Module 1

Investment Policy and Modern Portfolio Theory

0.660.520.680.590.931.19

Portfolio construction• Purpose: maximization of wealth by reaching a heuristic Reward-to-

risk

• How? Allocate, Select and Protect

• Illustration: realized and expected wealth?

Realized wealth = Expected wealth + Error

Heuristic Reward to risk = Allocation + Selection + protection

• It always starts with the Policy: Ask the right question! what risk? Thus, what allocation? Set the right allocation target in terms of objectives, constraints and

weight range monitoring

Choose a Portfolio strategy: Passive or Active

• No matter what, an investment strategy is based on four decisionsWhat asset classes to consider for investment What normal or policy weights to assign to each eligible class The allowable allocation ranges based on policy weights What specific securities to purchase for the portfolio

• Most (85% to 95%) of the overall investment return is due to the first two decisions, not the selection of individual investments

Asset allocation Security Selection

Active

(for pros)

Market

timing

Stock/Bond

picking

Passive

(for ind.)

Fixed

weights

Indexing

First, set the rules: the policy statement

• TOTAL RETURN= INCOME YIELD + CAPITAL GAIN YIELD

• Objectives – Think in terms of risk and return to find the “best” weights—i.e.,

• Capital preservation (high income, low capital gain) Low to moderate risk

• Balanced return (Balanced capital gains and income reinvestment)moderate to high risk

• Pure Capital appreciation (high capital gains, low to no income)High risk

• Constraints - liquidity, time horizon, tax factors, legal and regulatory constraints, and unique needs and preferences

• Management - Define an allowable allocation ranges based on policy weights

• Selection - Define guideline to pick securities to purchase for the portfolio (optional)

Examples of Investment Styles

Objectives Age/Risk MatrixRisk tolerance/

Time Horizon

0-5years (C/B/S)

6-10

(C/B/S)

11+

(C/B/S)

Higher 10/30/60 0/20/80 0/0/100

Moderate 20/40/40 10/40/50 10/30/60

Lower 50/40/10 30/40/30 10/50/40

C stands for CASH—i.e. money market securities

B stands for Bonds—i.e. corporate, municipal or treasury securities

S stands for Stocks—i.e. value, growth, international equity securities

Color code:• Capital preservation • Balanced return• Capital appreciation

YOUR TURN!Mr. Bob is 70 years of age, is in excellent health pursues a simple but active lifestyle, and has no children. He has interest in a private company for $90 million and has decided that a medical research foundation will receive half the proceeds now; it will also be the primary beneficiary of his estate upon his death. Mr. Bob is committed to the foundation ‘s well-being because he believes strongly that , through it, a cure will be found for the disease that killed his wife. He now realizes that an appropriate investment policy and asset allocation are required if his goals are to be met through investment of his considerable assets. Currently the following assets are available for building an appropriate portfolio:

$45 million Cash (from the sale of the private company interest, net of $45 million gift to the foundation)$10 million stocks and bonds ($5 million each)$9 million warehouse property not fully leased)$1 Million Bob residence

Build a policy statement for Mr. Bob!

Objectives (return)• Large liquid wealth from selling interest in the private

company• Income from leasing warehouse• Not burdened by large or specific needs for current income

nor liquidity.He has enough spendable income.• He will leave his estate to a Tax-exempted foundation• He has already offered a large gift to the foundation Thus, an inflation-adjusted enhancement of the capital

base for the benefit of the foundation will the primary minimum return goal.

• He is in the highest tax bracket (not mentioned but apparent)

Tax minimization should be a collateral goal.

Objectives (risk)

• Unmarried, Childless, 70 years old but in good health

Still a long actuarial life (10+), thus long term return goal.

• Likely free of debt (not mentioned, but neither the opposite)• Not skilled in the management of a large portfolio• Yet, not a complete novice since he owned stocks and bonds

prior to his wife’s death.• His heir—the foundation—has already received a large

asset base.

Long term return goal with a portfolio bearing above average risk.

Constraints

• Time--Two things: (1) long actuarial life and (2) beneficiary of his estate—the foundation– has a virtually perpetual life

• Taxes: highest tax brackets, investment should take this into consideration: tax-sheltered investments.

