example 8:
= Expected cash flow for time t ;
The initial outlay of a project is Rs. 1,00,000 and the expected cash flows are as
follows:
Table 12.25
Year
Expected cash flow (Rs.)
1
20,000
2
30,000
3
40,000
4
30,000
5
20,000
The risk adjusted discount rate for the project is 20%. Is the project worth accepting?
Sol:
NPV =
n
At
∑ (1 + r )
i =1
t
− I;
k
468
=
20, 000 30, 000 40, 000 30, 000 20, 000
+
+
+
+
− 1, 00, 000
1.2
(1.2) 2
(1.2)3
(1.2) 4
(1.2)5
= - Rs. 16,847
Hence the project is not worth accepting.
This method is employed frequently in practice. The firms use different discount rates for different
extent of risk associated with the projects. For routine replacement project, usually a low risk is
attached; moderate risk is attached with expansion projects whereas for new investments, risk
associated is high.
However, one of the major limitations of the method is the consistent estimation of dk, which may be
very difficult in practice and firms use arbitrary methods for estimation of dk. Another limitation is
the assumption that the risk is increasing constantly which is not possible in reality.
(iii)
Certainty equivalent method
Consider the following proposals made to a person:
Table 12.26
Outcome
Probability
Rs. 10,000
0.3
Rs. 5,000
0.7
Expected value of this offer is Rs. 6500 (0.3 × 10,000 + 0.7× 5,000). However the person agrees for
a certain sum in lieu of this offer. Let that sum be Rs. 3,000. Then this sum is the certainty
equivalent of this offer.
469
Graphically,
Expected return (Rs.)
Indifference curve
O
10,000
M
6500
P
5,000
3,000
N
Perceived risk
σP
σM
σO
Fig. 12.10
Since Rs. 3,000 is a sure sum so no risk is associated with this amount. The risk increases as the sum
to be received increases. A person may choose any point on the indifference curve. However as the
risk increases, the person becomes more and more reluctant towards taking risk.
The ratio of certainty equivalent to the expected value of the offer is called the certainty equivalent
coefficient. Certainty equivalent coefficient in this case is 0.4615 (3000 / 6500). This factor reflects
two things: variability of the outcomes and attitude towards risk.
If the certainty equivalent
coefficient is high, this means the person is more inclined towards risk and would not settle for a
smaller but certain amount. In place he would go for the gamble.
The certainty equivalent coefficient generally has value between 0 and 1. A certainty equivalent
coefficient equal to 1 signifies a certain cash flow. In other words the management is risk neutral.
449
Generally risks are involved with the projects and as a consequence, the value of a certainty
equivalent coefficient is less than one. This method is a conservative method of risk estimation.
Under the certainty equivalent method, the NPV of the project is calculated as follows:
NPV =
α t At
n
∑ (1 + i)
i =1
t
− I;
where
Example 9:
At
= Expected cash flow for time t ;
αt
= Certainty equivalent coefficient for the cash flow at time t ;
i
=
Risk-free rtae of interest; and
I
=
Initial certain investement.
An investment proposal involves an initial outlay of Rs. 50,00,000. The expected
cash flows and the certainty equivalent coefficients of the proposal are given below:
Table 12.27
Year
Expected cash flows (Rs.)
Certainty equivalent coefficients
1
10,00,000
0.90
2
15,00,000
0.80
3
25,00,000
0.80
4
30,00,000
0.75
The risk free rate of interest is 5%. Is the project worth accepting?
Sol:
For this proposal, NPV is given as
NPV =
n
i =1
=
α t At
∑ (1 + i)
t
− I;
(10,00,000)(0.90) (15,00,000)(0.80) (25,00,000)(0.80) (30,00,000)(0.75)
+
+
+
− 50, 00, 000
1.05
(1.05) 2
(1.05)3
(1.05) 4
= Rs. 524334
450
The project is worth accepting.
The certainty equivalent method is conceptually a superior method as compared with the adjusted
risk method because in this method the risk may vary over the years and this fact is visible in
different certainty equivalent coefficients for different years whereas the assumption in adjusted risk
method is that of increasing risk at a constant rate.
However, the specification of different risks and hence different certainty equivalent coefficients
may be difficult to be made. Since risks over different time periods can be calculated easily in
adjusted risk method so it is risk adjusted method that is preferred by the firms.
12.4
Return
Human beings need security, for their present and future. In order to secure their future, at least
financially, they think of several ways of getting a regular stream of income with minimum or no
chance of any sort of trouble.
One of such ways is to invest their earnings or savings in
shares/stocks. The objective of such an action is to reap maximum benefits in terms of bonus and
increase in the value of investment with minimal chances of loosing the money. This is the concept
of return and risk in share/ stock market. In fact, risk and return are the two key ingredient
phenomena of the stock market. Since both these factors are known in advance, we use the estimates
of these two, viz., expected risk and expected return respectively.
Return
Expressed as a rate percent, return on an investment is the ratio of annual income
received plus change in market price to the opening market price, i.e.,
R =
I t + Pt − Pt −1
Pt −1
I t = annual income at the end of time period t ;
Pt = price of the investment at the period t (closing price); and
Pt −1 = price of the investment at the period t − 1 (opening price at period t )
It
P − Pt −1
is the current yield; and t
is capital gain/loss.
Pt −1
Pt −1
451
Risk
Risk is defined as the degree of variability/deviation of actual return from its expected
value. A high value of risk means actual returns are at distance from the expected value and as such
the reliability of the return is less.
12.5
Risks and returns of portfolio
Till now, we have considered the risks and returns associated with a single asset. Such investments
exhibit risk through their deviations from their central values during the holding period. However, in
general, investors do not invest in a single asset but they diversify their investments in more than one
asset. Such diversifications are expected to provide some degree of income stability due to nonsynchronized price movements of different assets. In fact the business opportunity for different lines
are not the same and the returns from different industries vary in different directions. For example,
an above normal rain may be a cause of the boom in agro industries but the same may spoil the game
for construction business. In such situations, the variations in rates and the directions of returns from
investments made in different industries may balance each other with a result that the combined
deviation may be of order zero without affecting the central value.
The combination of two or more different investments is called a portfolio. The risk and return
characteristics of a portfolio are different from those of individual securities, which constitute a
portfolio.
In fact, Harry Markowitz, the proponent of the portfolio theory demonstrated that whereas the
portfolio return is a single aggregation of individual returns, the same is not true for the portfolio
risk.
Expected return and risk of a (two-asset) portfolio
As a simple case, we consider a portfolio consisting of two assets.
The expected rate of return
The expected rate of return on a portfolio is the weighted
average of the expected rates of individual returns of the assets comprising the portfolio, the weights
being the proportions of the investments in a single asset to the total investments.
