Notes for portfolio analysis

1. Risk and Return

2. Return
• Total income generated by the investment expressed as a percentage
of the cost of investment.
• Return = {Dividend or Interest Income from the asset + (Selling price –
Purchase price)}/Purchase price
• Types of Return;
• Average return (based on Arithmetic or Geometric mean)
• Expected return (based on probability distribution)
• Holding period return
• Effective Annualised return (EAR)

3. Average Return (Based on A.M.)
n
AR = ∑ri/n
i=1
Where,
ri : return earned in year i
n : number of years for which the investment is held

4. Ques: An investor wants to calculate average return of a share of ABC Ltd., currently available at Rs. 160
as on 30th
June 2024. The share price at the end of June’ 19, 20, 21, 22 and 23 was Rs. 100, 118, 130,
120, 140 and 160 respectively. Also, the stock paid dividends of Rs. 6, 5, 0, 10, 5 respectively during the
year 2017, 18, 19, 20 and 21 respectively. Compute the average return based on A.M.
Ans:
Year Stock Price (in Rs.) Dividend (in Rs.) Capital Gain (in Rs.)
2019 100 - -
2020 118 6 18
2021 130 5 12
2022 120 0 -10
2023 140 10 20
2024 160 5 20

5. AR (based on AM) = 14.71%
Year Stock Price (in Rs.) Dividend (in Rs.) Capital Gain (in Rs.) Return (%)
2019 100 - - -
2020 118 6 18 (6+18)/100 = 24%
2021 130 5 12 (5+12)/118 = 14.41%
2022 120 0 -10 (0-10)/130 = -7.69%
2023 140 10 20 (10+20)/120 = 25%
2024 160 5 20 (5+20)/140 = 17.85%

6. Average Return (Based on G.M.)
• A.M. doesn’t consider the impact of compounding and may give misleading results at times.
• Eg; An investor buys a share for Rs. 20. At the end of year 1, its price is Rs. 25 but the investor
holds it. At the end of year 2, it comes down again to Rs. 20. Here the average return using AM is
2.5%. This value is incorrect as the investor doesn’t earn anything over the past 2 years.
• Here, GM = [(1+0.25)(1-0.20)]1/2
– 1 = 0
AR (using GM) = {(1+r1)(1+r2)……(1+rn)}1/n
– 1
Where,
r1,2….,n : return earned in year 1,2….,n
n : number of years for which the investment is held

7. Expected Return
(Based on Probability distribution)
n
E(r)= ∑ piri
i=1
Where,
E(r)= Expected return on the security
pi = Probability in ‘ith
’ situation (where, i=1 to n)
ri = Return in ‘ith
’ situation

8. • Ques: You are given the return of a stock which depends on the state
of economy. Compute the expected return.
• Ans: E(r) = (18*0.4) + (15*0.3) – (5*0.3) = 10.2%
State of the Economy Return Probability
Good 18% 0.4
Normal 15% 0.3
Bad -5% 0.3

9. Holding Period Return (HPR)
• Total return earned during the holding period (investment horizon)
• Eg;
• An investor invests in a non-dividend paying share at the cost of Rs. 100 in the beginning of year 2011. At the end of
year 2024, it is sold at Rs. 150. Calculate its HPR on the share.
• HPR = (150-100)/100 = 50%
• It suffers from certain limitations;
• Does not consider the annual return
• If HP of 2 investments is different, one cannot compare them using HPR.
• To overcome these limitations, we use Effective Annualized Return (EAR)
HPR = {TI + (Pn – P0)}/ P0
Where;
TI : Total Income received during the period
Pn: Sale price at the end of the holding period
P0: Purchase price
n: No. of years for which shares are held (holding period)

10. Effective Annualised Return (EAR)
• Equivalent P.A. return earned on the investment
• Eg; An investor wants to invest in a zero-coupon bond having face value of Rs. 100. Three
different maturity period bonds are available as mentioned below. Which is the best investment
option?
EAR = (1+rt)1/t
– 1
Where;
rt: HPR
t: Holding period in years
Bond Time Horizon Price
A 6 months 97
B 1 year 95
C 16 years 22

