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Optimal Risky Portfolios- Asset Allocations

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Optimal Risky Portfolios- Asset Allocations

BKM Ch 7

- Idea
- from bank account to diversified portfolio
- principles are the same for any number of stocks

- A. bonds and stocks
- B. bills, bonds and stocks
- C. any number of risky assets

Bahattin Buyuksahin, JHU , Investment

- Market risk
- Systematic or nondiversifiable

- Firm-specific risk
- Diversifiable or nonsystematic

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- Portfolio risk depends on the correlation between the returns of the assets in the portfolio
- Covariance and the correlation coefficient provide a measure of the way returns two assets vary

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

= Variance of Security D

= Variance of Security E

= Covariance of returns for

Security D and Security E

Bahattin Buyuksahin, JHU , Investment

- Another way to express variance of the portfolio:

Bahattin Buyuksahin, JHU , Investment

Cov(rD,rE) = DEDE

D,E = Correlation coefficient of

returns

D = Standard deviation of

returns for Security D

E = Standard deviation of

returns for Security E

Bahattin Buyuksahin, JHU , Investment

Range of values for 1,2

+ 1.0 >r>-1.0

If r = 1.0, the securities would be perfectly positively correlated

If r = - 1.0, the securities would be perfectly negatively correlated

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

2p = w1212

+ w2212

+ w3232

+ 2w1w2

Cov(r1,r2)

Cov(r1,r3)

+ 2w1w3

+ 2w2w3

Cov(r2,r3)

Bahattin Buyuksahin, JHU , Investment

- Portfolio of 2 risky assets (cont’d)
- examples
- BKM7 Tables 7.1 & 7.3
- BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off)
- portfolio opportunity set (BKM7 Fig. 7.5)

- examples
- minimum variance portfolio
- choose wD such that portfolio variance is lowest
- optimization problem

- minimum variance portfolio has less risk
- than either component (i.e., asset)

- choose wD such that portfolio variance is lowest

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Standard deviation is smaller than that of either of the individual component assets

Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

Bahattin Buyuksahin, JHU , Investment

The relationship depends on the correlation coefficient

-1.0 << +1.0

The smaller the correlation, the greater the risk reduction potential

If r = +1.0, no risk reduction is possible

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Maximize the slope of the CAL for any possible portfolio, p

The objective function is the slope:

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- Finding the optimal risky portfolio: II. Formally
- Intuitively
- BKM7 Figs. 7.6 and 7.7
- improve the reward-to-variability ratio
- optimal risky portfolio tangency point (Fig. 7.8)

- Intuitively
- Formally:

Bahattin Buyuksahin, JHU , Investment

- formally (continued)

Bahattin Buyuksahin, JHU , Investment

- Example (BKM7 Fig. 7.8)
- 1. plot D, E, riskless
- 2. compute optimal risky portfolio weights
- wD = Num/Den = 0.4; wE = 1- wD = 0.6

- 3. given investor risk aversion (A=4), compute w*
- bottom line: 25.61% in bills; 29.76% in bonds (0.7439x0.4); rest in stocks

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- Security Selection
- First step is to determine the risk-return opportunities available
- All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

- Combining many risky assets & T-Bills
- basic idea remains unchanged
- 1. specify risk-return characteristics of securities
- find the efficient frontier (Markowitz)

- 2. find the optimal risk portfolio
- maximize reward-to-variability ratio

- 3. combine optimal risk portfolio & riskless asset
- capital allocation

- 1. specify risk-return characteristics of securities

- basic idea remains unchanged

Bahattin Buyuksahin, JHU , Investment

- finding the efficient frontier
- definition
- set of portfolios with highest return for given risk
- minimum-variance frontier

- take as given the risk-return characteristics of securities
- estimated from historical data or forecasts
- n securities ->n return + n(n-1) var. & cov.