• Unique circumstances: Large asset base, a foundation as a unique recipient some freedom in the building of the portfolio

Adapted Strategy• Majority in stocks (shield against inflation, above average

risk tolerance, and no real income or liquidity needs)• He already has 15% in real estate (house + warehouse) no

more needed, diversification effect achieved.• Additional freedom: Non-US stocks additional

diversification Target 75% equity (including Real Estate)• Fixed Income used to minimize income taxes—i.e.,

municipal and treasury securities. No need to look for YIELD nor downgrade the quality of the issues used.

• Additional freedom: Non-US fixed-income additional diversification effect.

Target 25% in fixed income

Proposed Allocation

Current Proposed Range

Cash / Money Market 70% 0% 0-5

US Stocks--LC 30% 30-40

US Stocks—SC 15% 15-25

Non US Stocks 15% 15-25

Total 7.5%? 60% 60-80

Real Estate 15% 15% 10-15

US Fixed Income 15% 10-20

Non-US Fixed Income 10% 5-15

Total Fixed Income 7.5%? 25% 15-35

In sum, the Importance of Asset Allocation

• An investment strategy is based on four decisions– What asset classes to consider for investment

– What normal or policy weights to assign to each eligible class

– The allowable allocation ranges based on policy weights

– What specific securities to purchase for the portfolio

• Most (85% to 95%) of the overall investment return is due to the first two decisions, not the selection of individual investments

• Summary:– Policy statement determines types of assets to include in portfolio

– Asset allocation determines portfolio return more than stock selection

– Over long time periods sizable allocation to equity will improve results

– Risk of a strategy depends on the investor’s goals and time horizon

What is “Investments”?

• Purpose: maximization of portfolio wealth through adequate Portfolio management

• Fair Reward-to-risk Ask the right question!

• Optimal portfolio management

= Allocation + Selection + Risk protection

Investment Vehicles

• Investments divided by asset class.1. Fixed-income investments (MM 27 & Bonds 49)2. Equity investments (stocks 140, COM.)3. Derivatives (Options and futures)4. Investment companies (MF 106, HF)5. Real estate6. Low-liquidity investments

Build a general culture on investments (Risk, Returns, Correlations)

• US asset classes

• Security markets size

• Government bond return

• Global equity returns

• Correlations

• Global Asset classes performance/correlation

• Investment companies performance

Alternative InvestmentsRisk and Return Characteristics

Computing Returns • The additional cents on the dollar invested…• R=(profit+additional cash flows)/initial investment• Over a period of time…average return• Average return=Σ(all returns)/nb of observations

• Why do returns matter?– $ does not mean much…alone

– Cross-comparison between markets

– Are “normally distributed”

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iknAverage

1

1

Example 1: Market Order• You buy a round lot (multiple of 100) of ABC stock at

$20. Brokerage fees are 3% on each transaction (3% for purchase and 3% for sale). You receive a year later $0.5 per share in dividends and sell the stock at $27. What is the rate of return on investment?

• Market Orders - buy or sell the stock at the best price at that time.

Solution 1R=Profit/investment

Return= Profit/initial investment = (Ending value - Beginning Value + Dividends -

Transaction costs on purchasing and selling) / (initial investment + transaction costs on purchasing)

• Beginning Value of Investment= $20n• Ending Value of Investment = $27nDividends = $0.5nTransaction Costs for purchase=3% x 20n=0.6nTransaction Costs for sale=3% x 27n=0.81n Profit = $27n - $20n + 0.5n-0.6n-0.81n = 6.09nInitial investment = 20n+0.6n = 20.6n

R= $6.09n/$20.6n = $6.09/$20.6= 29.56%

Example 2: Stop loss orders• Suppose you have 500 shares of ABC stock, bought at $50 and

priced at $60. You put a stop loss order at $55. Why would you do that? If the the price goes to $52, what would be your rate of return with and without the stop loss order?

• Special orders– Stop loss order: Implies that if the market price falls to or below a

specified price, the order becomes a market order and the stock will be sold at the prevailing price.

– Stop buy order: Used by short sellers to minimize losses if market price rises.

• Solution 2:• You are obviously satisfy with a profit of $5 per share.• With stop loss; R= (55-50)/50=10%• Without stop loss; R= (52-50)/50=4%

Example 3: Limit orders• Xyz stock is selling for $40. You have a limit buy order at $35. During the

year the stock goes to $30 then goes to $45. (1)What is R? (2)What would have R been with a simple market order? (3)What would R been is the limit buy order was at $25?