11. EAR = (1+rt)1/t
– 1
Where;
rt: HPR
t: Holding period in years
Bond Time Horizon Price HPR EAR
A (BEST) 6 months 97 (100-97)/97 =
3.09%
(1+0.0309)2
-1 =
6.27%
B 1 year 95 (100-95)/95 =
5.26%
(1+0.0526)1
-1
=5.26%
C 16 years 22 (100-22)/22 =
354.54%
(1+3.5454)1/16
– 1
= 6.24%

12. Portfolio Return
n
E(rp)= ∑ wxrx
x=1
Where,
E(rp)= Expected return on the portfolio
wx : Weight of security ‘x’ in the portfolio (A portfolio has ‘n’ number of securities, where the weight
of each security is the percentage of total funds invested in each security)
rx : Return of security ‘x’

13. Practice Questions
• Question 1:
An investor has Rs. 1,00,000 for investment in debt and equity. She expects to
earn return of 16% on his overall investment. Debt and equity are expected to
provide return of 15% and 20% respectively. What amount should she invest in
both the alternatives?
• Solution:
Assume he invests w % of funds in debt, and (1-w)% in equity
(W)(0.15) + (1-w)(0.2) = 0.16
W=80%= Rs. 80,000
1-w=20% = Rs. 20,000

14. • Question 2:
An investor has Rs. 90,000 for investment. He invested 30% of his funds in stock x, which is likely
to provide return of 14% and the remaining in stock y, which is expected to provide 19% return.
Find the expected return on his total investment.
• Solution:
(0.3*0.14) + (0.7*0.19) = 17.5%
Total return (in Rupee terms) = 17.5% * 90,000 = Rs. 15,750
• Question 3:
An investor has Rs. 1,00,000 for investment on which she wants to earn return of 16%. She has 2
sources available for investment; debentures offering return of 15% and equity shares offering
return of 20%. Find the amount invested in each security to achieve the target rate of return.
• Solution:
Assuming the percentage of funds invested in debt and equity as (w) and (1-w);
(w)(0.15) + (1-w)(0.20) = 0.16
w=0.8, 1-w=0.2
Thus, investment in debt = 80% of Rs. 1,00,000 = Rs. 80,000 and investment in equity is Rs.
20,000

15. Risk
• Deviation of actual outcome from the expected outcome.
• It is defined in terms of variability in expected return.
• All investments are subject to risk, however the level of risk differs.
• Risk of an investment can be computed using the following formula;
n
σ=√ ∑ pi[ri-E(r)]2
i=1
Where,
σ: Standard Deviation of the security
pi : Probability in ‘ith
’ situation (where, i=1 to n)
ri : Return in ‘ith
’ situation
E(r): Expected return of the security

16. • Ques: Compute the risk and return on the security with the following
probability distribution of returns:
• Ans: E(r) = 12%; S.D. = 10.05%
Probability Return (%)
0.1 20%
0.2 15%
0.2 -5%
0.3 10%
0.2 25%

17. Absolute vs Risk Adjusted Return
• Absolute return is the total return generated by an investment
without considering its risk.
• Risk adjusted return is return expressed in terms of per unit of
underlying risk.
• Risk adjusted return = (Expected Return – Risk free return)/Total Risk

18. Covariance and Correlation
σxy = ∑ pi[rxi-E(rx)][ryi- E(ry)]
Where,
σxy = Covariance between returns of x and y
Other variables have the same meaning as mentioned above
ρxy = σxy/(σx * σy)
Where,
ρxy = Correlation between returns of x and y
Other variables have the same meaning as mentioned above

19. Coefficient of Variation
(A Relative measure of Risk)
• Used to compare different investments
• Coefficient of Variation (CV) = Standard Deviation/Mean Return
• Ques: An investor is interested in investing his funds in securities of 2
companies A Ltd. and B Ltd. with the following expected return and risk:
Which stock should be preferred and why?
Sol: CVA = 0.5625, CVB = 0.5
Thus, he should prefer stock B as it has a lower risk to return ratio
A Ltd. B Ltd.
Expected Return 16% 12%
Risk (σ) 9% 6%

