MPT AND ASSET ALLOCATION

I. STEPS OF ASSET ALLOCATIONII. MUTUAL FUND THEOREMIII. SHAPE OF EFFICIENT FRONTIER

FOR VARIOUSIV. COMPOSITION OF MINIMUM

VARIANCE PORTFOLIO OF 2 RISKY ASSETS:

V. COMPOSITION OF MINIMUM VARIANCE PORTFOLIO OF 2 RISKY ASSETS WHEN ρ=-1

VI. OPTIMAL MIX OF STOCKS AND BONDS WHEN RISKFREE ASSET IS NOT AVAILABLE

VII. OBTAINING HIGHER RETURNS USING STOCKS AND BONDS WHEN RISKFREE ASSET IS NOT AVAILABLE

VIII. ASSET ALLOCATION: MANY RISKY ASSETS AND 1 RISKFREE ASSET

IX. THE RISK-REWARD TRADEOFF WHEN BORROWING RATE EXCEEDS THE LENDING RATE

X. PSYCHOLOGICAL FACTORSXI. PORTFOLIO THEORY: SUMMARYXII. NOTATIONS USEDXIII. TYPES OF PROBABLE QUESTIONS

We firstly deal with the optimal mix of two risky assets. Then we deal with asset allocation with the optimal risky portfolio (of two risky assets) and one riskfree asset. Finally we discuss asset allocation with many risky assets.

I. STEPS OF ASSET ALLOCATION1) Analyze needs and preferences of investor.2) Identify types of assets to hold.3) Determine capital market expectations.4) Derive efficient portfolio frontier.5) Identify optimal mix of risky assets.6) Determine effeicient mix of risky and risk free assets.

1) ANALYSIS OF NEEDS AND PREFERENCESa) Provide for necessities of life.b) Provide for emergency needs.c) Provide for large expenditures.d) Provide for insurance.e) Provide for retirement.f) Speculative investments after meeting the above tangible goals.

2) IDENTIFYING TYPES OF ASSETS TO HOLD

Assume that investor’s needs and preferences require her to invest in stocks, bonds and cash.

3) CAPITAL MARKET EXPECTATIONS

Form expectations about future rates of return by:

a) Using historical performance of assets.

b) Analyzing prospect of the economy, industries, companies and instruments.

Suppose the capital market expectations for stocks, bonds and T-Bills are as follows:

State of Prob. Stocks Bonds T-BillsEconomyBoom .40 45% 4.5% 5%Normal .40 8% 5.5% 5%Recession .20 -6% 25% 5%

Find expected returns E(R), std. dev. (σ) and the correlation coefficent of stocks and bonds (ρsb).

Expected Return

E(r) = p1r1 + p2r2 + p3r3

E(rs) = .4(.45) + .4(.08) + .2(-.06) = .20 or 20% = ¯rs

E(rb) = .4(.045) + .4(.055) + .2(.25) = .09 or 9% = ¯rb

rf = .05 or 5%

Standard Deviation: σ = Variance

First find varianceσ 2 = p1 (r1 – r)2 + p2 (r1 – r)2 + p3 (r3 – r)2

Stocks:Variance = σ2 = .4(.45 - .2)2 + .4(.08 - .2)2 + .2(-.06 - .2)2

= 0.04428Standard deviation = σ = 0.210428 or, 21.04%

Bonds:Variance = σ2 = .4(.045 - .09)2 + .4(.055 - .09)2 +

.2(.25 - .09)2 = 0.00642Standard deviation = σ = 0.080125 or, 8.01%So, we have the following capital market expectations:

E(r) SDStocks: 20% 21.04%Bonds: 9 8.01T-Bills: 5 0

Both stocks and bonds are risky in isolation.

What can happen if we combine them to form a portfolio?

Two statistics can tell us about risk reduction potential: covariance and correlation coefficient.

Covariance:

Measures extent to which two variables move together.

Cov can be positive or negative.

If positive: The variables move up and down together.