• Limit Orders - customer specifies highest purchase or lowest sell price. (Time specifications for order may vary: Instantaneous - “fill or kill”, part of a day, a full day, several days, a week, a month, or good until canceled “GTC”)- limit buy: specifies the highest price investor is willing to pay.- limit sell: specifies the lowest price investor is willing to accept.

• Solution 3: (1) When market declined to $30, your limit order was executed $35 (buy), then the price went to $45.

Rate of return = ($45 - $35)/$35 = 28.6%.•  (2)Assuming market order @ $40: Buy at $40, price goes to $45 Rate of return = ($45 - $40)/$40 = 12.5 %.

•  (3) Limit order @ $25: Since the market did not decline to $25 (lowest price was $30) the limit order was never executed.

Example 4: Margin Transactions

Buy 200 shares at $50 = $10,000 position

Borrow 50%, investment of $5,000

If price increases to $60, position– Value is $12,000– Less - $5,000 borrowed – Leaves $7,000 equity for a– $7,000/$12,000 = 58% equity position – Return on investment?– R=profit/initial investment=(12000-10000)/5000=40%

Example 5: Margin Transactions

Buy 200 shares at $50 = $10,000 position

Borrow 50%, investment of $5,000

If price decreases to $40, position– Value is $8,000– Less - $5,000 borrowed – Leaves $3,000 equity for a– $3,000/$8,000 = 37.5% equity position– Return on investment?– R=profit/initial investment=(8000-10000)/5000=-40%

Example 6: Margin Transactions• In the previous example, how far can the stock price

fall, before you receive a margin call? Assume a maintenance margin of 25%

• A call occurs when the proportion of equity = minimum maintenance margin,i.e.

• 25%=(200P-5000)/200P

• So P=5000/(200-25% x 200)= $33.33

Example 7: margin transactions

• You buy a round lot (multiple of 100) of ABC stock at $20 on 55% margin. The broker charges 10% on the borrowed money Brokerage fees are 3% on each transaction (3% for purchase and 3% for sale). You receive a year later $0.5 per share in dividends and sell the stock at $27. What is the rate of return on investment?

Solution 7R=Profit/investment

Return= Profit/initial investment = (Ending value - Beginning Value + Dividends - Transaction

costs on purchasing and selling - interests paid on borrowed money) / (initial investment + transaction costs on purchasing)

• Beginning Value of Investment= $20n• Ending Value of Investment = $27nDividends = $0.5nTransaction Costs for purchase=3% x 20n=0.6nTransaction Costs for sale=3% x 27n=0.81n Interests on amount borrowed: 10% x 45% x 20n =0.9nProfit = $27n - $20n + 0.5n-0.6n-0.81n-0.9n = 5.19nInitial investment = 55% x 20n+0.6n = 11.6n

R= $5.19n/$11.6n = $5.19/$11.6= 44.74%

Short sale example 8

• You sell short 200 shares of ABC, which is priced at $120. The margin requirement is 40%. Commissions on sale are $113. During the year, dividends of $2.9 are paid. At the end of the year you repurchase the stock at $90 (you close your position!) and you are charged $109 plus 10% on the money borrowed.

• What is you return on investment?

Solution 8 R=Profit/investment

*Profit on a Short Sale = Beginning Value - Ending Value- Dividends - Transaction Costs - Interest

• Beginning Value of Investment= 200 x $120 shares= $24,000(which is sold under a short sale arrangement)

•   Ending Value of Investment = 200 x $90 = $18,000 (Cost of closing out position)

Dividends = $2.9 x 200 shares = $580Transaction Costs = $113 + $109 = $222Interest = .1 x (.6 x 24000) = $1,440Profit = $24,000 - $18,000 - $580 - $222 - $1440 = $3,758Your investment = margin requirement + commission= (.40 x $24,000) + $113= $9600 + $113= $9,713

R= $3,758/$9,713 = 38.69%

Example 9: Computation of the Expected Return for Risky Assets

0.20 0.10 0.02000.30 0.11 0.03300.30 0.12 0.03600.20 0.13 0.0260

E(Rpor i) = 0.1150

Expected Portfolio

Return (Pi X Ri)

Expected Security

Return (Ri)

Probability (Pi)

iasset for return of rate expected the )E(R

iasset in portfolio theofpercent theW

:where

RP)E(R

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i

1ipor

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iii

Risk• We need to think in terms of estimates in an uncertain

world: Estimate=average return +/- some volatility

• Uncertainty or volatility of returns• Standard deviation of returns

• Measured in %• What does it mean?

