PORTFOLIO RISK AND RETURN

1. McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Portfolio Risk and
Return: Part I

2. 5-2
Rates of Return
• Holding-Period Return (HPR)
• HPR of a share of stock depends on the increase (or decrease) in
the price of the share over the investment period as well as on any
dividend income the share has provided.
• The rate of return is defines as dollars earned over the investment
period (price appreciation as well as dividends) per dollar invested:
HPR=
𝐸𝑛𝑑𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 −𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒+𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
This definition of HPR assumes that the dividend
is paid at the end of the holding period.

3. 5-3
• You put up $50 at the beginning of the year
for an investment. The value of the
investment grows 4% and you earn a
dividend of $3.50. Your HPR was…
a) 4%
b) 3.5%
c) 7%
d) 11%
Example: Holding Period Return

4. 5-4
• Solution:
4% + $3.50/$50 = 11%
Answer (d) is correct
Example: Holding Period Return

5. 5-5
Rates of Return
• Measuring Investment Returns over
Multiple Periods
• Arithmetic average
• Sum of returns in each period divided by number of periods
• Geometric average
• Single per-period return; gives same cumulative performance as
sequence of actual returns
• Compound period-by-period returns; find per-period rate that
compounds to same final value
• Dollar-weighted average return
• Internal rate of return on investment

6. 5-6
Quarterly Cash Flows/Rates of Return of a Mutual Fund
1st
Quarter
2nd
Quarter
3rd
Quarter
4th
Quarter
Assets under management at start of
quarter ($ million)
1 1.2 2 0.8
Holding-period return (%) 10 25 −20 20
Total assets before net inflows 1.1 1.5 1.6 0.96
Net inflow ($ million) 0.1 0.5 −0.8 0.6
Assets under management at end of
quarter ($ million)
1.2 2 0.8 1.56

7. 5-7
Example
Calculate arithmetic average, geometric average and
dollar-weighted return.
Arithmetic average = (10+25-20+20)/4 = 8.75%
Geometric average =
(1+0.10)(1+0.25)(1-0.20)(1+0.20)=(1+r)^4
r = 7.19%
Dollar-Weighted Average Return
0 = −1.0 +
−0.1
1+𝐼𝑅𝑅 1 +
−0.5
(1+𝐼𝑅𝑅)2 +
0.8
(1+𝐼𝑅𝑅)3 +
−0.6+1.56
(1+𝐼𝑅𝑅)4
IRR = 3.38%

8. 5-8
Example: AR, GR & DWR
• A fund begins with $10 million and reports
the following three-month results
• Compute the arithmetic, geometric and
dollar-weighted average return.
Months 1 2 3
Net inflows (end of month, $
million)
3 5 0
HPR(%) 2 8 (4)

9. 5-9
Example: AR, GR & DWR
• Solution:
a. The arithmetic average is (2+8-4)/3 = 2%
per month
b. The geometric average is:
[(1+0.02)(1+0.08)(1-0.04)]^1/3 -1 = 0.0188
or 1.88% per month

10. 5-10
Example: AR, GR & DWR
• Solution: Dollar-Weighted Average (IRR)
Months
1 2 3
AUM at beginning of month 10.0 13.2 19.256
Investment profits during the month
(HPR*Assets)
0.2 1.056 (0.77)
Net inflows during the month 3.0 5.0 0.0
Assets under management at end of
month
13.2 19.256 18.486
Time
0 1 2 3
Net Cash Flow -10 -3.0 -5.0 +18.486
IRR is 1.17% per month

11. 5-11
Rates of Return
• Conventions for Annualizing Rates of
Return
• APR = Per-period rate × Periods per year
• 1 + EAR = (1 + Rate per period)
• 1 + EAR = (1 + Rate per period)n = (1 + )n
• APR = [(1 + EAR)1/n – 1]n
• Continuous compounding: 1 + EAR = eAPR
APR
n

12. 5-12
Rates of Return
Q1. If weekly return is 0.2%, then calculate
the compound annual return.
Q2. If the return for 15 days is 0.4%, the
annualized return is?
Q3. What is the annualized return for an 18-
month return of 20%?