If negative: The variables move inversely.

Cov(rs , rb) = p1 (rs1 – rs) (rb1 – rb) + p2 ( rs2 – rs)(rb2 – rb) +

p3 ( rs3 – rs) (rb3 – rb)

rs1 = return for stocks in the first scenario

rb1 = return for bonds in the first scenario

Cov(rs,rb) = .4(.45-.2)(.045-.09) + .4(.08-.2)(.955-.09)+

.2(-.06-.2)(.25-.09)

= .4(.25)(-.045) + .4(-.12)(-.035) + .2(-.26)(.16) = - 0.01114

Problem with covariance: It has infinite range: can range from + to - does not give us much information.

Statisticians have set bounds to the infinite range of the covariance as follows and that gives us correlation coefficient:

Correlation Coefficient = ρbs = Cov (rs , rb) = σs , σb

= -0.01114 = -0.6607 (0.210428)(0.080245)

Negative correlation implies: ___________________Range of correlation:ρ = +1: Two variables are perfectly positively correlated.If one goes up by 10%, the other also goes up by 10%.

ρ = -1: Perfectly negatively correlated.If one goes up by 10%, the other goes down by 10%.

ρ = 0: No correlation between two variables. Example?

Cov (rs, rb) = ρbs . σs . σb

4) DERIVING EFFICIENT PORTFOLIO FRONTIER

Wb Ws E(rp) σp

1.0 0 9% 8%.9 .1 10.1 6.8 .2 11.2 4.8.7 .3 12.3 5.6 .4 13.4 6.4.5 .5 14.5 8.4.4 .6 15.6 10.8.3 .7 16.7 13.3.2 .8 17.8 15.8.1 .9 18.9 18.40 1.0 20 21

E(rp) = (rs) + Wb(rb)

σp = Ws2σs

2 + Wb2σb

2 + 2(WsWb)Cov(rs,rb)

Investment Opportunity Set

02468

10121416182022

0 10 20 30

Sd

E(r

)

Find expected return of optimal portfolio O*, expected return E(r*) and its standard deviation σ*.

E(r*) = Ws* E(rs) + Wb

* E(rb)= 0.2904*0.20 + 0.7096*0.09= 0.1219 or 12.19%

Find standard deviation (σ*) of O* portfolio

σ* = Ws2σs

2 + Wb2σb

2 + 2(WsWb)Cov(rs,rb) σ * = .(290385)2(.044279985)+(.709615)2(.00641999)+

2(.290385)(.709615)(-.01114) = 0.0487401 or 4.87%

6) DETERMINING EFFICIENT MIX OF RISKY AND RISK FREE ASSET

6.1(a) What is the most efficient (least risky) way to get 9% from investing in stocks, bonds and T-Bills?

E(rc) = rf + y[E(r*) – rf].09 = .05 + y[.1219 - .05]y = 0.556 (in O*)

The composition of the portfolio:yWs

* = 0.556(.2904) = 0.1615 proportion in stocksyWb

* = 0.556(.7096) = 0.3945 proportion in bonds1-y = 1-.5560 = 0.4440 proportion in T-bills

1.0000

(b) What is the standard deviation of the above portfolio?

σp = yσ* = .556(.0487) = .02709, or 2.71%

(c) The optimal mix (y*) of the optimal portfolio (O*) and T-Bills is the proportion of O* which maximizes investor utility. If the proportion y in 6.1(a) is your optimal mix of O* and T-Bills, find investor’s coefficient of risk aversion.A = Coefficient of risk aversionif y = y* = E(r*) – rf

Aσ*2

0.556 = .1219 - .05 A = 54.53 A(.0487)2

6.2(a) Suppose it turns out that investor’s coefficient of risk aversion, A = 6. Find the optimal mix of O* and T-Bills.

y* = E(r*) –rf

Aσ*2

= .1219 - .05 = 5.05 6(.0487)2

(b) Find the expected return of the portfolio

E(rp) = rf + y [E(r*) – rf]

= .05 + 5.05 [.1219 - .05] = .4131 or 41.31%

(c) Find the Standard deviation of the portfolio

σp = yσ* = 5.05(.0487) = .2459 or 24.59%

6.3 Suppose investor can tolerate σc = 2%. Find y (proportion she should invest in O*) and E(rc) of portfolio. Find the mix of stocks, bonds and cash. (Homework).