452
For a two-asset portfolio
E (rp ) = w1 E (r1 ) + w2 E (r2 )
= w1 E (r1 ) + (1 − w1 ) E (r2 )
For an n-asset portfolio
n
∑ w E (r )
E ( rp ) =
i =1
i
i
rp = Expected return on the portfolio;
ri = Expected return on the i th security of the portfolio;
wi = weight associated with the i th security of the portfolio;
n = the number of assets in the portfolio; and
n
∑w
i =1
Portfolio risk
= 1 .
i
Portfolio risk is the variance of the returns from the expected portfolio return.
⎛
⎞
n
σ p2 = Var ⎜ ∑ wi ri ⎟
⎝ i =1
⎠
n
⎛ n
⎞
= E ⎜ ∑ wi ri − ∑ wi E (ri ) ⎟
i =1
⎝ i =1
⎠
⎛ n
⎞
= E ⎜ ∑ wi ( ri − E (ri ) ) ⎟
⎝ i =1
⎠
2
2
i =1
i
i
n
n
i =1 j =1
j ≠i
i
n
∑ wi2σ 12 + ∑∑ wi w jσ ij
i =1 j =1
j ≠i
i=1
n
=
i
n
n
=
2
∑ w E ( r − E (r ) ) + ∑∑ w w E ( r − E (r ) ) E ( r
n
=
2
∑w σ
2
i
i=1
n
2
i
n
+ ∑∑ wi w jσ iσ j ρij ;
i =1 j =1
j ≠i
where σ i2 = Var (ri ), i = 1, 2,...n
σ ij = Covar (ri , rj ) and
ρij = Cor(ri , rj ).
453
j
i
i
j
− E ( rj ) )
For a two-asset portfolio
σ p2 = w12σ 12 + (1 − w1 )2 σ 22 + 2w1 (1 − w1 )σ 1σ 2
This expression shows that the portfolio risk depends on
(i)
Variance of the individual asset; and
(ii)
The covariance between the different assets of a portfolio.
Among these factors, a portfolio manager/investor can exercise control only over the weights, i.e.,
the proportions of the different investments to the total investment. Thus portfolio theory helps
managers in deciding the proportion of each security in the portfolio.
Consider a portfolio P consisting of two assets PL, (low risk, low return) and PH (high risk, high
return), having expected returns of 12% and 18% respectively. The standard deviations of the assets
respectively are 16% and 20%. Now, we prepare the following table, which give the portfolio return
and portfolio risk for different values of weights wL, and wH and the correlation ρLH:
454
Table 12.28
σp when ρ =
PL
PH
E (P)
1.0
0.5
0
-0.5
0.1
100
0
12.0
16.00
16.00
16.00
16.00
16.00
95
5
12.3
16.20
15.72
15.23
14.73
14.20
90
10
12.6
16.40
15.50
14.54
13.51
12.40
85
15
12.9
16.60
15.32
13.93
12.38
10.60
80
20
13.2
16.80
15.20
13.41
11.34
8.80
75
25
13.5
17.00
15.13
13.00
10.44
7.00
70
30
13.8
17.20
15.12
12.71
9.71
5.20
65
35
14.1
17.40
15.16
12.54
9.18
3.40
60
40
14.4
17.60
15.26
12.50
8.91
1.60
55
45
14.7
17.80
15.42
12.59
8.90
0.20
50
50
15.0
18.00
15.62
12.81
9.17
2.00
45
55
15.3
18.20
15.88
13.15
9.68
3.80
40
60
15.6
18.40
16.18
13.60
10.40
5.60
35
65
15.9
18.60
16.53
14.15
11.29
7.40
30
70
16.2
18.80
16.92
14.80
12.32
9.20
25
75
16.5
19.00
17.35
15.52
13.45
11.00
20
80
16.8
19.20
17.82
16.32
14.66
12.80
15
85
17.1
19.40
18.32
17.17
15.94
14.60
10
90
17.4
19.60
18.85
18.07
17.26
16.40
5
95
17.7
19.80
19.41
19.02
18.61
18.20
0
100
18.0
20.00
20.00
20.00
20.00
20.00
We observe the following points.
(i)
It is possible to combine two assets in such a way that the portfolio risk will be less than the
individual risk of each asset. For example, for ρ = 0.5, PL = 95 to 45 and PH = 05 to 55,
portfolio standard deviation σp is throughout less than the individual standard deviations σL
and σH.
449
(ii)
For any given pair of weights wL, and wH, the portfolio standard deviation σp declines as ρ
varies from +1.0 to –1.0.
(iii)
For perfect positive correlation, portfolio risk increases when the higher risk asset gets the
higher weightage.
(iv)
For perfect negative correlation, portfolio risk first decrease, reaches a minimum value and
then starts increasing.
(v)
For less than perfect correlation, some combinations are more efficient than the others. For
example, for ρ
= 0.5, the expected returns increased from 12.0 to 13.8 but the risk
decreased from 16.0 to 15.12. There is no return-risk trade-off.
(vi)
For every correlation, there is a minimum risk portfolio that has a risk lower than the risk of
the individual assets.
To find the weights to be assigned to different assets in order to have the minimum risk, we proceed
as follows
σ p2 = (ωLσ L ) 2 + (ωH σ H )2 + 2ωLωH σ Lσ H ρ LH
= (ωLσ L ) 2 + ( (1 − ωL )σ H ) + 2ωL (1 − ωL )σ Lσ H ρ LH
2
For σ 2p to be minimum
∂σ p2
∂ω L
= 0
⇒ ω Lσ L2 - (1 − ω L )σ H2 +(1 − 2ω L )σ Lσ H ρ LH
⇒ ωL* =
= 0
σ H2 − σ Lσ H ρ LH
σ L2 + σ H2 − 2σ Lσ H ρ LH
and ωH* = 1-ωL* =
σ L2 − σ Lσ H ρ LH
σ L2 + σ H2 − 2σ Lσ H ρ LH
where ωL* and ωH* are the optimal weights corresponding to the asset i, i =1,2.
The following table lists weights corresponding to the minimum risk for each value of the correlation
coefficient.
450
Table 12.29
ρLH
wL
wH
σP
1
500
-400
0
0.5
71
29
15.12
0
61
39
12.49
-0.5
57
43
8.87
-1
56
44
0.00
Example 10(a): Calculate portfolio risk on a portfolio comprising of two assets with the same
variance 25% and zero covariance when the assets are combined in
(i)
Equal proportion
(ii)
In the ratio 2:3.
(b)
Is it possible to reduce portfolio variance to zero?
(c)
For what value(s) of weights, will the portfolio variance be minimum?