13. 5-13
Rates of Return
Q1. r(annual) = (1+0.002)^52 -1
= 10.95%
Q2. r(annual) = (1+0.004)^365/15 – 1
=10.20%
Q3. r(annual) = (1+ 0.20)^2/3 – 1
=12.92%

14. 5-14
Example: EAR & APR
• Suppose you buy a $10,000 face value
Treasury bill maturing in one month for
$9900. On the bills maturity date, you
collect the face value. Since there are no
other interest payment, calculate the
holding-period return, APR and EAR.

15. 5-15
Example: EAR & APR
• HPR = $100/$9900 = 0.0101 or 1.01%
• APR = 1.01%*12 = 12.12%
• EAR: (1+0.0101)^12 = (1+EAR)
EAR = 12.82%

16. 5-16
Other Return Measures
• Gross Return: is the return earned by an asset manager prior to deductions for
management expenses, custodial fees, taxes or any other expenses that are not directly
related to the generation of returns but rather related to the management and
administration of an investment.
• Net Return: is a measure of what the investment vehicle (mutual fund etc.) has earned
for the investor and accounts for all managerial and administrative expenses that reduce
an investor’s return.
• Pre-tax and After-Tax Nominal Return: Returns computed prior to and after adjustment
of taxes on realized gain.
• Real Return & Nominal Return: A nominal return (r) consist of three components: a real
risk-free return as compensation for postponing consumption (rf), inflation as
compensation for loss of purchasing power (π) and a risk premium for assuming risk (rp).
Thus, nominal return and real return can be expressed as:
1 + 𝑟𝑛𝑜𝑚𝑖𝑛𝑎𝑙 = 1 + 𝑟𝑓 ∗ 1 + 𝜋 ∗ 1 + 𝑟𝑝
(1+𝑟𝑟𝑒𝑎𝑙) = (1 + 𝑟𝑓)* 1 + 𝑟𝑝
1 + 𝑟𝑟𝑒𝑎𝑙 = 1 + 𝑟𝑛𝑜𝑚𝑖𝑛𝑎𝑙 / 1 + 𝜋

17. 5-17
Risk and Risk Premiums
• Scenario Analysis and Probability
Distributions
• Scenario analysis: Possible economic scenarios;
specify likelihood and HPR
• Probability distribution: Possible outcomes with
probabilities
• Expected return: Mean value of distribution of
HPR
• Variance: Expected value of squared deviation
from mean
• Standard deviation: Square root of variance

18. 5-18
Example: Scenario Analysis for the Stock Market

19. 5-19
Risk and Risk Premiums
• Deviation from Normality and Value at Risk
• Kurtosis: Measure of fatness of tails of probability
distribution; indicates likelihood of extreme outcomes
• Skew: Measure of asymmetry of probability distribution
• Using Time Series of Return
• Scenario analysis derived from sample history of returns
• Variance and standard deviation estimates from time
series of returns:

20. 5-20
Value at Risk (VaR)

21. 5-21
Value at Risk
• It is defined as the maximum dollar amount
expected to be lost over a given time horizon, at a
pre-defined confidence level.
• For example, if the 95% one-month VAR is $1
million, there is 95% confidence that over the next
month the portfolio will not lose more than $1
million.

22. 5-22
Var Calculation
• Suppose you worry about large investment losses in worst-case scenario for
your portfolio. You might ask: “How much would I lose in a fairly extreme
outcome, for example, if my return were in the fifth percentile of the
distribution?”
• You can expect your investment experience to be worse than this value only
5% of the time and better than this value 95% of the time.
• In investment parlance, this cutoff is called the value at risk (VaR). A loss-
averse investor might desire to limit portfolio VaR, i.e. limit the loss
corresponding to a probability of 5%.
• For normally distributed returns, VaR can be derived from the mean and
standard deviation of the distribution.
• Excel’s standard normal function =NORMSINV(0.05) computes the fifth
percentile of a normal distribution with a mean of zero and a variance of 1,
which turns out to be -1.64485.
• In other words, a value that is 1.64485 standard deviations below the mean
would correspond to a VaR of 5%