20. Portfolio Risk
N n n
σp
2
= ∑ wx
2
* σx
2
+ ∑ ∑ wx * wy * σxy
x=1 x=1 y=1
x ≠y
σp = √ σp
2
Where,
σp
2
= Portfolio Variance
σp = Portfolio Standard Deviation
σx
2
= Variance of security ‘x’ (Portfolio has N number of securities)
wx , wy= Weight on security x and y respectively
σxy = Covariance between x and y
 Note: the portfolio risk cannot be the weighted average of risk of individual securities as
every pair of securities co-vary with each other. Thus, it is important to consider the
covariance/correlation amongst each pair of securities in the portfolio.

21. Portfolio Risk
• The earlier mentioned formula is generalised for any number of
securities in the portfolio. Portfolio variance can be simply written for
2 securities portfolio in the following manner:
• Similarly, it can be written for 3 securities portfolio as;
σp
2
=(w1
2
*σ1
2
)+(w2
2
*σ2
2
)+(2*w1*w2*σ12)
σp
2
=(w1
2
*σ1
2
)+(w2
2
*σ2
2
)+(w3
2
*σ3
2
)+ (2*w1*w2*σ12)+(2*w1*w3*σ13)+(2*w2*w3*σ23)

22. Questions
• Ques 1: Following are a set of returns for stocks x and y;
1. Estimate the return and risk of stocks x and y
2. Which stock should be preferred?
3. Find the covariance and correlation between x and y
4. Assuming that an investor holds a portfolio of stocks x and y, with 50% of his
funds invested in both, find the portfolio return and risk
Probability 0.2 0.2 0.2 0.2 0.2
Return on
stock x
0.11 0.09 0.25 0.07 -0.02
Return on
stock y
-0.03 0.15 0.2 0.2 0.06

23. Solution:
1. E(rx) = 10%, E(ry) = 11.6%, σx = 8.7%, σy = 8.9%
2. To evaluate which investment is preferable, we must look at the
return to risk ratio or risk to return ratio (coefficient of variation)
Coefficient of Variation (CV) = [σ/E(r)]*100
CVx = 87.18%, CVy = 76.84%
Thus, the investor should prefer stock y as it has a lower risk to
return ratio
3. σxy = 0.003, ρxy = 0.386
4. E(rp)= 10.8%, σp
2
= 0.0054, σp = 7.3%

24. • Ques 2: An investor is considering investment in stocks A and B, whose
details are given below:
Find the expected return and risk of the portfolio with 60%
investment in A and 40% in B.
• Solution:
E(rp) = 16%
σp = 3.6%
Stock A Stock B
E(R) 14% 19%
σ 3% 6%
ρAB (Correlation coefficient
between A and B)
0.5

25. • Ques 3: An investor has a portfolio of 5 securities whose expected returns and
amount invested are as follows:
Find the expected % return of the portfolio.
Ans: 13.15%
• Ques 4: An investor invests Rs. 200000 in a asset X which has expected return of
8.5%, Rs. 280000 in asset Y which has an expected return of 10.2% and Rs.
320000 in asset Z which has an expected return of 12%. What is the expected
return of the portfolio (in % and absolute terms)?
Ans: 10.495%; Rs. 83,960
Security 1 2 3 4 5
Amount
(in Rs.)
150000 250000 300000 100000 200000
Expected
return
12% 9% 15% 18% 14%

26. • Ques 5: A investor is considering the following 2 investment proposals. Returns
from both the proposals are the same but their probabilities differ. Compute the
expected return and risk of the following 2 proposals and advice the investor.
Ans: E(r)= 21.5%, E(Y) = 15.25%, SD (X) = 8.81%, SD (Y) = 13.36%, CV (X) = 40.98%,
CV (Y) = 87.6%
Prefer X as it has lower CV
Return (%) Prob. (Stock X) Prob. (Stock Y)
-10 0.05 0.20
15 0.15 0.20
20 0.30 0.25
25 0.25 0.25
30 0.25 0.1