Sol:
(a)
As the covariance between the two assets is zero so
σ p2 = (ω1σ 1 ) 2 + (ω2σ 2 ) 2
(i) In this case,
w1 = w2 =
⇒ σ p2 =
1
2
1 2 1
σ1 + σ 22
4
4
=
1
( 25+25)
4
=
50
4
= 12.5%
(ii) In this case,
w1 =
⇒ σ p2 =
=
2
3
; w2 =
5
5
4
9
σ 12 +
σ 22
25
25
13
× 25
25
= 13%
449
(b)
For portfolio variance to be zero, we must have
(ω1σ 1 ) 2 + (ω2σ 2 ) 2 = 0
⇒ ω12 + (1-ω1 ) 2 = 0
⇒ 2ω12 − 2ω1 + 1 = 0
2± 4 - 8
4
⇒ ω1 =
Thus for variance to be zero, we must have imaginary weights, which is not possible. So it
is not possible to have a zero variance portfolio in case of uncorrelated assets.
(c)
For portfolio variance to be minimum, we must have
∂σ p2
∂ω1
= 0
2ω1σ 12 - 2(1 − ω1 )σ 22
=
⇒
ω1 (σ 12 + σ 22 )
⇒
ω1
=
= ω2
σ 22
1
=
and ω2
2
2
σ +σ2
2
1
=
σ 12
1
=
2
2
σ +σ2
2
1
Thus for portfolio variance to be minimum, the assets should be combined in equal
proportion.
12.6
Portfolio risk and the correlation between assets
We have seen that for any given pair of weights, portfolio risk changes with the correlation. This is
due to the fact that as correlation changes from plus unity to minus unity, the magnitude of the comovement in the same direction decreases. Thus decrease in price of one asset is compensated by
the increase in the price of the other. This diversification (investing in assets of different nature)
reduces the portfolio risk and the portfolio risk is minimum when the correlation between the assets
is perfectly negative.
Since the portfolio risk lies between the risks corresponding to ρ = +1.0 and ρ = -1.0, we examine
the behavior of the risk for ρ = +1.0, 0, -1.0
450
(i)
Perfect positive correlation (ρ = +1.0)
In this case
σ p2 = (ωLσ L ) 2 + (ωH σ H ) 2 + 2ωLωH σ Lσ H
= (ωLσ L + ωH σ H ) 2
⇒ σP
= ω Lσ L + ω H σ H
Thus the portfolio standard deviation is the weighted average of the standard deviations of the two
assets.
In this case, there is a direct and linear relationship between returns and risk. Lower risk corresponds
to lower returns and higher returns mean higher risk. Thus there is a trade-off between the returns
and risk.
451
(ii)
Perfect negative correlation (ρ = -1.0)
In this case
σ p2 = (ωLσ L ) 2 + (ωH σ H ) 2 − 2ωLωH σ Lσ H
= (ωLσ L − ωH σ H ) 2
⇒ σP
=
ω Lσ L − ω H σ H
σP = 0 ⇔
or, ωL* =
ωL
σL
=
ωH
σH
σH
σH +σL
; and ω *H
=
σL
σH +σL
For these values of weights, the portfolio risk will reduce to zero.
In our case,
ωL* =
ω*
H
=
σH
σH +σL
σL
σH +σL
=
20
16 + 20
= 0.56 ; and
=
16
16 + 20
= 0.44
σ P = ωL*σ L − ω * σ H = 0.56 *16 − 0.44 * 20 = 0.16
H
The difference is the approximation error.
Graphically,
E(rρ) (%)
B
ρ = -1.0
ρ =0
ρ <1
A
ρ = 1.0
ρ = -1.0
C
σρ2 (%)
Fig 12.11
452
For ρ = -1.0, portfolio risks corresponding to different weights exhibit a V-shaped pattern, the tip of
V lying on the line of the expected returns.
The pattern is having a clockwise movement showing that with the increase in the weight of the
higher risk asset, the overall risk is gradually decreasing and overall expected return is increasing.
This phenomenon continues till the risk is completely eliminated (point A). After this point, as the
high return, high risk asset is getting the higher weightage, the expected return is increasing along
with the increase in the risk., thus markin a trade off between the return and the risk.
The graph corresponding to ρ = 1.0 and ρ = -1.0 forms a triangle ABC, B and C being points
corresponding to the pure assets. This triangle marks the bounds for portfolio diversification and all
the feasible portfolios will have a risk lying in the plane of the triangle ABC.
(iii)
Zero correlation (ρ = 0)
In this case, the assets comprising of the portfolio are
completely uncorrelated. In this case
σ p2 = (ωLσ L ) 2 + (ωH σ H )2
⇒ σP
=
(ωLσ L ) 2 + (ωH σ H ) 2
A look at the column corresponding to ρ = 0 shows that diversification will help in reduction of risk
as for some values of WL and WH, σP2 < σL2 and σP2 < σH2.
Example 11:
A portfolio consists of two assets X and Y. The following information is available
regarding the assets
Table: 12.30
Information
Portfolio (X,Y)
X
Y
Expected return (%)
20
25
Variance (%)
100
144
Covariance (X, Y) (%)
120
453
(a)
Find the correlation between the two assets.
(b)
Find the expected return and risk of a portfolio when the assets are combined in ratio 1:3.
Sol:
Cov( X , Y )
ρ XY =
(a)
σ XσY
=
120
= 1
10 × 12
Thus the two assets are perfectly correlated.
(b)
E (rp ) = ω X E (rX ) + ωY E (rY )
=
1
3
(20) + (25)
4
4
=
95
%
4
= 23.75%
Further
σ p2 = ω X2 σ X2 + ωY2σ Y2 + 2ω X ωY σ X σ Y
=
1
9
1 3
(100) + (144) + 2 × × × 10 × 12
16
16
4 4
= 132.14%
σp
Example 12:
= 11.5%
Consider the following two assets
Table 12.31
(i)
(ii)
Portfolio
Expected return (%)
Variance (%)
X
20
25
Y
100
144
The correlation coefficient between X and Y is –0.5
(a)
For what values of weights w1 and w2, will the portfolio variance be minimum?
(b)
What is the minimum value of the portfolio variance?
(c)
What will be the portfolio variance if the weights are equal?
Repeat the above parts (a), (b) and (c) if correlation coefficient between X and Y is 0.5.