2

1

)(1

1tan

vartan

averagekn

DeviationdardS

ianceDeviationdardS

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Example 10: Risk Computation of Monthly Rates of Return

Closing ClosingDate Price Dividend Return (%) Price Dividend Return (%)

Dec.00 60.938 45.688 Jan.01 58.000 -4.82% 48.200 5.50%Feb.01 53.030 -8.57% 42.500 -11.83%Mar.01 45.160 0.18 -14.50% 43.100 0.04 1.51%Apr.01 46.190 2.28% 47.100 9.28%May.01 47.400 2.62% 49.290 4.65%Jun.01 45.000 0.18 -4.68% 47.240 0.04 -4.08%Jul.01 44.600 -0.89% 50.370 6.63%

Aug.01 48.670 9.13% 45.950 0.04 -8.70%Sep.01 46.850 0.18 -3.37% 38.370 -16.50%Oct.01 47.880 2.20% 38.230 -0.36%Nov.01 46.960 0.18 -1.55% 46.650 0.05 22.16%Dec.01 47.150 0.40% 51.010 9.35%

E(RCoca-Cola)= -1.81% E(Rhome Depot)=E(RExxon)= 1.47%

Stdev(Coca-Cola)= 6.06% Stdev(Home Depot)= 10.62%

Variance (Standard Deviation) of Returns for an Individual Investment

n

i 1i

2ii

2 P)]E(R-R[)( Variance

where Pi is the probability of the possible rate of return, Ri

Standard deviation is the square root of the varianceVariance is a measure of the variation of possible rates

of return Ri, from the expected rate of return [E(Ri)]

Example 11: Variance (Standard Deviation) of Returns for an Individual Investment

Possible Rate Expected

of Return (Ri) Return E(Ri) Ri - E(Ri) [Ri - E(Ri)]2 Pi [Ri - E(Ri)]

2Pi

0.08 0.11 0.03 0.0009 0.25 0.0002250.10 0.11 0.01 0.0001 0.25 0.0000250.12 0.11 0.01 0.0001 0.25 0.0000250.14 0.11 0.03 0.0009 0.25 0.000225

0.000500

Variance ( 2) = .00050

Standard Deviation ( ) = .02236

Example 12:

• What is the probability for long-term government bonds to return more than 0%?

• Z=(5.6%-0)/9.2%=0.61P=72.9%

• What is the probability to make more than 10% with small caps?

• Z=(17.7%-10%)/33.9%=0.23P=59.1%

Risk and Return• How to compare assets?• Coefficient of variation = measure of relative

risk• CV =Total risk/return

CS 1.56SCS 1.91CB 1.41TB 1.64Rf 0.84

• Which one do you pick?• What is the problem here?

Covariance of Returns

• A measure of the degree to which two variables “move together” relative to their individual mean values over time

For two assets, i and j, the covariance of rates of return is defined as:

Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}

Covariance and Correlation

Correlation coefficient varies from -1 to +1

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ij

R ofdeviation standard the

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returns oft coefficienn correlatio ther

:where

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j

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•The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations

Portfolio effect• Portfolio Return is the weighted average return of each asset in the

portfolio

• Portfolio Risk is not the weighted average risk of each asset in the portfolio. Portfolio risk has to do with each asset’s weight and risk, but also the degree to which they move together (corr)

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Mathematical Explanation

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Summary Portfolio effect

Portfolio return (RP) =

Average return of all securities

Portfolio risk (σP) ≈

Average risk of all securities

Minus

the propensity of those securities to be unrelated (…returnwise!)

)W( ii

)corrWW( j,ijiji

)RW( ii

Portfolio risk and return…in EnglishPortfolio return=

(weighted) average assets’ return

Portfolio risk =(weighted) average assets’ risk

- (weighted) average assets prices’ propensity to move in opposite direction

OrPortfolio risk =

(weighted) average assets’ risk- Benefits from diversification

Combining Stocks with Different Returns and Risk

• Assets may differ in expected rates of return and individual standard deviations

• Negative correlation reduces portfolio risk

• Combining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal

Example 13:ii E( )E(R Asset

1 .10 .07

2 .20 .1

Case W1 W2 E(Ri) E( port) E( port) E( port) E( port) E( port)