27. Types of Risk
• Depending upon the causes or sources of risk, we can classify total risk
of a security as;
• Systematic (Non-Diversifiable) Risk:
• That part of total risk which is caused by factors beyond the control of a specific
company.
• All investments are subject to systematic risk.
• It cannot be diversified away by holding a number of securities in the portfolio.
• Unsystematic (Diversifiable) Risk:
• Caused by factors within the control of a specific company.
• Can be diversified by holding securities which are least (or not) correlated.

28. Systematic Risk (Non-Diversifiable)
• Market Risk: It is the tendency of security prices to move together. Its
caused due to the herd mentality of investors.
• Interest rate Risk: Arises due to changes in market interest rate. Rise
in interest rates has a negative impact on security prices.
• Purchasing power (Inflation) Risk: Purchasing power of investor’s
income reduces in times of rising prices.
• Exchange Rate risk: Affects companies with global exposure. This is
classified under systematic risk as a large number of firms have
foreign exchange exposure.

29. Unsystematic Risk (Diversifiable)
• Business Risk: This refers to the operating risk, which is measured
using DOL.
• Financial Risk: This is associated with the firm’s financing decisions
and is measured using DFL.

30. Diversification
• Investors prefer to hold a combination of different securities rather than investing all the
funds in a single security.
• It is based on the basic idea that ‘Don’t put all your eggs in the same basket’.
• Diversification helps to reduce the variability of returns and reduces risk of the total
investment.
• In a well-diversified portfolio, only unsystematic risk can be reduced.
• Portfolio risk reduces with lower correlation.
• Maximum benefits of diversification be attained when correlation is -1.
• In order to diversify risk an investor should hold a well-diversified portfolio of different and
unrelated securities.
• In general, the larger the number of securities with lesser correlation, greater will be the
diversification benefit due to lower unsystematic risk.

32. Mean-Variance Frontier
(Case of 2 risky assets)
• Two securities can be combined in multiple ways by changing the
proportion of investment.
Some combination of 2 stocks will provide
smaller risk than either of the stock taken alone,
as long as ρ < (σx/ σy), where σx< σy.

33. Minimum Variance Portfolio
• Portfolio with a particular combination of investment in securities which gives the least possible risk.
• It can be estimated using the following formula:
• When correlation between the securities is -1,
• W1 = σ2/(σ1+σ2)

34. Question
• An investor is interested in 2 stocks-X and Y. Their standard deviations
are 20% and 39%. Their returns have a correlation of 0.3. What
portfolio allocation between these 2 investments minimizes the
variance? Also calculate the risk of this portfolio.
• Wx = 89%, Wy = 11%
• σ =19%

35. Case of N-Securities and Efficient Frontier

36. Optimal Portfolio
Steeper the utility curve,
greater is the level of risk
aversion.

37. Capital Market Theory
• Portfolio theory describes how rational investors must build efficient portfolios.
• Capital Market Theory tells us how assets must be priced in the capital markets if
everyone behaved in the way portfolio theory suggests.
• Introduction of the risk-free asset allows the development of capital market theory
from the portfolio theory.
• We can have different portfolio possibility lines by combining the risk-free asset and
the risky portfolios on the efficiency frontier.
• We can keep drawing further lines from Rf to efficient frontier at higher points, until
we reach the point of tangency (M).
• M is the market portfolio. It must include all risky assets in the market. If a risky asset
is not in this portfolio in which everyone wants to invest in, there would be no demand
for it and hence it would have no value.