454
Sol:
(i)(a)
σ p2 = ωX2 σ X2 + ωY2 σ Y2 + 2ρ XY ω X ωY σ X σ Y
= ωX2 σ X2 + (1 − ω X ) σ Y2 + 2ρ XY ω X (1 − ω X ) σ X σ Y
2
= 100ω X2 + 144 (1 − ω X ) − 120ω X (1 − ω X )
2
For σ 2p to be minimum
∂σ 2p
∂ω X
⇒ ωX
⇒ 25ω X -36(1 − ω X ) -15(1 − 2ω X ) = 0
= 0
51
40
; ωY =
91
91
=
(b)
2
2
⎛ 51 ⎞
⎛ 40 ⎞
⎛ 40 ⎞ ⎛ 51 ⎞
⎟ + 144 ⎜ ⎟ − 120 ⎜ ⎟ ⎜ ⎟
⎝ 91 ⎠
⎝ 91 ⎠
⎝ 91 ⎠ ⎝ 91 ⎠
σ p2 = 100 ⎜
= 29.67%
(c)
σ 2p =
100 144 120
+
−
4
4
4
= 25 + 26 - 30 = 31%
(ii)(a)
σ p2 = 100ω X2 + 144 (1 − ω X ) + 120ω X (1 − ω X )
2
For σ p2 to be minimum
∂σ p2
∂ω X
⇒ ωX
= 0
=
⇒ 25ω X - 36(1 − ω X ) +15(1 − 2ω X ) = 0
21
10
; ωY =
31
31
(b)
⎛ 21 ⎞
2
⎛ 10 ⎞
2
⎛ 10 ⎞ ⎛ 21 ⎞
σ 2p = 100 ⎜ ⎟ + 144 ⎜ ⎟ + 120 ⎜ ⎟ ⎜ ⎟
⎝ 31 ⎠
⎝ 31 ⎠
⎝ 31 ⎠ ⎝ 31 ⎠
= 87.09%
455
(c)
σ 2p =
100 144 120
+
+
4
4
4
= 25 + 26 + 30 = 91%
12.7
Impact of portfolio diversification- limits of diversification gain
We have seen that
(i)
Portfolio risk can be made less than the individual risk of any of the asset in the portfolio;
and
(ii)
Portfolio risk varies with the size of the correlation between the assets of the portfolio.
(iii)
Portfolio risk lies between the limits set by the risks corresponding to ρ = 1.0 and ρ = -1.0.
For the first case, diversification does no good and for the second case, it is possible to
eliminate the risk completely.
In reality, both ρ = 1.0 and ρ = -1.0 are the extreme situations which are not possible to meet and
the feasible portfolios contain asset which are less then perfectly correlated. Such portfolios may
have the best possible risk but the risk cannot be completely eliminated.
At this point, a natural question arises:
What happens if we add more and more assets to the
portfolio? Will the diversification still be beneficial? If yes, can the portfolio risk be reduced to
zero? The answers to these queries have their key in the nature of the diversification.
Naïve diversification:
In this diversification, a portfolio consists of the assets chosen at random.
Since the individual fluctuations are random in nature, they tend to cancel out each other thus
reducing the portfolio risk. Now, we know that the sample variance varies inversely with the size of
the sample so addition of the new assets will reduce the portfolio risk. However, the risk cannot be
completely eliminated. In fact,
σ 2p =
n
n
n
∑ (ω σ ) + ∑∑ ω ω σ
2
i =1
i
i
i =1 j =1
j ≠i
i
j
ij
1
, then
n
n
n(n − 1)
(average variance) +
(average covariance)
2
n
n2
If ω1 = ω 2 = ... = ω n =
σ 2p =
=
1
⎛ 1⎞
(average variance) + ⎜ 1 − ⎟ (average covariance)
n
⎝ n⎠
As n → ∞, σ p2 → average covariance
This is the lower limit to which portfolio risk can be reduced. This is called the systematic error.
456
Systematic error
This is the overall market risk that affects all the securities. This risk
cannot be eliminated by any amount of diversification. Studies have shown that this limit can be
reached with 10 to 15 securities.
Non-systematic or diversifiable risk is specific and can be completely eliminated.
σρ2 (%)
50
40
Diversifiable (non-systematic risk)
30
20
Un diversifiable (systematic risk)
10
0
.
. .
.
.
.
.
5
10 15
20
25
30
35
.
40 Number of assets
Fig 12.12
Markowitz diversification
In this diversification, instead of reducing risk through inclusion of more assets, the risk is reduced
through inclusion of those assets, which have a high degree of (negative) covariance between them.
By having assets with strong negative correlations between them, it is possible to reduce the risk
below the level of systematic risk. We have seen that it is possible to have a portfolio with only two
assets that have perfect negative correlation between them such that the expected portfolio risk is
equal to zero.
Portfolio selection
Harry Markowitz developed a procedure to select the optimal portfolio on the basis of two aspects of
portfolio selection.
(i)
Technical aspect
This aspect of portfolio selection relates to the determination of
the set of efficient portfolios from the available feasible set.
457
(ii)
Personal aspect
This aspect of portfolio selection relates to the determination of
the best (risk-return) opportunities from the efficient portfolio set, in consonance with the investor's
attitude towards risk.
Efficient portfolios
The objective of the portfolio theory is to determine the portfolios with maximum possible returns
for a given risk or to determine the portfolios with minimum risk for give return. Such portfolios are
called efficient or dominant portfolios.
The first step in selection of efficient portfolios is the determination of feasible or opportunity set of
portfolios.
With three assets to invest in, the possible alternatives are as follows
E ( Rp )
IV
II
III
I
σˆ p
Fig 12.13
Here curve I represents combination of assets A and B; curve II represents the combinations of assets
B and C; curve III represents the combination of assets C and A and curve IV represents all the three
assets in the portfolio.
With the increased number of securities in a well-diversified portfolio, full set of feasible portfolios
can be shown as follows
458
E ( Rp )
Efficient set
Feasible set
Fig 12.14
From among all these portfolios of the feasible set, the portfolios, which maximize the expected
return for a given standard deviation, are said to be efficient. Since individual investors differ with
respect to their attitude towards risk, an optimal portfolio can be selected by super imposing the set
of investors' indifference curves on the above feasible set. The point of tangency marks the highest
level of satisfaction an investor can attain.
For example, consider a feasible set of six portfolios A, B, C, D, E and F. Graphically the situation
can be represented as follows:
459
σˆ p
Expected return (%)
25
Efficient portfolios
E
20
D
C
15
.
.
Global maximum return portfolios
.
.
.
10
F
Global minimum variance portfolios
B
A
5
.
Dominated portfolios
0
.
.
.
.
.
.
.
.
5
10
15
F
20
25
30
35
40
σρ2 (%)
Fig 12.15
The curve represents the portfolio opportunity set. The highest point of the curve represents the
global maximum return portfolio and the extreme left point is the global minimum variance
portfolio. The portfolios between the global maximum return portfolio and the global minimum
variance portfolio are the efficient (dominated) portfolios and the line segment joining the two is
called the efficient frontier. The portfolios lying below the efficient frontier are the dominated
portfolios in the sense that for the same return, they have larger variance or for the same variance,
they have the smaller return than that of efficient portfolios.
The efficient frontier is convex towards the axis of expected returns (vertical axis) as for any given
return, risk cannot be lowered than a specific value and for any given risk, and return cannot be
increased infinitely.
