FINA305 Lecture Notes PDF

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FINA305 LECTURE NOTES1. RISK, RETURN AND CAPITAL ALLOCATIONRisk and return: A trade-off  Most investors prefer an investment with high returns with low risks  High expected returns tend to be followed with high risks  There is a trade-off, depending on how you value returns and risk (risk prefere...

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FINA305 LECTURE NOTES 1. RISK, RETURN AND CAPITAL ALLOCATION Risk and return: A trade-off  Most investors prefer an investment with high returns with low risks  High expected returns tend to be followed with high risks  There is a trade-off, depending on how you value returns and risk (risk preference) Return Rates of return: Single period



HPR = P1 – P0 + D1/P0



o HPR = Holding period return o P0 = Beginning price o P1 = Ending price o D1 = dividend during period one Continuous compounded return: Use log difference: HPR=Ln(P1+D)-ln(P0)=ln[(P1+D)/P0].

Expected Return and Standard Deviation  Expected returns

o o o 

p(s) = Probability of a state r(s) = Return if a state occurs s = State

Variance (VAR):



Standard deviation (STD):

Time Series Analysis of Past Rates of Return   

True means and variances are unobservable because we don’t actually know possible scenarios like the one in the examples So we must estimate them (the means and variances, not the scenarios) We observe a series r: o r(1), r(2),..., r(n) o Monthly returns of the last 10 years (120 observations)

Returns using arithmetic and geometric averaging  Arithmetic Average



Geometric (Time-Weighted) Average

= Terminal value of the investment

σ

Estimating variance and standard deviation  Estimated variance o Expected value of squared deviations



Unbiased estimated standard deviation

The Reward-to-Volatility (Sharpe) Ratio  Risk premium: The difference between the expected HPR on a risky asset and the risk-free rate  Sharpe Ratio = risk premium/SD of excess returns Risk preference and risk-return trade off  Representative investors are risk averse.  Investment attractiveness increases with expected return and decreases with risk.  What happens when return increases with risk? o Investors need to make a trade-off between risk and return. o One quantitative way is through utility. Table 6.1 Available Risky Portfolios

Each portfolio receives a utility score to assess the investor’s risk/return trade off Risk aversion and utility values  Utility function o U = Utility o E(r) = expected return on the asset or portfolio o A = coefficient of risk aversion  If large, will be more risk averse/sensitive to the risk  If A =0, utility only depends on expected return, so investor doesn’t care about risk anymore, just return (risk neutral)  If A < 0, the investor is risk lover (then coefficient becomes positive) o σ2 = variance of returns o ½ = a scaling factor Table 6.2 Utility scores of portfolios with varying degrees of risk aversion

σ Risk averse and portfolio dominance  Mean-variance (M-V) Criterion o Portfolio A dominates portfolio B if:

and

Capital allocation across risky and risk-free portfolios  Asset allocation: the choice among broad asset classes that represents a very important part of portfolio construction  The simplest way to control risk is to manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets Basic asset allocation example Let:  Y = weight of the risky portfolio, P, in the complete portfolio  (1-y) = weight of risk-free assets

Portfolios of one risky assets and a risk-free asset Let:  Y = portion allocated to the risky portfolio, P  (1-y) = portion to be invested in risk-free asset, F

E(rc) = rf + σc/σp[E(rp) – rf] Var(ax + by) = a2 var(x) + b2 var(y) + 2ab[cov(x,y)] = y2 σp2 + (1-y) x 0 + 2y(1-y) cov (rp, rf) = y2σp2 The expected return on the complete portfolio = E(rc) = 7 + y(15 – 7) The risk of the complete portfolio: σ c = y σP = 22y Re-arrange and substitute y = σc/ σp  Slope = Sharpe ratio =

Figure 6.4 The Investment Opportunity Set *every point along line is one possible choice *different points along line means different portfolio choice In square: 0 < y < 1 and 0< (1-y) < 1 (savings and invest) Right hand-side: y>1 and (1-y) r > +1  Case 1: if p = 1, the securities are perfectly positively correlated, don’t need to introduce new asset, no diversification needed o When ρDE = 1, there is no diversification (just a linear function) o σP = wEσE + wDσD  Case 2: if p = -1, the securities are perfectly negatively correlated, no uncertainty, is possible, means you can get 100% diversification, can reduce all risk to zero. Variance = 0. All risk is non-systematic. o When ρDE = -1, a perfect hedge is possible w

=

=1 - w

D

+ o  Case 3: if - 1 < p < 1 (partly systematic), not linear Example E

D

D

E

Debt

Equity

σ Expected return ,E(r)

8%

13%

Standard deviation,

12%

20%

Covariance

72

Correlation

0.30

Figure 7.3 Portfolio Expected Return

Figure 7.4 Portfolio Standard Deviation: Inverse parabola

Figure 7.5 Portfolio expected return as a function of standard deviation

-1 < p < 1 (blue line  most common case. Any point along the line is a choice. D = wD = 1, wE = 0 E = wD = 0, wE= 1 Most common case is blue line (between the two