Rho--> 1 0.5 0 -0.5 -1f 0.00 1.00 0.20 0.1000 0.1000 0.1000 0.1000 0.1000g 0.20 0.80 0.18 0.0940 0.0878 0.0812 0.0740 0.0660h 0.40 0.60 0.16 0.0880 0.0779 0.0662 0.0520 0.0320i 0.50 0.50 0.15 0.0850 0.0740 0.0610 0.0444 0.0150j 0.60 0.40 0.14 0.0820 0.0710 0.0580 0.0410 0.0020k 0.80 0.20 0.12 0.0760 0.0682 0.0595 0.0492 0.0360l 1.00 0.00 0.10 0.0700 0.0700 0.0700 0.0700 0.0700

Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = +1.00

1

2With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either single asset

Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

f

gh

ij

k1

2With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset

Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = +0.50

f

gh

ij

k1

2With correlated assets it is possible to create a two asset portfolio between the first two curves

Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = -0.50

Rij = +0.50

f

gh

ij

k1

2

With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset

Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = -1.00

Rij = +0.50

f

gh

ij

k1

2

With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk

Rij = -0.50

Exhibit 7.13

Numerous Portfolio Combinations of Available Assets

Efficient Frontier for Alternative Portfolios

Efficient Frontier In Practice (all equity markets of the world: 1981-2001)

00.05

0.10.15

0.20.25

0.3

0 0.2 0.4 0.6

Standard Deviation

Ret

urn

MerrillLynch

LehmanBrothrs

NikkoSec.

DaiwaEur.

CreditAgr.

Robeco BankJ. Baer

UBS CICM

Equity 60 50 65 58 70 50 45 38 50Bond 30 25 35 37 25 40 43 54 50Cash 10 20 0 5 5 10 12 8 0

9 different Institutional efficient Benchmarks

Asset Allocation and cultural Differences

• Mindset, Social, political, and tax environments

• U.S. institutional investors average 45% allocation in equities

• In the United Kingdom, equities make up 72% of assets

• In Germany, equities are 11%

• In Japan, equities are 24% of assets

R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Cash

Bond

Equity

Conclusion: What is the use of an efficient set?

• Goal: find an optimal mix (weight) so that the ratio of compensation for risk to risk (or reward to risk) is optimal for your level of risk tolerance.

• Your inputs: Expected returns, standard deviations and correlations (for each asset class)

• Your output: Optimal weight in each asset class (how much should you put in each asst class?)

Can we do better than the efficient set?

• Imagine two portfolio: (1) a risky “best of the best” portfolio with an expected return of Rm and a standard deviation of σm and (2) a riskless portfolio of t-bills with an expected of Rf and a standard deviation close to zero.

• You allocate Wrf in the riskless portfolio and (1-Wrf) in the risky (best of the best portfolio)

• The standard deviation and expected return of this portfolio shall be:

σp=(1-Wrf) x σm or Wrf=1- σp/σm, then

Rp=Wrf x Rf + (1-Wrf) x Rp replace Wrf by 1- σp/σm

RP= Rf +(Rm –Rf) /σm x σpCapital Market Line (CML)

Rp= intercept + slope x σp

What does it mean?

It means that…

• We know how to get the composition of the “best-of-the-best” portfolio (M) It has the highest

reward to risk –i.e., (Rm –Rf)/σm

• Then, we know how to get Rm and σm

• Finally, for the risk we are willing to take (indifference curve policy statement), we can find our optimal asset allocation by mixing the best of the best portfolio with cash!

• Cool (I mean sweeeeet) huh?• Application: efficient frontier analysis

Example 14

• Describe step by step how to build an efficient set and choose a portfolio that fits your risk tolerance.

The selection process: Risk and Diversification

Return = expected + unexpected

Risk (return)= 0 + market risk + business risk

The trick: if you hold many securities, the particularities of each security becomes irrelevant…thus, in a well diversified portfolio business-specific risk is irrelevant!

Risk and Return

• The higher the risk, the greater the expected return.

• Ri=Real rate +Inflation premium + Risk premium

• Ri=risk free rate +compensation for risk

• Compensation for risk=risk premium=compensation for a high standard deviation…

Risk that matters…

• If only market risk matters, then the risk premium of a security should be related (somehow) to the market risk premium!