23. 5-23
Var Calculation (Continued)
• Thus, VaR = E(r) + (-1.64485)σ
• We can obtain this value directly from Excel’s nonstandard normal function =
NORMINV(0.05, E(r),σ)
• If the given sample of returns is not normally distributed then 5% VaR is estimated
directly as the fifth percentile rate of return.
• For instance, for a sample of 100 returns, first the returns are ordered from high to low
and then count the fifth observation from the bottom, that will give you the value of
VaR.
• If the 5% of the observations, don’t make an integer; then interpolation is required.
Suppose we have 72 monthly observations so that 5% of the sample is 3.6
observations.
• Then we approximate the VaR by going 0.6 of the distance from the third to the fourth
rate from the bottom.
• Further assume, the third and fourth observations are -42% and -37%, then the
interpolated value for VaR is -42 + 0.6(42-37) = -39%

24. 5-24
Figure 5.1 Normal Distribution with Mean Return 10% and
Standard Deviation 20%

25. 5-25
Normal Distribution Properties Relevant for
Investment Management
Two special properties of the normal distribution lead to
critical simplifications of investment management when
returns are normally distributed:
1. The return on a portfolio comprising two or more assets
whose returns are normally distributed also will be
normally distributed
2. The normal distribution is completely described by its
mean and standard deviation. No other statistic is
needed to learn about the behaviour of normally
distributed return.

26. 5-26
Example: VaR
Suppose the current value of a stock portfolio is $23 million.
A financial analyst summarizes the uncertainty about next
year’s holding-period return using the scenario analysis in
the following table. What are the annual holding-period
returns of the portfolio in each scenario?
a) Calculate the expected holding-period return and the
standard deviation of returns. Also, calculate the VaR of
the portfolio with normally distributed returns with the
same mean and standard deviation as this stock?
b) Suppose that the worst three rates of return in a sample
of 36 monthly observations are -17%, -5% and 2%.
Estimate the VaR.

27. 5-27
Example: VaR
Scenario Business
Conditions
Probability End-of-Year
Value
($ million)
Annual Dividend
($ million)
1 High Growth 0.30 35 4.40
2 Normal
Growth
0.45 27 4.00
3 No Growth 0.20 15 4.00
4 Recession 0.05 8 2.00

28. 5-28
Example Solution: VaR
Scen
ario
Prob. Ending
Value($
million)
Dividend
($
million)
HPR HPR*Prob Deviation
: HPR -
Mean
Prob*Dev
iation
Squared
1 0.30 35 4.40 0.713 0.214 0.405 0.049
2 0.45 27 4.00 0.348 0.157 0.04 0.001
3 0.20 15 4.00 -0.174 -0.035 -0.482 0.046
4 0.05 8 2.00 -0.565 -0.028 -0.873 0.038
Sum 0.308 0.134

29. 5-29
Example Solution:VaR
a) Expected HPR = 0.308 or 30.8%
• Variance = 0.134
• Standard Deviation = √0.134 = 36.6%
• For the corresponding normal distribution, VaR
would be 30.8% - 1.645*36.6% = -29.43%
b) With 36 returns, 5% of the sample would be
0.05*36 = 1.8 observations. The worst return is
-17% and second worst is -5%. Using interpolation,
we estimate the fifth percentile return as:
-17% + 0.8(17-5) = -7.4%

30. 5-30
Portfolio Risk
• Portfolio return : when two individual assets are combines in a
portfolio, we can compute the portfolio return as a weighted average
of the return of the two assets. The equation is given as :
• 𝑅𝑝 = 𝑤1𝑅1 + 𝑤2𝑅2
• Portfolio Risk: can be calculated by taking the variance of both assets
and including the covariance between the assets. The equation is
given by:
• 𝜎𝑝 = 𝑤1
2
𝜎1
2
+ 𝑤2
2
𝜎2
2
+ 2𝑤1𝑤2𝐶𝑜𝑣 𝑅1, 𝑅2
• OR
• 𝜎𝑝 = 𝑤1
2
𝜎1
2
+ 𝑤2
2
𝜎2
2
+ 2𝑤1𝑤2𝜌12𝜎1𝜎2
• Wherein , 𝐶𝑜𝑣 𝑅1, 𝑅2 = 𝜌12𝜎1 𝜎2