Example 13:
Consider the following information regarding seven portfolios
Table 12.32
Portfolio
Expected return (%)
Standard deviation (%)
A
10
18
B
12
15
C
20
28
W
13
19
X
18
15
Y
16
20
Z
14
25
460
Classify the portfolios as efficient or dominated portfolios.
Sol:
An efficient portfolio is one that has the smallest standard deviation for a given return or the
largest return for a given standard deviation and a dominated portfolio is one that has a larger
standard deviation for a given return or a smaller return for a given standard deviation.
Then we have the following table of efficient and dominated portfolios.
Table 12.33
Portfolio
Expected return (%)
Standard deviation (%)
Category
A
10
18
Dominated (by B, X)
B
12
15
Dominated (by X)
C
20
28
Efficient
W
13
19
Dominated (by X)
X
18
15
Efficient
Y
16
20
Dominated (by X)
Z
14
25
Dominated (by X, Y)
Markovitz portfolio analysis is based on certain assumptions about the behavior of the investor. The
assumptions are listed below.
(i)
For investors, the rate of return is the most important concept and they perceive different
rates of returns as a probability distribution before making investment decisions.
(ii)
In general, investors are risk averse and they seek the highest return for a given risk. Risk is
measured in terms of the variability of the expected returns.
Capital allocation line (CAL)
A capital allocation line is a portfolio opportunity set with one
risk-free asset.
Risk-free asset (security)
A risk free security is a security with zero variance (and hence
standard deviation). It has been pointed out that
(i)
Portfolios which consist of one risk-free asset generate investment opportunities with linear
relationship between expected return and risk;
461
(ii)
Such a portfolio opportunity set will dominate the portfolios consisting of only risky assets.
Mathematically, define
F = A risk-free portfolio;
M = A risky portfolio;
C = A complete portfolio (a combination of F and M );
w = weight of M in C ;
1 − w = weight of F in C ;
rc =
return on C ;
=
return on F ;
rf
rm =
return on M ;
Then,
E (rc ) = wE (rm ) + (1 − w)rf
= rf + w ( E (rm ) − rf
)
(12.1)
E ( rm ) − rf denotes the risk premium of the risky portfolio.
Again,
σ c = S.D. of C
= wσ m ; σ m is the S.D. of risky portfolio M .
(12.2)
From (12.1) and (12.2), we have
E (rc ) = rf +
σc
( E (rm ) − rf
σm
)
⎛ E (rm ) − rf ⎞
= rf + σ c ⎜
⎟
σm
⎝
⎠
(12.3)
This is the equation of capital allocation line (a portfolio opportunity set with one risk free asset).
The intercept of this line is rf which represents the minimum assured return and its slope is
E ( rm ) − rf
σm
which represents the increase in risk premium for a unit increase in risk.
Consider the following situation
462
Expected return (%)
CAL (M)
M
CAL (Z)
Z
rf
F
X
CAL (X)
σ (%)
Fig 12.16
Three capital allocation lines CAL (X), CAL (Z), and CAL (M) originate from the point F, which
represents the risk-free investment return. In other words, F is a pure portfolio of 100% risk free
holdings and the expected rate of return rf. The curve XMZ represents a minimum variance frontier,
with X as the lower most point (the one having the least risk for a given expected return). The
highest point Z demonstrates the portfolio with the maximum return. The highest and steepest CAL
and the efficient frontier meet tangentially at point M. The combination of M with the risk-free
portfolio F is the best risk-return trade-off. Any combination of F and M on the curve segment FM
will offer the best combination of portfolios.
Market portfolio
In the portfolio constructed above, the investor is using his own funds to
invest. As a result, the efficient frontier ends at the point M, which represents the maximum capital
available with the investor. Had more capital been available, higher returns could be generated by
constructing more efficient portfolios.
This becomes possible if the investor can avail funds by borrowing at a risk free rate of interest rf.
The same funds are then invested in the risky asset M. The portfolio constructed in this manner is
called a borrowing portfolio. On the other hand, the portfolio of the lender is called a lending
portfolio.
463
Since the investment in the risky asset is more than the capital of the investor, the weight of the risky
asset will exceed one. For example, if an investor has Rs. 4,00,000 available with him and he
borrows Rs. 1,00,000 from the market and invests Rs. 5,00,000 in a risky asset, then the total weight
of the risky asset is 1.25 (5,00,000/4,00,000). Since this weight is more than one, the risk free asset
is assigned a value –0.25 (reflecting the proportion of borrowing to his own capital).
Now consider the following situation.
Expected return (%)
Borrowing portfolio
CAL
Efficient frontier of risky assets
Lending portfolio M
rf
F
Market portfolio
σ (%)
Fig12.17
The portfolio M is the best risk-return combination of assets and hence is universally desirable. This
is called the market portfolio. To construct such a portfolio, the portfolios of all individual investors
are added. Since some these portfolios are borrowing portfolios, the others are lending portfolios
and as a result, they cancel out each other. Thus the aggregate value of a market portfolio is the
entire wealth of the economy. The return on the market portfolio is the weighted average of return
on all capital assets.
Market portfolio is a theoretical concept. It is very difficult to construct such a huge portfolio due to
a very large number of capital assets. Also determination of their exact proportions in the portfolio is
difficult to determine. As a proxy for market portfolio, a portfolio containing all securities can be
used.
464
The concept of market portfolio can be extended to define capital market line.
Capital market line (CML)
A CAL with one-month treasury bills as a risk free asset and a
market index portfolio (for example, Dow Jones, NYSE, or NSE) as a risky asset is called a capital
market line (CML).
A CML indicates the risk-return relationship of efficient portfolios.
Optimal portfolios
Investors invest in different portfolios according to their risk preference
(or risk aversion preference). A risk adverse investor would opt for risk-free opportunities in spite of
a lower return. He would opt for risky opportunities only when the risk premium is high. As a
result, the portfolio of a risk-averse investor will be on the lower end of the efficient portfolio
frontier.
Investors' attitude towards risk can be explained by his utility curve (or indifference curve). All
portfolios along a utility curve have the same utility for an investor. A higher curve means a higher
level of satisfaction of the investor.
An optimal portfolio is one, which is tangential to the highest attainable indifference curve
449
Expected return (%)
CML
Indifference curves
Market portfolio
rf
Optimal portfolio
σ (%)
Fig. 12.18
Example 14:
Find the weights to be assigned to a risk free asset having return of 6% an a market
portfolio with expected return of 15% and standard deviation 10% if the required rte of return is
20%.
Sol:
E (rp ) = w f rf + wm rm = w f rf + (1 − w f )rm
20 =
6w f + 16(1 − w f )
2
7
⇒ w f = − ; wm =
5
5
Thus equivalent to 40% of the investor’s capital should be borrowed to invest in the risky asset, i.e.,
the market portfolio to get the desired return.