• Let’s assume that those risk premiums are proportional:

security risk premium=β x market risk premium • This β is a multiplier which has to do with the relative

risk premium of a security to the market risk premium…it is a relative Market (systematic) Risk

SML• Ri=RF + RRP, then…

– Security risk premium = (Ri- RF)

– Market risk premium = (Rm- RF)

• If security risk premium=β x market risk premium

• Then, (Ri- RF) =β x (Rm- RF)

• That is,

Ri = RF +β x (Rm - RF)

This is also known as the SML (market equilibrium), a component of the CAPM

• As a result, any security’s return can calculated using β, RRF, and Rm

Graph of SML

•What if the observed returns are different from the theoretical returns?

•The Alpha-strategy consists of finding securities with abnormal excess return.

Example 15: SML Questions

• What is the market “relative” risk (β)?

• What does a β of 2 mean?

• What does a β of –1 mean?

• How do we get β?

• What is the β of a portfolio?

1. Beta coefficients are not stable for individual securities.

2. Performance evaluation depends upon the choice of the market proxy.

3. T-bills are not exactly risk-free

4. Unpleasantries have been neglected (taxes and transaction cost)

Problems With SML

Example 16: Questions

• What is the difference between the CML and SML? Why are the measures of risk different?

RP= Rf +(Rm –Rf) /σm x σp CML (allocation)

Ri= Rf +(Rm-Rf) x σi/ σm x ρi,m SML (selection)

Ri= Rf +(Rm-Rf) x βi,m

Example 17: what is the separation theorem?

How is the concept of leverage included in the CML?

Wrf>0 Wrf<0

Wrf=0

Where, Wrf=1-(P/M)

Example 18: What is the alpha strategy?

• Is it possible to find an asset which is above the CML? Then how can we use the SML to select underpriced securities?

• According to the SML:

Ri-Rf = 0 + β x (Rm-Rf)

In a regression format: (Ri-Rf)= α + β x (Rm-Rf) + εThen (alpha strategy):

if α is not significantly different from 0, security is fairly priced

if α is significantly greater 0, security is underpriced

if α is significantly smaller than 0, security is overpriced

Alpha Strategy

x A x B

α

Alpha-strategy– SML: (Ri – Rf) = alpha + beta x (Rm-Rf) + ε

• Example: EXTR

Alpha Beta R2

EXTR

(t-stat)

0.019

(2.28)

1.47

(4.57)

0.25

Example 19: A simple illustration

Assume: RFR = 6%

RM = 12%

Stock Beta

A 0.70B 1.00C 1.15D 1.40E -0.30

RFR)-(RRFR)E(R Mi i

E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%

E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%

E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%

E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%

E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%

Comparison of Required Rate of Return to Estimated Rate of Return

Stock (Pi) Expected Price (Pt+1) (Dt+1) of Return (Percent)

A 25 27 0.50 10.0 %B 40 42 0.50 6.2C 33 39 1.00 21.2D 64 65 1.10 3.3E 50 54 0.00 8.0

Current Price Expected Dividend Expected Future Rate

Stock Beta E(Ri) Estimated Return Minus E(Ri) Evaluation

A 0.70 10.2% 10.0 -0.2 Properly ValuedB 1.00 12.0% 6.2 -5.8 OvervaluedC 1.15 12.9% 21.2 8.3 UndervaluedD 1.40 14.4% 3.3 -11.1 OvervaluedE -0.30 4.2% 8.0 3.8 Undervalued

Required Return Estimated Return

Plot of Estimated Returnson SML Graph

)E(R i

Beta0.1

mRSML

0 .20 .40 .60 .80 1.20 1.40 1.60 1.80-.40 -.20

.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02

AB

C

D

E

Let’s conclude and summarize now…• Develop an investment policy statement

– Identify investment needs, risk tolerance, and familiarity with capital markets

– Identify objectives and constraints– Investment plans are enhanced by accurate formulation of a policy

statement• ALLOCATION: determine the market/sector weights

– Asset allocation determines long-run returns and risk, which success depends on construction of the policy statement

– (1) EFFICIENT FRONTIER and (2) CML– CML ≈ EFFICIENT FRONTIER when T-Bill is included in the

efficient set• SELECTION: determine undervalued securities

– Actual (observed of predicted) Return Vs. SML (fair) return– Alpha Analysis: Is the SML significantly violated?– Optimal allocation between selected securities with the efficient

frontier

Example 20: CASE• You gather the following information about two stocks A

and B, the SP500 and the treasury bill:

State Prob. E(Ra) E(Rb) E(SP500) Rtbill

Bad 25% 20% -20% 0% 2%

Average 40% 10% 20% 5% 2%

Good 35% -5% 40% 10% 2%Covariance A B SP500 Tbill

A 0.009619

B -0.02133 0.0531

SP500 -0.0037374 0.00865 0.0014749

Tbill 0 0 0 0

Example 16: Continued1.What is the probability to break-even if you

invest in A?