31. 5-31
Relationship between Return & Risk
Compute the expected return and standard deviation for a
portfolio comprising of two assets- Asset 1 with an annual
return of 7% and annualized risk of 12%;
Asset 2 has an annual return of 15% and annualized risk of
25%. s
If the correlation between the two assets is +1, 0.5, 0.2 and
-1.0.
(Assume different combinations of weights
(0%-100%, 10%-90%, 20%-80%, 30%-70%..etc) between
the two assets and plot the relationship on a graph)

32. 5-32
Relationship between Return & Risk
The figure shows the portfolio return for four correlation coefficients ranging
from -1 to +1 and weights ranging between two assets from 0% to 100%.
Portfolio risk becomes smaller with each successive decrease in the
correlation coefficient with the smallest risk when 𝜌12=-1.
(refer to excel for computations)

33. 5-33
Portfolio of Many Risky Assets
• If in the portfolio, the number of risky assets
are ‘N’, then expected return and standard
deviation of the portfolio can be written as:

34. 5-34
Avenues for Diversification
a. Diversify with asset classes or among countries
b. Diversify with index funds
c. Diversify by not owning your employer’s stock
d. Buy insurance for risky portfolios (e.g. buying put option or any commodity having
negative correlation with equity)
e. Evaluate each asset before adding to a portfolio : A general rule to evaluate whether a
new asset should be included to an existing portfolio is based on the following return-
risk trade-off relationship:
E(𝑅𝑛𝑒𝑤) = 𝑅𝑓 +
𝜎𝑛𝑒𝑤𝜌𝑛𝑒𝑤,𝑝
𝜎𝑝
∗ [E 𝑅𝑝 − 𝑅𝑓]
Where E(𝑅𝑛𝑒𝑤) 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡, 𝑅𝑓 is the return on the risk-free asset, σ is the standard
deviation, ρ is the correlation coefficient and the subscripts ‘new’ and ‘p’ refers to the new and existing
portfolio.
If the new asset’s risk-adjusted return benefits the portfolio, then the asset should be included. The
condition can be rewritten using the Sharpe ratio on both sides of the equation as:
𝐸 𝑅𝑛𝑒𝑤 − 𝑅𝑓
𝜎𝑛𝑒𝑤
>
𝐸 𝑅𝑝 −𝑅𝑓
𝜎𝑝
∗ 𝜌𝑛𝑒𝑤,𝑝

35. 5-35
Concept of Risk Aversion & Portfolio Selection
• Risk aversion is related to the behaviour of
individuals under uncertainty.
• Assume that an individual is offered two
alternatives: one where he will get $50 for
sure and other is a gamble with a 50%
chance that he gets $100 and 50% chance
that he gets nothing.
• The expected value in both cases is $50.
• What will an investor choose?

36. 5-36
Concept of Risk Aversion
• So, there are three possibilities in front of investor: to gamble, not to gamble
or be indifferent.
• If the investor chooses to gamble, then the investor is said to be risk loving
or risk seeking.
• If the investor is indifferent about the gamble or the guaranteed outcome,
then the investor may be risk neutral. Risk neutrality means that the investor
cares only about the return and not about risk, so higher return investments
are more desirable even if they come with higher risk.
• If the investor chooses the guaranteed outcome, he/she is said to be risk
averse because the investor does not want to take the chance of not getting
anything at all.
• In general, investors are likely to shy away from risky investments for a lower
but guaranteed return. A risk neutral investor would maximize return
irrespective of risk and a risk-seeking investor would maximize both risk and
return.