A measure for risk tendency of the investor is given by the risk aversion index of the investor. Then
a utility score (U) for a portfolio has been defined as
U = E (r ) − 0.005 Aσ 2
where
E (r ) = Expected return;
A = Index of the investor's risk aversion;
450
σ 2 = Variance of the return; and
0.005 = Scaling factor that allows expected return and standard deviation
in the equation to be expressed as percentages.
It is possible to have different portfolios that have the same utility. All these portfolios lie along the
same indifference curve. Consider the following table
Table 12.34:
E(r) (%)
A = 1.5
σ (%)
U = E ( r ) − 0.005 Aσ 2
= E ( r ) − 0.0075σ 2
Example 15:
5
10
4.25
10
12
8.92
15
15
13.3125
20
20
17
25
30
18.25
30
40
18
Consider the following information
A = 2.5
u = 5%
σ = 8%
Find the required rate of return by an investor with the given risk aversion index.
Sol:
U = E (r ) − 0.005 Aσ 2
⇒ E (r ) = U + 0.005 Aσ 2
= 5 + 0.005(2.5)(64)
= 5.8%
451
12.8
Capital asset pricing model (CAPM)
This is an equilibrium model portraying relationship between expected portfolio return and the
systematic risk associated with the portfolio.
Assumptions of the model
1.
Number of investors is large so that prices are not affected by a single investor.
2.
The investors choose portfolios according to Markowitz scheme.
3.
The holding period of investments for all the investors is the same.
4.
The expectations of all the investors are homogeneous.
5.
There are no hidden costs or charges on investments.
Salient features of the model
The two basic elements of the model are capital market lime (CML) and security market line (SML)
(i)
Capital market line (CML)
As already explained, CML is a CAL consisting of one
month T-bills as a risk free asset and a market-index portfolio as a risky asset.
(ii)
Security market line (SML)
We know that the total risk consist of two components
(a) systematic risk and (b) unsystematic risk. Systematic risk is inherent to the portfolio and cannot
be eliminated whereas unsystematic risk can be eliminated through diversification.
A measure of systematic risk is the β - coefficient of the risk. The CAPM is the relationship between
this β - coefficient and the required return. A geometric representation of this relationship is called a
security market line.
452
Expected return (%)
SML
M
rM
Market portfolio
rf
0
1.0
Defensive securities
β
Aggressive securities
Fig. 12.19
Mathematically,
E (ri ) = rf + β ( E (rm ) − rf
)
where
E (ri )
rf
= Expected return of security i;
= Expected return of risk-free security;
E (rm ) = Expected return on market portfolio; and
β
= Systematic risk of the security i.
The equation can be rewritten as
E ( ri ) − rf
= β ( E ( rm ) − rf
)
(12.4)
The equation (12.4) portrays the relationship between the expected return and the risk of a security.
The three components of the risk are (i) Risk-free rate of return; (ii) risk premium on market
portfolio; and (iii) coefficient of systematic riskβ.
(i)
Risk-free rate of return Risk-free rate of return is a theoretical concept and refers to
absolutely certain return. As a proxy for this return, the returns on assets like T-bills and bank
deposits; and the rates prevailing in money market, may be used. The maturity period of the T-bills
453
or the bank deposits is taken as less than one year. Although the risk associated with theses assets is
almost negligible, under inflationary conditions the real return may be zero or even negative.
Risk premium on market portfolio E ( rm ) − rf is
Risk premium on market portfolio
(ii)
the difference between the expected return on the market portfolio and the risk-free rate of return.
Theoretically, a market portfolio is the most desirable portfolio of the risky assets. It contains all the
securities in the proportion of their values in the portfolio. For this portfolio, the risk premium is
proportional to the risk σ m2 and the degree of its aversion.
Coefficient of systematic risk β
(iii)
βi =
E (ri ) − rf
E (rm ) − rf
Here βi is the ratio of the premium on the ith security to the risk premium on the market portfolio.
Also,
βi =
Cov(ri , rm )
σ
2
m
= ρ im
σi
σm
βm = 1 .
Then
β i > 1 , the risk premium on the ith asset is more than that on the market portfolio. Such
If
assets are called aggressive assets.
If
β i < 1 , the risk premium on the ith asset is lessn that on the market portfolio. Such assets are
called defensive assets.
Now, for a portfolio with n assets,
E (rp ) = E ( w1r1 + w2 r2 + ... + wn rn )
where wi = the weight of the i th asset in the portfolio;and
ri = the return on the i th asset in the portfolio;
i = 1, 2,...n
Also,
σ 2p =
j −1 n
n
∑ (w r ) + ∑∑ w w r r
2
i =1
i i
i =1 j = 2
454
i
j i j
For a two asset portfolio with one asset risk free and the other s market portfolio
= β ( E (rm ) − rf ) ; and
E (ri ) − rf
σi
= βσ m
Given σ a = 10%; σ m = 5%; and ρ a , m = 0.6 , find β a .
Example 16:
Sol:
βa =
σ aσ m ρ a , m
σ m2
=
10 × 5 × 0.6
25
= 1.2
12.9
Using CAPM in capital budgeting
The CAPM suggests that the required rates of return on any risky asset consists of a risk-free rate and
a risk premium to commensurate the size of the 'systematic risk' that the asset possesses. Then the
present value of the project can be computed as
E (r ) = rf + β ( E (rm ) − rf
PV =
n
CFt
∑ (1 + r )
t =1
t
=
)
n
∑
t =1
(1 + r
CFt
f
+ β ( E (rm ) − rf
))
t
But CAPM provides β for a single period, then the β - values must be appropriated for the life of the
project to discount the full stream of benefits.
If the current rate of return is r0 with risk β0, then the acceptance rule is to accept the project
if
E ( R ) − rf
β
≥
ro − rf
β0
; otherwise reject it.
455
CAPM and pricing of assets
We have seen that SML is the relationship between the risk premium of an individual asset and the
risk premium of the market portfolio. If the assets are priced correctly, they will lie exactly on SML.
However in case of mispricing (over pricing or under pricing) of assets, they will be deviated from
SML.
If the assets lie above the SML, they are offering higher expected return for a given value of β. This
obviously means that the asset is under priced. Since for the same class of risk, these assets are
offering a higher return, so these are attractive assets (to buy). The buying pressure from the
interested investors will push up the price of these assets. As a result, the returns will decline. The
process continues till the equilibrium is reached and the assets are priced correctly.
Similarly, the assets lying below SML are over priced and are offering lower return in the same risk
class. As a result these assets are unattractive (to buy or to retain). The selling pressure will push
down the prices of these assets till equilibrium is reached.
