The Sharpe model provides a simpler approach to portfolio optimization compared to the Markowitz model. It assumes the return of individual securities is linearly related to a single market index. This allows estimation of systematic and unsystematic risk for individual stocks based on their beta coefficient. An optimal portfolio is constructed by selecting stocks with the highest excess returns over the risk-free rate relative to their beta, up to the cutoff point where this ratio begins declining. The percentage invested in each stock is based on its beta and unsystematic risk. This results in a portfolio with the highest expected return for a given level of risk.
Many investors mistakenly base the success of their portfolios on returns alone. Few consider the risk that they took to achieve those returns. Since the 1960s, investors have known how to quantify and measure risk with the variability of returns, but no single measure actually looked at both risk and return together. Today, we have three sets of performance measurement tools to assist us with our portfolio evaluations. The Treynor, Sharpe and Jensen ratios combine risk and return performance into a single value, but each is slightly different. Which one is best for you? Why should you care? Let's find out.
Portfolio performance measures should be a key aspect of the investment decision process. These tools provide the necessary information for investors to assess how effectively their money has been invested (or may be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot possibly see the whole investment picture, which may inadvertently lead to clouded investment decisions.
Many investors mistakenly base the success of their portfolios on returns alone. Few consider the risk that they took to achieve those returns. Since the 1960s, investors have known how to quantify and measure risk with the variability of returns, but no single measure actually looked at both risk and return together. Today, we have three sets of performance measurement tools to assist us with our portfolio evaluations. The Treynor, Sharpe and Jensen ratios combine risk and return performance into a single value, but each is slightly different. Which one is best for you? Why should you care? Let's find out.
Portfolio performance measures should be a key aspect of the investment decision process. These tools provide the necessary information for investors to assess how effectively their money has been invested (or may be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot possibly see the whole investment picture, which may inadvertently lead to clouded investment decisions.
noorulhadi Lecturer at Govt College of Management Sciences, noorulhadi99@yahoo.com
i have prepared these slides and still using in mylectures, Reference: Portfolio management by S kevin and online sources
noorulhadi Lecturer at Govt College of Management Sciences, noorulhadi99@yahoo.com
i have prepared these slides and still using in mylectures, Reference: Portfolio management by S kevin and online sources
Capital Asset Pricing Model (CAPM) was introduced in 1964 as an extension of the Modern Portfolio Theory which seeks to explore the diverse ways by which investors can construct investment portfolios through means that can possibly minimize risk levels and at the same time ensure maximization of returns.
Asset Pricing and Portfolio Theory
I have presented a unique analysis which showcases the concepts of Aggregate & Aggregate lending and the numerical aspects of CAPM theory
1CHAPTER 6Risk, Return, and the Capital Asset Pricing Model.docxhyacinthshackley2629
1
CHAPTER 6
Risk, Return, and the Capital Asset Pricing Model
2
Topics in Chapter
Basic return concepts
Basic risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
1
Value = + + +
FCF1
FCF2
FCF∞
(1 + WACC)1
(1 + WACC)∞
(1 + WACC)2
Free cash flow
(FCF)
Market interest rates
Firm’s business risk
Market risk aversion
Firm’s debt/equity mix
Cost of debt
Cost of equity
Weighted average
cost of capital
(WACC)
Net operating
profit after taxes
Required investments
in operating capital
−
=
Determinants of Intrinsic Value:
The Cost of Equity
...
For value box in Ch 4 time value FM13.
4
What are investment returns?
Investment returns measure the financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
5
An investment costs $1,000 and is sold after 1 year for $1,100.
Dollar return:
Percentage return:
$ Received - $ Invested
$1,100 - $1,000 = $100.
$ Return/$ Invested
$100/$1,000 = 0.10 = 10%.
2
6
What is investment risk?
Typically, investment returns are not known with certainty.
Investment risk pertains to the probability of earning a return less than that expected.
The greater the chance of a return far below the expected return, the greater the risk.
2
7
Probability Distribution: Which stock is riskier? Why?
8
Consider the Following
Investment AlternativesEcon.Prob.T-BillAltaRepoAm F.MPBust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0Avg. 0.40 8.0 20.0 0.0 7.0 15.0Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00
9
What is unique about the T-bill return?