37. 5-37
Utility Theory & Indifference Curves
• Utility is a measure of relative satisfaction from consumption of
various goods and services or in the case of investments, the
satisfaction that an investor derives from different portfolios.
• Utility theory allows us to quantify the rankings of investment choices
using risk and return.
• Utility function is given by:
U= 𝐸 𝑟 −
1
2
𝐴𝜎2
Where ‘U’ is the utility of an investment, E(r) is the expected return, 𝜎2
is the variance of the investment and ‘A’ is the measure of risk aversion,
which is measured as the marginal reward that an investor requires to
accept additional risk.
Thus, ‘A’ is higher for more risk-averse individuals.

38. 5-38
Key Takeaways from Utility Function
a) Utility is unbounded on both sides. It can be highly positive or
highly negative.
b) Higher return contributes to higher utility.
c) Higher variance reduces the utility but the reduction in utility gets
amplified by the risk aversion coefficient.
d) Utility does not indicate or measure satisfaction. It can be useful
only in ranking various investments. For example, a portfolio with
utility of 4 is not necessarily two times better than a portfolio with a
utility of 2.
e) The risk aversion coefficient ‘A’ is greater than zero for a risk-
averse investor, zero for risk neutral investor and negative for risk
loving investor
f) A risk-free investor (𝜎2 =0) generates the same utility for all
individuals.

39. 5-39
Concept & Application of Indifference Curves(IC)
in Portfolio Management
• An indifference curve plots the combination of risk-return
pairs that an investor would accept to maintain a given
level of utility.
• IC are thus defined in terms of a trade-off between
expected rate of return and variance of the rate of return.
• Because an infinite number of combinations of risk and
return can generate the same utility for the same investor,
indifference curves are continuous at all points.
• By definition, all pints on any one of the three curves have
the same utility. An

40. 5-40
Indifference Curves for Risk Averse Investors
• The utility of risk-averse investor always increases as one moves from
northwest – higher return with lower risk.
• The IC are convex because of diminishing marginal utility of return.
• The upward-sloping convex IC has a slope coefficient closely related to the
risk aversion coefficient.
• The greater the slope, the higher is the risk aversion of the investor as a
greater increment in return is required to accept a given increase in risk

41. 5-41
Indifference Curves for Various Types of Investors
The most risk-averse investor has an indifference curve with the greatest slope
(in this case it is indifference curve labelled – ‘Ip’).
The risk-loving investor’s indifference curve, however, exhibits a negative slope,
implying that the risk-lover is happy to substitute risk for return
The indifference curves of risk-neutral investors are horizontal because the utility
is invariant with risk.

42. 5-42
Application of Utility Theory to Portfolio Selection
• The simplest application of utility theory and risk aversion is to a portfolio of two
assets, a risk-free asset and a risky asset.
• The risk-free asset has zero risk and a return of Rf.
• The risky asset has a risk of σi(>0) and an expected return of E(Ri).
• Further, E(Ri)>Rf
• Construct a portfolio of these two assets with expected return of E(Rp) and
standard deviation of σp. Further, give ‘w1’ weight to risk-free asset and (1-w1)
weight to risky asset.
• Thus, the E(Rp) = w1*Rf + (1-w1)*E(Ri)
• 𝜎2
𝑝 = 𝑤12∗
𝜎𝑓2
+ (1 − 𝑤1)2
∗ 𝜎𝑖2
+ 2 ∗ 𝑤1 ∗ 1 − 𝑤1 ∗ 𝜌𝑓𝑖 ∗ 𝜎𝑓 ∗ 𝜎𝑖
• As the σf =0 (i.e. std. deviation of risk-free asset is zero), the first and third term in
the above formula for variance are zero leaving only the second term.
• Hence, the std. deviation of the portfolio is given by σ𝑝 = (1-w1)*σi

43. 5-43
Application of Utility Theory to Portfolio Selection
Assuming only two assets are available in the economy and the risky asset
represents the market, the line in above exhibit is called the ‘CAPITAL
ALLOCATION LINE (CAL)’.
The CAL represents the portfolios available to an investor.
CAL has an intercept of Rf and a slope of
(𝐸 𝑅𝑖 −𝑅𝑓)
𝜎𝑖
, which is the additional;
required return for every increment in risk and is sometimes referred to as the
market price of risk.