Thus SML can be used to predict 'fair return' in a balanced and well functioning market. We
illustrate this fact with the help of some examples:
Consider a market where risk free rate of return is 6% and the risk premium on the market portfolio
is 8%.
(i)
Consider a defensive security A with β = 0.6. The security is expected to yield a return of
7%.
Using SML, we find the required rate of return is 10.8% (6 + 0.6 × 8). Thus the security is over
priced.
Now, we use CAPM to construct a portfolio that would return 10.8%
The expected return on market portfolio is
E (rm ) = 6 + 1× 8 = 14% (β is 1 for a market portfolio)
456
If the portfolio to be constructed consists of a risk free security with weight ω and a market portfolio,
then
10.8 = 6 × ω + 14 × (1 − ω )
= 14 − 8ω
⇒
ω = 0.4
and 1 − ω = 0.6
Thus a portfolio consisting of a risk free asset and a market portfolio in ratio of 40:60 will be priced
correctly for the given risk.
(ii)
Now consider an aggressive security B with β = 1.5. The security is expected to yield a
return of 17%.
Using SML, we find the required rate of return is 18% (6 + 1.5 × 8). Thus the security is over priced.
To get the required return at correct price, the portfolio is to be constructed of a risk free security
with weight ω and a market portfolio such that
18 = 6 × ω + 14 × (1 − ω )
= 14 − 8ω
⇒
ω = − 0.25
and 1 − ω = 1.25
Thus by borrowing 25% of his original capital and investing the whole capital in the market
portfolio, predicted return can be obtained.
Expected return (%)
Actual expected return (%)
SML
Predicted return (%)
α
β
Fig 12.20
449
The difference between the predicted return (using SML) and the actual expected return is called α
of the asset. If α < 0, ⇔ the asset is over priced. If α > 0, ⇔ the asset is under priced.
Evaluation of CAPM
CAPM is an equilibrium model that has been used widely. The merits of the model are listed below
(i)
The model establishes a direct proportional relationship between risk and the expected
return.
(ii)
CML in the model indicates the relation between risk and return of the assets whereas SML
indicates the role of undiversifiable systematic risk that must be taken into consideration
when pricing securities and portfolios. It is the systematic risk component of the risk that is
relevant.
(iii)
The coefficient of systematic risk β in CAPM is the ratio of risk premium on a security to
the risk premium on market portfolio. As such this ratio is easy to compute and interpret.
(iv)
The model helps investors to decide the composition of their portfolios such as the
proportion of investment in risk-free assets, risky assets, and the number of assets.
However, like other models, CAPM is also a simplified representation of facts. As such, some of the
real life aspects of the problems may be overlooked by the model. We list below some of the
limitations of the model
(i)
The model emphasizes for a risk free asset, which would yield a risk free return. However,
the rate of interest of such assets depends upon the maturity period of these assets, e.g.,
bank deposits have different rates of interest for different time periods.
(ii)
Another problem related to the model is the use of an appropriate market index. Since
proxy portfolios are used for perfect market portfolio, the problem is the choice of
appropriate proxy portfolio.
450
Problems
1.
Define 'systematic' and 'unsystematic' risks in context of portfolio theory.
2.
Consider the data given below:
Table 12.35
Company
Expected return (%)
S.D. (%)
A
10
15
B
8
12
C
15
25
Also,
ρ ( A, B ) = 0.2; ρ ( A, C ) = 0.6; and ρ ( B, C ) = 0.5
Construct a portfolio with the equal weight age.
3.
The following table gives the annual rates of return of two securities and an index that can
be a proxy for a market portfolio. The risk less rate of interest is 5%.
Table 12.36
Time period
Security A
Security B
Proxy index
1
14.4
121.2
11.9
2
-22.2
-33.9
0.4
3
47.5
3.7
26.9
4
7.7
3.1
-8.5
5
42.6
17.2
22.6
6
30.5
-16.9
16.5
7
14
-32
12.5
8
32.5
-30.4
-10
9
30.4
114
23.9
10
1.8
-3.7
11.2
11
-6.2
-33
-8.5
12
22
33.2
3.9
13
4.3
21.6
14.3
14
6.5
17.8
19.1
15
-37.8
7.8
-14.7
16
-26.6
-62.3
-26.5
17
97.1
65.4
37.4
18
45
-28
2.8
19
-11
-6
-7
20
-4.7
27
12.5
451
4.
(i)
Calculate the systematic and non-systematic risk for the two securities.
(ii)
For a risk free rate of return 5%, draw a CML and SML.
(iii)
Calculate the equilibrium risk premium on the two securities.
XYZ Ltd. has market value of its equity shares at Rs. 56,00,000 and the total debt at Rs.
44,00,000. An estimate of the beta of the stock is 1.25. The expected risk premium on the
market is 12% whereas the risk free rate of interest is 8%.
What is the beta of the company’s existing portfolio of assets. Estimate the company’s cost
of capital and the discount rate.
5.
The expected returns and the standard deviations of the shares of two companies ABC Ltd.
and XYZ Co. are as follows:
Table 12.37
Company
Expected return (%)
Standard deviation (%)
ABC Ltd.
15
40
XYZ Co.
10
25
If the correlation coefficient between the two shares is
(i)
-1
(ii)
0; and
(iii)
-1
Compute the expected returns and the standard deviations of the following portfolios:
6.
(a)
ABC Ltd.
100%
(b)
XYZ Co.
100%
(c)
ABC: XYZ
3:2
(d)
ABC: XYZ
1:1
Consider the following information:
rm = 15%; rf = 6%; σ m = 25%
452
For this information,
7.
(i)
Find the equation and slope of CML
(ii)
Calculate the expected return on an asset with
(a)
σ = 10%
(b)
σ = 20%
(c)
σ = 30%
For the following data, apply CAPM to compute expected market return.
rf = 2.25%;
β = 1.75;
E (r ) = 15%
8.
It is given that
rf = 7.25%;
E (rm ) = 1575%
For each of the following securities, compute the required rate of return:
Table 12.38
9.
Security
β
A
0.5
B
1.0
C
1.5
D
2.0
A portfolio consists of two assets X and Y. following information is available regarding the
portfolio
Table 12.39
Assets
Expected return (%)
Standard deviation
w(%)
(%)
X
14
18
40
Y
20
25
60
453
(i)
Find the correlation coefficient ρ XY if the standard deviation of the portfolio is
20%
10.
(ii)
Find the portfolio standard deviation if X: Y:: 2:3.
(iii)
Repeat (i) if the desired deviation is 12%, and 16%.
Consider the following investment options
Table 12.40
Portfolio
Expected return (%)
Standard deviation (%)
A
25
25
B
8
0
If the risk aversion coefficient A of an investor were 3.5, which portfolio would he choose?