6.4 Equation of Capital Market Line (CML)

E(rc) = rf + [E(rm) – rf]σp

σm

Assuming O* Portfolio to be the market portfolio,

E(rc) = rf + [E(r*) – rf] σp

σm

E(rc) = .05 + [.1219 - .05 ] σp

.0487

E(rc) = .05 + 1.48 σp

II. MUTUAL FUND THEOREM

All investors, no matter what their risk preferences, will want to hold risky assets in the same relative proportions as implied in the optimal portfolio, as long as they share the same capital market expectations.

This implies: asset allocation and security selection are two separate decisions (independent of each other); hence called “separation property”.

Security selection decision is made for investors via the optimal portfolio of risky assets. Investor risk preference does not play any role in security selection. This is a purely technical problem.

Asset allocation decision (how much to invest in various assets classes) depends on risk preferences of investors. This is a personal choice of the best mix of the risky portfolio and the riskfree asset.

A portfolio manager will offer the same optimal risky portfolio to all clients, no matter what their degree of risk aversion. Risk aversion comes into play when clients select their desired point on the highest capital allocation line.

III. SHAPE OF EFFICIENT FRONTIER FOR VARIOUS ρ

ρ = -1 ρ = +1 ρ = 0

IV. COMPOSITION OF MINIMUM VARIANCE PORTFOLIO OF RISKY ASSETS

Wmin(s) = σb2 – Cov(rs,rb) _

σb2 + σs

2 – 2 Cov (rs, rb)

= (.0801)2 – (-.01114) _ = (.0801)2 + (.2104)2 – 2(-.01114)

0.2406

Wmin(b) = 1 – 0.2406 = 0.7594

σmin = = 4.68%E(rmin) = (.2406)(.20)+(.7594)(.09) =.1165, or

11.65%

)01114.)(7594)(.2406(.2)00642(.)7594(.)04428(.)2406(. 22

V. COMPOSITION OF MIN. VARIANCE PORT. OF RISKY ASSETS WHEN = -1

Assume that in the above example the correlation between S and B is –1.

Ws (ρ = -1) = σb _ = 0.080125 _ σs + σb 0.210428 + 0.080125

= 0.2752Wb(ρ = -1) = (1-0.2757) = 0.7243

E(rp) = 0.2757(.20) + .7243(.09)= 0.1203, or 12.03%

σp = ?

VI. OPTIMAL MIX OF STOCKS AND BONDS WHEN RISKFREE ASSET IS NOT

AVAILABLE

VII. OBTAINING RETURNS MORE THAN 20% USING STOCKS AND BONDS WHEN RISKFREE ASSET IS NOT AVAILABLE

E(rc) = .2 = y E(rs) + (1-y) E(rb) Solve for y.

Recall, E(rs) = 20%, E(rb) = 9%.

VIII. ASSET ALLOCATION: MANY RISKY ASSETS AND 1 RISKFREE ASSET

- Process is similar to the case of two risky assets.- Complexity:

X. PSYCHOLOGICAL FACTORS

1. Risk aversion

You face the choices:(a) 100% chance of winning $3,000(b) 80% chance of winning $4,000 & 20% chance of

winning nothing

- Which one do you pick?- In the long run, you come out ahead by making the

second choice consistently.

2. Fear of loss

A. You face the choices: (a) a certain loss of $3,000(b) an 80% chance of losing $4,000 and 20% chance of losing nothing

- Most people will gamble on the second one though it is riskier than the first: its expected value is $3,200 [.8($4,000) + .2(0)]=$3,200. They will gamble on second one because it has a 20% chance of not losing anything.