The T-bill will return 8% regardless of the state of the economy.
Is the T-bill riskless? Explain.
5
10
Alta Inds. and Repo Men vs. the Economy
Alta Inds. moves with the economy, so it is positively correlated with the economy. This is the typical situation.
Repo Men moves counter to the economy. Such negative correlation is unusual.
7
11
Calculate the expected rate of return on each alternative.
r = expected rate of return.
rAlta = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
^
^
n
∑
r =
^
i=1
riPi.
12
Alta has the highest rate of return. Does that make it best?^rAlta 17.4%Market15.0Am. Foam13.8T-bill 8.0Repo Men 1.7
13
What is the standard deviation
of returns for each alternative?
σ = Standard deviation
σ = √ Variance = √ σ2
n
∑
i=1
= √
(ri – r)2 Pi.
^
14
= [(-22 - 17.4)20.10 + (-2 - 17.4)20.20
+ (20 - 17.4)20.40 + (35 - 17.4)20.20
+ (50 - 17.4)20.10]1/2
= 20.0%.
Standard Deviation of Alta Industries
11
15
T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam = 18.
This is the fifth presentation for the University of New England Graduate School of Business course GSB711 Managerial Finance, offered by Dr Subba Reddy Yarram. This presentation examines risk, return and the Capital Asset Pricing Model (CAPM).
Risk and Return: Portfolio Theory and Assets Pricing ModelsPANKAJ PANDEY
Discuss the concepts of portfolio risk and return.
Determine the relationship between risk and return of portfolios.
Highlight the difference between systematic and unsystematic risks.
Examine the logic of portfolio theory .
Show the use of capital asset pricing model (CAPM) in the valuation of securities.
Explain the features and modus operandi of the arbitrage pricing theory (APT).
Investment returns measure financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
($) dollar terms.
(%) percentage terms.
Typically, investment returns are not known with certainty.
Investment risk pertains to the probability of earning a return less than expected.
Greater the chance of a return far below the expected return, greater the risk
2. Need for Sharpe Model
In Markowitz model a number of co-variances have
to be estimated.
If a financial institution buys 150 stocks, it has to
estimate 11,175 i.e., (N2 – N)/2 correlation
co-efficients.
Sharpe assumed that the return of a security is
linearly related to a single index like the market
index.
3. Single Index Model
Casual observation of the stock prices over a
period of time reveals that most of the stock
prices move with the market index.
When the Sensex increases, stock prices
also tend to increase and vice – versa.
This indicates that some underlying factors
affect the market index as well as the stock
prices.
4. Stock prices are related to the market index and this
relationship could be used to estimate the return of
stock.
Ri = αi + βi Rm + ei
where Ri — expected return on security i
αi — intercept of the straight line or alpha co-efficient
βi — slope of straight line or beta co-efficient
Rm — the rate of return on market index
ei — error term
5. Risk
Systematic risk = βi2 × variance of market index
= β i2 σ m 2
Unsystematic risk= Total variance – Systematic risk
ei2
= σi2 – Systematic risk
Thus the total risk= Systematic risk + Unsystematic risk
= β i2 σ m 2 + e i2
6. Portfolio Variance
2
N
2 N 2 2
σ 2 = ∑ x i β i ÷ σ m + ∑ x i e i
p
i =1
i =1
7. where
σ2p = variance of portfolio
σ2m = expected variance of market index
e2i= Unsystematic risk
xi = the portion of stock i in the portfolio
8. Example
The following details are given for x and y companies’
stocks and the Sensex for a period of one year.
Calculate the systematic and unsystematic risk for the
companies stock. If equal amount of money is allocated
for the stocks , then what would be the portfolio risk ?
X stock
Y stock
Sensex
Average return
0.15
0.25
0.06
Variance of return 6.30
5.86
2.25
Βeta
0.71
0.27
9. Company X
Systematic risk
= βi2 × variance of market index
= βi2 σm2 = ( 0.71)2 x 2.25 = 1.134
Unsystematic risk= Total variance – Systematic risk
ei2
= σi2 – Systematic risk = 6.3 – 1.134 =5.166
Total risk= Systematic risk + Unsystematic risk
= βi2 σm2 + ei2 = 1.134 + 5.166 = 6.3
10. Company Y
Systematic risk
= βi2 × variance of market index
= βi2 σm2 = ( 0.27)2 x 2.25 = 0.1640
Unsystematic risk= Total variance – Systematic risk
ei2
= σi2 – Systematic risk = 5.86 – 1.134 =5.166
12. Corner portfolio
The entry or exit of a new stock in the portfolio
generates a series of corner portfolio.