What is the maximum level till he would prefer the risky portfolio?
11.
Consider the following mutually exclusive investment proposals
Table 12.41
Year
Proposal A (Rs. ’00,000)
Proposal B (Rs.’00,000)
0
(500)
(800)
1
150
100
2
150
120
3
150
130
4
150
150
5
150
160
6
150
150
7
-
120
8
-
120
9
-
115
10
-
105
454
The company employs the risk-adjusted methods for evaluating the projects. The required rates of
return are as follows
Table 12.42
Project pay-back
Required rate of return (%)
< 1 year
10
1-5 years
12
5-10 years
15
10-15 years
18
Which project should the firm select?
12.
Consider the following information regarding an investment proposal
Table 12.43
Year
Expected CFAT (Rs. ‘000)
Certainty equivalent quotient
0
200
1.0
1
160
0.8
2
140
0.7
3
120
0.6
4
100
0.5
5
80
0.4
6
80
0.3
The firm's cost of equity capital is 18%. The risk-free rate of interest is 6%. What should
be the decision of the firm?
12
13
In example 5, find the change in NPV if
(a)
VC increases by 12%;
(b)
VC and SP both increase by 8%.
A company is considering a project which has the following features which influence the
NPV of the project
455
Table 12.44
Particulars
Value
Initial investment (I)
Rs. 50,000
Cost of capital (k)
12%
Units manufactured and sold per annum
1500
Selling price per unit (P)
Rs. 30
Variable cost per unit (V)
Rs. 22
Fixed cost (F)
Rs. 5,000
Depreciation (D)
Rs. 2,000
Tax rate (T)
40%
Life of the project (n)
6
Net salvage value (S)
0
The relationship between these factors and the NPV is as follows
NPV =
∑
( Q( P − V ) − F − D )(1 − T ) + D
(1 + k )t
+
S
− I
(1 + k ) n
Carry out the sensitivity analysis with reference to Q, P, and k.
14
A project involves an initial outlay of Rs. 50 lakh. The following cash flows are associated
with the project
Table 12.45
Year 1
Cash flow
Year 2
Probability
(Rs. ’00,000)
Cash flow
Year 3
Probability
(Rs. ’00,000)
Cash flow (Rs.
Probability
’00,000)
14
0.4
15
0.4
20
0.3
16
0.5
16
0.4
21
0.4
18
0.1
20
0.2
22
0.3
456
Assume that the cash flows are independent, calculate the expected NPV and σ NPV . The
risk-free rate of interest is 5%.
15
An investor is considering an investment with an initial outlay of Rs. 2,00,000. The
expected value and the standard deviation of the cash flows are as follows
Table 12.46
Year
Expected value (Rs. ‘000)
S.D. (Rs. ‘000)
1
12
5
2
10
6
3
8
7
4
8
6
The cash flows are perfectly correlated. Calculate the expected NPV and σ NPV . The riskfree rate of interest is 8%.
16
A firm is considering an investment project which ahs an estimated life of four years. The
cost of project is Rs. 50,000 and the possible cash flows are given below
Table 12.47
Year 1
Cash flow
Year 2
Prob.
(Rs.)
Cash flow
Year 3
Prob.
(Rs.)
Cash flow
Year 4
Prob.
(Rs.)
Cash flow
Prob.
(Rs.)
10,000
0.2
15,000
0.4
20,000
0.3
20,000
0.2
20,000
0.5
24,000
0.3
30,000
0.5
30,000
0.4
30,000
0.3
31,000
0.3
35,000
0.2
40,000
0.4
457
The cash flows of various years are independent. The risk free rate of interest is 6%.
(a)
Find the expected NPV.
(b)
If the NPV is approximately normally distributed, what is the probability that
NPV will be zero or negative.
(c)
17
What is the probability that the profitability index will be more then 1.1?
A firm has determined its risk profile – the maximum standard deviation acceptable for a
given expected value of the profitability index. The risk profiles are as follows
Table 12.48
Expected profitability index
Maximum standard deviation
1.00
0.00
1.05
0.05
1.10
0.08
1.15
0.15
1.20
0.28
1.25
0.35
1.30
0.38
1.35
0.40
An investment proposal is there before company for consideration. The initial outlay of the
proposal is Rs. 50 lakh. The distribution of the NPV has been estimated to be as follows
Table 12.49
NPV (Rs. lakh)
Probability
-5
0.02
0
0.03
5
0.08
10
0.20
15
0.30
18
0.15
20
0.10
21
0.9
22
0.3
Should the company accept the project?
458
18
Consider the following information gathered by a company on two mutually exclusive
projects
Table 12.50
Particulars
Project A
Project B
Initial outlay
Rs. 20,000
Rs. 50,000
Life of the project
6
7
Cash flows
Rs. 5,000
Rs. 8,500
Required rate of return (%)
12
14
Which of the projects should be selected by the company?
19
Following are the expected cash flows of a project
Table 12.51
Year
Cash flow (Rs.)
0
50,000
1
8,000
2
10,000
3
12,000
4
9,000
5
10,000
The certainty equivalent factor is given by the expression
α t = 0.5 + 1.2t
Calculate the NPV of the project if the risk-free rate of interest is 6%.
459
20
For an initial investment of Rs. 20,000, the following proposal is there before a company
Table 12.52
Year1
Year 2
Year 3
CFAT (Rs.)
Probability
CFAT (Rs.)
Probability
CFAT (Rs.)
Probability
5,000
0.1
10,000
0.2
12,000
0.1
8,000
0.2
12,000
0.3
15,000
0.3
10,000
0.3
15,000
0.3
18,000
0.4
15,000
0.4
20,000
0.2
20,000
0.2
The cash flows are independent of each other. If the risk free rate of interest is 5%, answer
the following questions
(a)
What is the expected NPV of the project?
(b)
What is the standard deviation of the expected NPV?
(c)
With what probability will the NPV less than zero?
(d)
What is the probability that the NPV will lie between (i) 0 and Rs. 5,000 (ii) Rs.
5,000 to Rs. 10,000; and (iii) more than Rs. 10,000?
21
A company has the following three proposals before it for consideration
Table 12.53
Project A
Project B
Project C
NPV (Rs.)
Probability
NPV (Rs.)
Probability
NPV (Rs.)
Probability
(3,000)
0.05
(2,000)
0.01
(5000)
0.05
(2,000)
0.10
0
0.04
(1,000)
0.10
0
0.15
500
0.15
2,000
0.40
2,000
0.20
1,500
0.20
5,000
0.20
5,000
0.25
2,000
0.30
10,000
0.25
6,000
0.15
2,500
0.15
-
15,000
0.09
3,000
0.10
-
18,000
0.01
4,000
0.05
-
Rank the projects with respect to return and risk.
449