- use an optimization program
- to compute the efficient frontier (Markowitz)
- subject to same constraints

- definition

Bahattin Buyuksahin, JHU , Investment

- Finding the efficient frontier (cont’d)
- optimization constraints
- portfolio weights sum up to 1
- no short sales, dividend yield, asset restrictions, …

- optimization constraints
- Individual assets vs. frontier portfolios
- BKM7 Fig. 7.10
- short sales -> not on the efficient frontier
- no short sales -> may be on the frontier
- example: highest return asset

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

We now search for the CAL with the highest reward-to-variability ratio

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- The separation property tells us that the portfolio choice problem may be separated into two independent tasks
- Determination of the optimal risky portfolio is purely technical
- Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- Remember:
- If we define the average variance and average covariance of the securities as:
- We can then express portfolio variance as:

Bahattin Buyuksahin, JHU , Investment

Loss: payout = $100,000

p = .001

No Loss: payout = 0

1 − p = .999

Bahattin Buyuksahin, JHU , Investment

Consider the following:

Bahattin Buyuksahin, JHU , Investment

- Consider the variance of the portfolio:
- It seems that selling more policies causes risk to fall
- Flaw is similar to the idea that long-term stock investment is less risky

Bahattin Buyuksahin, JHU , Investment

When we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:

Bahattin Buyuksahin, JHU , Investment

- What does explain the insurance business?
- Risk sharing or the distribution of a fixed amount of risk among many investors

Bahattin Buyuksahin, JHU , Investment

- Perfect hedges (portfolio of 2 risky assets)
- perfectly positively correlated risky assets
- requires short sales

- perfectly negatively correlated risky assets

- perfectly positively correlated risky assets

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

CHAPTER 8

Index Models

Bahattin Buyuksahin, JHU , Investment

- Idea
- the same factor(s) drive all security returns

- do not look for equilibrium relationship
- between a security’s expected return
- and risk or expected market returns

- between realized stock return
- and realized market return

Bahattin Buyuksahin, JHU , Investment

- Formally
- stock return
- = expected stock return
- + unexpected impact of common (market) factors
- + unexpected impact of firm-specific factors

- stock return

Bahattin Buyuksahin, JHU , Investment

- Factor model
- problem
- what is the factor?

- problem

- solution
- market portfolio proxy
- S&P 500, Value Line Index, etc.

Reduces the number of inputs for diversification

Easier for security analysts to specialize

Bahattin Buyuksahin, JHU , Investment

ßi = index of a securities’ particular return to the factor

m = Unanticipated movement related to security returns

ei = Assumption: a broad market index like the S&P 500 is the common factor.

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Regression Equation:

Expected return-beta relationship:

Bahattin Buyuksahin, JHU , Investment

- Risk and covariance:
- Total risk = Systematic risk + Firm-specific risk:
- Covariance = product of betas x market index risk:
- Correlation = product of correlations with the market index

Bahattin Buyuksahin, JHU , Investment

- Portfolio’s variance:
- Variance of the equally weighted portfolio of firm-specific components:
- When n gets large, becomes negligible

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Macroeconomic analysis is used to estimate the risk premium and risk of the market index

Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i )

Developed from security analysis

Bahattin Buyuksahin, JHU , Investment

- The market-driven expected return is conditional on information common to all securities
- Security-specific expected return forecasts are derived from various security-valuation models
- The alpha value distills the incremental risk premium attributable to private information

- Helps determine whether security is a good or bad buy

Bahattin Buyuksahin, JHU , Investment

- Risk premium on the S&P 500 portfolio
- Estimate of the SD of the S&P 500 portfolio
- n sets of estimates of
- Beta coefficient
- Stock residual variances
- Alpha values

Bahattin Buyuksahin, JHU , Investment

- Maximize the Sharpe ratio
- Expected return, SD, and Sharpe ratio:

Bahattin Buyuksahin, JHU , Investment

- Combination of:
- Active portfolio denoted by A
- Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M
- Modification of active portfolio position:
- When

Bahattin Buyuksahin, JHU , Investment

The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

- Beta books
- Merrill Lynch
- monthly, S&P 500

- Merrill Lynch
- Value Line
- weekly, NYSE

- regression analysis

Bahattin Buyuksahin, JHU , Investment

- Example (Merrill Lynch differences, Table 8.3)
- total (not excess) returns
- slopes are identical
- smallness

- total (not excess) returns
- percentage price changes
- dividends?
- S&P 500

- beta = (2/3) estimated beta + (1/3) . 1
- sampling errors, convergence of new firms

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment

Bahattin Buyuksahin, JHU , Investment