- People’s horror of losses exceeds even their aversion to risk. Risk aversion is not always the guiding light of decision making.

B. Heads : you win $150, tails : you lose $100. Given this choice, will you bet? The potential payoff is 1.5 times the possible loss.

XI. PORTFOLIO THEORY: SUMMARY

1. Portfolio theory/management is concerned with an investor’s portfolio – the combination of assets invested in and held by an investor. Risk of a security is not to be considered in isolation; it is important to consider how much the risk of a portfolio will reduce when a risky security is added to it.

2. Basic portfolio theory originated with Harry Markowitz. It is based on the expected return and risk characteristics of securities.

3. Investors seek efficient portfolios, defined as those with maximum return for a specified level of risk, or, the minimum risk for specified level a return. The efficient set (frontier) of portfolios can be calculated from equations for the expected return and risk for a portfolio.

4. The expected return for a portfolio is a weighted average of the individual securities’ expected return.

5. Portfolio risk is not a weighted average of individual security risks because it is necessary to account for the covariations between the returns on securities. Once determined, the weighted covariance term can be added to the weighted variance of the securities to determine portfolio risk. To determine the covariations between securities’ returns, it is necessary to calculate the covariance or the correlation coefficient, either of which can be positive, negative, or zero. Investors seek to reduce as much as possible, positive correlation, which is typical.

6. Portfolio risk depends not only on the variances and covariances, but also on the weights for each security (as does the portfolio expected return). The weights are the variables to be manipulated in solving the Markowitz portfolio model.

7. After generating the efficient set of portfolios, an investor chooses one based on the point of tangency between the efficient frontier and the investor’s highest indifference curve.

XII. NOTATIONS USED

0* = Optimal portfolio of risky assets

W*(s) = Composition of stocks in the optimal portfolio

W*(b) = Composition of bonds in the optimal portfolio

E(r*) = Expected return of the optimal portfolio

σ* = Standard deviation of the optimal portfolio

E(rc) = Expected return of complete portfolio of risky and riskfree assets

σc = Standard deviation of the complete portfolio

y = Investment in the optimal risky portfolio

1 – y = Investment in riskfree asset

y* = Optimal investment in the optimal risky portfolio given ‘A’

1 – y* = Optimal investment in riskfree asset

XIII. TYPES OF PROBABLE QUESTIONS

(1) Given expected returns and SD’s of 2 risky assets and return of a riskless asset:

(i) Find the optimal pf of risky assets using W*(s) formula.W*(b) = 1 – W*(s)

(ii) Find the E(r*) of 0* pf: E(r*) = W*(s) E(rs) +

W*(b) E(rB)

(iii) Find standard deviation of 0* pf:σ* = [Ws

2σs2 + W2

Bσ2B + 2WsWBσsσB ]1/2

(iv) Find S* (slope of CML): E(r*) – rf

σ*

(2) Lowest risk way of achieving (say) 9% E(r) from S, B & Cash.

(i) Given E(r), SD of S&B, find optimal pf of S & B using W* (s) formula.

(ii) Find E(r*) of 0* pf.

(iii)Find y (investment in 0* pf needed to get 9%) E(rc)

E(rc) = .09 = rf + y [E(r*)-rf] then solve for ‘y’.

(1-y) = investment in T Bills (Cash).

(iv) Find investment proportions in S, B & cash

S: W*(s) * y =B: W*(b) * y =C: = 1-y

1.0(3) You can tolerate a level of risk of (say) 12%. How

much should you invest in 0* (S and B) and how much in cash?

(4) Your risk aversion coefficient ‘A’ is (say) = 2.

(a) What would be your optimal investment in 0* (S and B) and in Cash?

(b) What is the E(rp) and SD of your optimal portfolio?

Optimal mix of 0* and cash: y* = E(r*) – rf

Aσ*2

= Investment in 0* given value for ‘A’1-y* = Optimal investment in cash.