44. 5-44
Indifference Curve & CAL

45. 5-45
Indifference Curve & CAL
• Overlaying each individual’s indifference curves on the CAL will provide us with the optimal portfolio
for that investor.
• Points under the CAL may be attainable but are not preferred by any investor because the investor
can get a higher return for the same risk by moving up to the allocation line.
• Points above the CAL are desirable but not attainable with available assets.
• In the exhibit, Curve1 is above the capital allocation line, Curve 2 is tangential to the line and Curve 3
intersects the line at two points.
• Curve 1 has the highest utility and Curve 3 has the lowest utility. Because Curve 1 lies completely
above the CAL, points on Curve 1 are not achievable with the available assets on the CAL.
• Point m is clearly superior to point n as the investor is able to earn more return at the same level of
risk.
• At point a and b, the IC curve intersects the CAL, but the investor can invest at point a or b to derive
the risk-return trade-off and utility associated with Curve 3.
• Thus, point ‘m’ and the utility associated with curve 2 is the best that the investor can do because
he/she cannot move to a higher utility IC.
• Optimal portfolio, essentially is the point of tangency between the CAL and the indifference
curve. Further, the optimal portfolio maximizes per unit of risk and gives maximum
satisfaction to the investor.

46. 5-46
Indifference Curve & CAL

47. 5-47
Indifference Curve & CAL
• The above exhibit shows two ICs for two different
investors: Suppose Kate has a risk aversion coefficient of
2 and John has a risk coefficient of 4.
• The IC for Kate is to the right of the IC for John because
Kate is less risk averse than Jane and accept a higher
amount of risk.
• Accordingly, their optimal portfolios are different: point K
is the optimal portfolio for Kate and Point J is the optimal
portfolio for John.
• Further, the slope of John’s curve is higher than Kate’s
suggesting that John needs greater incremental return as
compensation for accepting an additional amount of risk
compared with Kate.

48. 5-48
Efficient Frontier & Investor’s Optimal Portfolio
• If two assets are perfectly correlated, the risk-return opportunity set is
represented by a straight line connecting those two assets.
• If the two assets are not perfectly correlated, the portfolio’s risk is less
than the weighted average risk of the components, and the portfolio
formed from the two assets bulges on the left as a curve with ρ less
than 1.
• The addition of new assets to this portfolio creates more and more
portfolios that are either a linear combination of the existing portfolio
and the new asset or a curvilinear combination, depending on the
correlation between the existing portfolio and the new asset.
• As the number of available assets increases, the number of possible
combinations increases rapidly.
• When all investible assets are considered, and there are hundreds
and thousands of them, we can construct an ‘opportunity set of
investments’.

49. 5-49
Efficient Frontier & Investor’s Optimal Portfolio
INVESTOR OPPORTUNTIY SET

50. 5-50
Efficient Frontier & Investor’s Optimal Portfolio
• The ‘investment opportunity set’ as shown in previous slide
shows the effect of adding a new asset class, such as
international assets.
• As long as the new asset is not perfectly correlated with the
existing asset class, the investment opportunity set will expand
out further to the northwest, providing a superior risk-return
trade-off.
• The investment opportunity set with international assets
dominates the opportunity set that includes only domestic
assets.
• Adding other asset classes will have the same impact on the
opportunity set. Thus, we should continue to add asset classes
until they do not further improve the risk-return trade-off.

51. 5-51
Minimum-Variance Portfolios

52. 5-52
Minimum-Variance Portfolios
• Consider points A, B and X in exhibit (shown in the previous slide) and assume
that they are on the same horizontal line by construction.
• Thus, the three points have the same expected return E(Rp) as do all other
points on the imaginary line connecting A,B and X.
• Given a choice, a, investor will choose the point with the minimum risk, which is
Point X. Point X, however, is unattainable because it does not lie within the
investment opportunity set.
• Thus, the minimum risk that we can attain for E(Rp) is at point A and point B and
all points to the right of point A are feasible but they have higher risk.
• Similarly, point C is the minimum variance point for the return earned at C and
points to the right of C have higher risk.
• In all cases, the ‘minimum variance portfolio’ is the one that lies on the solid
curve and the entire collection of these minimum-variance portfolios is referred to
as the ‘minimum-variance frontier’.
• The minimum variance frontier defines the smaller set of portfolios in which
investors would want to invest.