In an one stock portfolio, it itself is the corner
portfolio .
In a two stock portfolio, the minimum risk and the
lowest return would be the corner portfolio.
As the number of stocks increases in a portfolio, the
corner portfolio would be the one with lowest return
and risk combination.
14.
For each security αi and βi should be
estimated
Portfolio return is the weighted average of the
estimated return for each security in the
portfolio.
The weights are the respective stocks’
proportions in the portfolio.
16. A portfolio’s beta value is the weighted
average of the beta values of its component
stocks using relative share of them in the
portfolio as weights.
βp is the portfolio beta.
18.
The selection of any stock is directly related to its
excess return to beta ratio.
where Ri = the expected return on stock i
Rf = the return on a risk less asset
βi = Systematic risk
19. Optimal Portfolio
The steps for finding out the stocks to be
included in the optimal portfolio are as:
Find out the “excess return to beta” ratio for each
stock under consideration
Rank them from the highest to the lowest
Proceed to calculate Ci for all the stocks according
to the ranked order using the following formula
20. (R i − R f )βi
2
σ ei
i =1
N β2
2
i
1 +σ m ∑ 2
i =1 σ ei
2
m
N
σ ∑
Ci
21. σm2 = variance of the market index
σei2 = stock’s unsystematic risk
22. Cut-off point
The cumulated values of Ci start declining
after a particular Ci and that point is taken
as the cut-off point and that stock ratio is
the cut-off ratio C.
σ ei2
- Unsystematic risk
(Ri – Rf) / βi – Excess return
to Beta
23. Example
Data for finding out the optimal portfolio are given below
Security number Mean return Excess return Beta σ ei2
Ri
Ri – Rf
1
19
14
1.0 20
23
18
1.5
30
12
3
11
6
0.5 10
12
4
25
20
2.0
5
13
8
1.0
6
9
β
14
2
(Ri – Rf) / βi
7
4
0.5
40
20
50
14
10
8
8
9
1.5
30
6
The riskless rate of intrest is 5% and the market variance is 10.
Determine the cut – off point .
24.
C1 = (10 x .7)/ [ 1 + ( 10 x .05)] =4.67
C2 = (10 x 1.6)/ [ 1 + ( 10 x .125)] =7.11
C3 = (10 x 1.9)/ [ 1 + ( 10 x .15)] =7.6
C4 = (10 x 2.9)/ [ 1 + ( 10 x .25)] = 8.29* Cut-off point
C5 = (10 x 3.3)/ [ 1 + ( 10 x .3)] = 8.25
C6 = (10 x 3.34)/ [ 1 +( 10 x .305)] = 8.25
C7= (10 x 3.79)/ [ 1 + ( 10 x .38)] = 7.90
25.
The highest Ci value is taken as the cut-off
point i.e C*.
The stocks ranked above C* have high
excess returns to beta than the cut off Ci and
all the stocks ranked below C* have low
excess return to beta.
Here the cut off rate is 8.29.
Hence the first four securities are selected.
26.
If the number of stocks is larger there is no
need to calculate ci values for all the stocks
after the ranking has been done.
It can be calculated until the C* value is found
and after calculating for one or two stocks
below it, the calculations can be terminated.
27. Construction of the optimal portfolio
After determining the securities to be
selected, the portfolio manager should find
out how much should be invested in each
security.The percentage of funds to be
invested in each security can be estimated as
follows .
Zi = (βi / σ2ei ) x [ (Ri – Rf / βi) – C ]
29.
So the largest investment should be made in
security 1 ( 0.38%) and the smallest in
security 4 ( 0.12%).
The characteristics of a stock that make it
desirable can be determined before the
calculations of an optimal portfolio is begun.
The desirability of any stock is solely a
function of its excess return to beta ratio.