38. Capital Market Theory
• Every investor will invest in a
combination of risk-free asset and the
tangency portfolio (M).
• CML (Capital Market Line) is the
efficient frontier in the presence of
risk-free asset.
• Equation of CML (Expected return of
an efficient portfolio:
• Slope of CML:

39. Security Market Line (SML)
• At the point of tangency,
• Slope of MV Frontier = Slope of CML
• On equating the two, we get the following
equation:
• E(Ri) = Rf + β (E(Rm)-Rf)
• Where,
• Ri is the return on any risky asset
• β is Cov.(i,m)/Var.(m)
• Rf is the return on risk free asset
• This relationship holds for all expected
portfolio/security returns.
• This is plotted as the Security Market Line
(SML) which describes the returns for all
assets (efficient as well as inefficient) and is
popularly referred to as CAPM.

40. • CAPM is used to calculate the required or expected rate of return on a
security.
• If the estimated return or actual return from the security is different,
it is under or over valued.

41. Beta as a measure of systematic risk
• Systematic risk is captured by the sensitivity of a security’s return with respect to market return.
• It is measured with Beta (β)
• The return on each security can be defined using the Sharpe’s Single Index Model using the following equation:
• If β > (<) 1; then the security is more (less) risky than the market and is termed as aggressive (defensive) stock.
• If β = 1, the security is as risky as market portfolio.
• β = Cov(ri,rm)/var(rm)
• Systematic Risk = β2
* σ2
m
• Unsystematic Risk (σ2
ei)= Total Risk – Systematic Risk = (σ2
i)– (β2
* σ2
m)
Ri = α + β Rm + ei
Where,
Ri: Return on a particular security i
Rm: Return on market index
α, β: Intercept and Slope of the regression equation respectively

42. • Ques. 1: following information is available in respect of a security G
and the market portfolio M. Find out the β of the security. If the
market return rises by 10 points, what will be the change in return of
stock M?
• Ans: β = 0.67, Return on stock G will rise by 6.7 points.
Probabilities Return of Security G Return of Market Portfolio
0.3 10% 12%
0.4 12% 15%
0.3 14% 18%

43. • Ques 2: Following information in available in respect of a security A
and the market portfolio M. Find out beta of the security.
• Ans: Beta = -0.7
Security A (return in %) Market Portfolio M (return in %)
10 15
18 12
14 18
20 16
11 19

44. • Ques 3: You have the following 5 stocks in an equally weighted
portfolio. Market standard deviation is 0.22.
• Find the systematic and unsystematic risk for individual securities.
• Find the return, systematic risk and unsystematic risk for the entire
portfolio.
Stocks A B C D E
Return 0.25 0.27 0.24 0.18 0.16
Beta 1.65 1.35 1.15 0.95 1.12
Variance 0.2916 0.1764 0.1225 0.0484 0.0625

45. • Q4. Which of the following stocks are undervalued, overvalued or
fairly valued?
• Rf = 8%, Rm = 15%, βa = 0.68, βb=-0.86, βc=1.29
• The forecast returns for stocks a,b,c in the next year are as follows:
State of the
Economy
Probability Ra(%) Rb(%) Rc(%)
Recession 10% 10 28 -22
Below Average 20% -10 14.7 -2
Average 40% 7 0 20
Above Average 20% 45 -10 35
Boom 10% 30 -20 50

46. • Ra (Undervalued):
• Estimated Return: 13.8%
• Required Return (as per CAPM): 12.76%
• Rb (Overvalued):
• Estimated Return: 1.74%
• Required Return (as per CAPM): 1.98%
• Rc (Undervalued):
• Estimated Return: 17.4%
• Required Return (as per CAPM): 17.03%

47. Value at Risk (VaR)
• Value at risk (VaR) is a statistic that quantifies the extent of possible
financial losses within a portfolio over a specific time frame.
• It is defined as the maximum dollar amount expected to be lost over a given time
horizon, at a pre-defined confidence level.
• It quantifies the worst case scenarios.
• Investors use VaR to understand and manage the downside risk associated with their
investments.
• E.g.: Your portfolio has a VaR of -$1000 at 90% confidence level. This means that:
• There is a 90% probability that the portfolio will not loose more than $1000 over the specified
time frame.
• There is a 10% chance that the portfolio could loose more than $1000 within the same period.
• -$1000 is the maximum expected loss you would face 90% of the time.