Find Z=(mean-X)/standard deviation X=0%; then you need the expected return and the standard deviation of A

E(R)= 25% x 20%+40%x10%+35%x(-5%)=7.25%

(A)=(0.009619)1/2=9.8%

Z=7.25%/9.8%=0.74

P(0.74)=77% chance

Example 20: Continued2. What is diversification? Illustrate using a portfolio consisting of A

and B.• Refer to book and slides for the first part. For the second part use a

three-case scenario as in example 6, i.e.,

E(R) COV(A,B)

A 7.25% 9.8% -0.02133

B 17% 23.04%

Case W1 W2 E(Ri) E(port)

1 0.00 1.00 17% 23.04%2 0.50 0.50 12.125% 7.08%3 1.00 0.00 7.25% 9.80%

0.00

0.05

0.10

0.15

0.20

0.00 0.05 0.10 0.15 0.20 0.25

A

B

Example 20: Continued3. In the previous question, what would the allocation to A and B if you

chose the minimum risk portfolio?

If Wa=W then Wb=1-W and the variance of the portfolio is

%4.29

%6.70

,

%6.7002133.020531.0009619.0

02133.00531.0

),(2

),(

),(4),(22220dW

d

is that ,0dW

d means variance

),(2),(22

),()1(2)1(

),(2

22

2

2222p

2p

22222222

22222

22222

B

A

BA

B

BBA

BBBAp

BAp

BABBAAp

W

W

Thus

BACOV

BACOVW

BAWCOVBACOVWW

Minimum

BACOVWBAWCOVWWW

BACOVWWWW

BACOVWWWW

Example 20: Continued4.Which stock would you consider as an addition to a portfolio made of the

SP500? Which stock would you consider for stand-alone portfolio?

Stock to consider as an addition to a portfolio made of the SP500Get the alpha of each stock:

First get the theoretical (CAPM) return, then subtract it to the expected return. To get CAPM return:Ra=Rf+BETA(A) x (Rm-Rf)Rb=Rf+BETA(B) x (Rm-Rf)Rf is the treasury bill return=2%Rm is the sp500 return=5.5% (it is the weighted average return for sp500)M=(0.0014749)1/2=3.84%BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85ThenRa=Rf+BETA(A) x (Rm-Rf)=2%-2.53*3.5%=-6.86%Rb=Rf+BETA(B) x (Rm-Rf)=2%+5.85*3.5%=22.48%ALPHA(A)=7.25%-(-6.86%)=14.11%UndervaluedALPHA(B)=17%-22.48%= -5.48%OvervaluedThen you would A to a well-diversified portfolio like A

Example 20: Continued

Stock to consider for stand-alone portfolio get the Coefficient of Variation• Calculate the Coefficient of Variation:

• CV(A)=9.8%/7.25%=1.35

• CV(B)=23.04%/17%=1.35

• There are basically equivalent in terms of reward to risk in a stand-alone portfolio

5.What is the difference between the CML and SML?Look at slides and book

Example 20: Continued

6. How much (proportions) would you invest in A and B in order to get a portfolio as risky as the market?

The market has a beta of 1; Solve a system of two equations:Wa x BETA(A)+Wb x BETA(B)=1Wa+Wb=100%Then, Wb=[1-BETA(A)]/[BETA(B)-BETA(A)]BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85Wb=42%So, Wa=58%

Example 20: Continued

7.You have created your AB portfolio, then you decide to sell A and invest the proceed in T-bills. What the new portfolio Expected return, standard deviation and beta?

Wb=42%; Wa=58% sell A, buy TB Wrf=58%BETA(new portfolio)=Wb x BETA(B) +Wrf x BETA(Rf)And of course BETA(Rf)=0; Rf = 0So, BETA(new portfolio)=.42 x 5.85=2.46E(new portfolio)=.42 x 17% + .58 x 2%=8.3%(new portfolio)=.42 x 23.04%=9.68% (from the portfolio

risk equation with 2 assets, it simplifies a lot as Rf = 0)

Example 20: Continued8. What is the separation Theorem?

Answer in Book