53. 5-53
Global Minimum-Variance Portfolio
• The left-most point on the minimum-
variance frontier is the portfolio with the
minimum variance among all portfolios of
risky assets and is referred to as the
‘global minimum-variance portfolio’.
• An investor cannot hold a portfolio
consisting of risky assets that has less risk
than that of the global minimum-variance
portfolio.

54. 5-54
Minimum-Variance Portfolio: Markowitz Efficient Frontier
• Consider point A and C on the minimum-variance frontier, both of them have the
same risk.
• Given a choice, an investor will choose Portfolio A because it has a higher
return.
• This applies to all the points on the minimum-variance frontier that lie below the
global minimum-variance portfolio.
• The curve that lies below and to the right of the global minimum-variance
portfolio is referred to as the ‘Markowitz Efficient Frontier’ because it contains all
portfolios of risky assets that rational, risk-averse investors will choose.
• As we move right from the global minimum-variance portfolio, there is an
increase in risk with a concurrent increase in return.
• The increase in return with every unit increase in risk, however, keeps
decreasing as we move from left to the right because the slope continues to
decrease.
• Thus, investors obtain decreasing increase in returns as they assume more risk.

55. 5-55
Capital Allocation Line & Optimal Risky Portfolios

56. 5-56
Capital Allocation Line & Optimal Risky Portfolios
• All portfolios on the efficient frontier are candidates for being combined with
the risk-free asset. Two combinations are shown in the exhibit and presented
by point P and A.
• Comparing CAL –A and CAL-P, reveals that there is a point on CAL(P) with a
higher return and same risk for each point on CAL(A).
• Thus, the portfolios on CAL(P) dominate the portfolios on CAL(A).
• Lets compare point X and Y, it is quite clear that pint X is on the efficient
frontier and has the highest return of all risky portfolios for its risk. However,
point Y on CAL-P is achievable by leveraging portfolio P and it lies above
point X and has the same risk but higher return.
• CAL-P dominates both CAL-A and Markowitz efficient frontier of risky assets.
• Thus, CAL-P is the optimal capital allocation line and Portfolio P is the optimal
risky portfolio.

57. 5-57
Optimal Investor Portfolio

58. 5-58
Optimal Investor Portfolio
• The location of an optimal investor portfolio depends on the investor’s
risk preferences.
• Moving from the risk-free asset along the capital allocation line, we
encounter investors who are willing to accept more risk.
• At point P, the investor is 100% invested in the optimal risky portfolio.
• Beyond point P, the investor accepts even more risk by borrowing
money and investing in the optimal risky portfolio.
• Portfolio P is the optimal risky portfolio that is selected without regard
to investor preferences.
• To identify the optimal portfolio, we overlay the indifference curve on
the capital allocation line and find that point C on CAL-P is the
optimal portfolio.

59. 5-59
Two-Fund Separation Theorem
• The two-fund separation theorem states that all investors regardless of taste,
risk preferences and initial wealth will hold a combination of two portfolios or
funds: a risk-free asset and an optimal portfolio of risky assets.
• The separation theorem allows us to divide an investor’s investment problem
into two distinct steps: the investment decision and the financing decision.
• In the first step, the investor identifies the optimal risky portfolio from
numerous risky portfolios without considering the investor’s preferences.
Further, the CAL connects the optimal risky portfolio and the risk-free assets.
• In the second step, depending on each investor’s risk preference, using
indifference curves, determines the investor’s allocation to the risk-free asset
(lending) and to the optimal risky portfolio.
• Portfolios beyond the optimal risky portfolio are obtained by borrowing at the
risk-free rate.