Figure 7.6 The opportunity set of the debt and equity funds and two feasible CALs

B and A not comparable as B is not on north-west side. But B provides the chance to obtain F. (F strictly better than A as its on the northwest, dominates A. F

Optimal = tangency (portfolio tangent with curve)  highest sharpe ratio  need to maximize the slope

The Sharpe Ratio  Maximize the slope of the CAL for any possible portfolio, P  The objective function is the slope:

S

p

=

E (r p ) - r f

σ

p

 

The slope is also the Sharpe Ratio Steeper slope = higher sharpe ratio = more compensation for risk

Figure 7.7 Debt and equity funds with the optimal risky portfolio

Investor’s optimal choice: need tangency between indifference curve and CAL

Y1* and Y2*  common property: both come from c and rf – coming from the same source, same risky asset and riskfree asset. The difference between Y1* and Y2*: Only difference is in the different weight of c and rf. CAL is totally independent of investor’s risk aversion which means construction of optimal risky asset is purely technical. c and rf is purely technical  does not rely on any assumption of investor’s risk preference. Weight you put on these two depends on preference. Separation property  Can separate asset allocation into two steps: 1. Purely technical (independent of investors risk preference) = CAL, 2. Use investors risk preference to determine which point along line to choose Figure 7.8 Determination of the optimal overall portfolio

Optimal complete portfolio  comes from tangency between indifference curve and capital allocation line.

N assets The expected return: n

E (r p ) = å w i E ( ri ) i=1

The variance: 2 p

n

=å

n

å

i = 1 j =1

w i w j C o v (r i , r j

)

σ Risky portfolio choice under N assets  From the portfolios that have the same expected return, choose the one that has minimum variance.  Min =å å w w C ov (r ,r ) n

n

2 p

i

j

i

j

i =1 j =1



S. t. E (rp ) =



n

å

w i E ( ri )

=a. By changing a, we could have optimal risky portfolio for different expected return, which could then be plotted. i =1

Markowitz Portfolio Optimization Model  Security selection o The first step is to determine the risk-return opportunities available o All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations Figure 7.10 The Minimum-Variance Frontier of risky assets For any portfolios that have the same expected return, choose the portfolio with the minimum variance/standard deviation. (most left) Choose highest return with lowest risk. Not all options are efficient. The investor will only choose from the top part of the curve.

 

Search for the CAL with the highest reward-to-variability ratio. Everyone invests in P, regardless of their degree of risk aversion o More risk averse investors put more in the risk-free asset o Less risk averse investors put more in P

MRA (More Risk Averse) investors will hold a mix of tangency portfolio and T-bills. LRA (Less Risk Averse) investors will borrow at the riskless rate and invest the proceeds in the tangency portfolio.

 

Capital Allocation and the Separation Property Portfolio choice problem may be separated into two independent tasks o Determination of the optimal risky portfolio is purely technical o Allocation of the complete portfolio to risk-free versus the risky portfolio depends on personal preference



The power of diversification o Remember: 2 p

o

=

n

n

i= 1

j= 1

w i w jC o v (ri , r j

åå

)

If we define the average variance and average covariance of the securities as: 2

=

C ov =

1 n

n

å

2 i

i =1 n

1 n (n -1

)

n

åå

j =1 i =1 j¹i

C o v ( ri , r j )

σ

o

2 p

=

1 n

2

+

n -1 C ov n

We can then express portfolio variance as  The first component will converge to zero if n converges to infinity. This component is firm-specific risk. If n goes to infinity  1/n converges to 0 (percentage becomes smaller and smaller)  The systematic risk of a diversified portfolio depends on the covariance of the returns, which is a function of the systematic factors in the economy.  Variance = firm-specific risk  Covariance = systematic risk 1. Decrease in sigma = increase in n 2. When p decreases, better diversification effect 3. Marginal decline: marginal effect is decreasing When n increases from 1 to 2, big decline in total risk Most investors hold portfolio between n = 20 and 30.

Capital Asset Pricing Model (CAPM): Resulting Equilibrium  All investors will hold the same portfolio (optimal risky asset) for risky assets – market portfolio (separation property)  The optimal risky asset is constructed from all the available individual risky assets on the market. n → ∞.   

Investor competes for better risk-return compensation On equilibrium: o Firm-specific risk is diversifiable, and thus does not receive compensation o Only systematic risk receives compensation On equilibrium: o Investors sell securities with low risk-return ratio and buy securities with high risk-return ration o Risk-return ratio of all risky assets should equal to each other

Total risk = non-systematic risk/systematic risk Example: two risky assets  If portfolio well diversified, non-systematic risk is 0. All components for market portfolio captured by systematic risk.  Return (rm) = E(rm) – rf (risk premium)  Systematic risk-return ratio = E(rM)/ σM2 Example: individual asset  Non-systematic risk is not equal to 0  Means you cannot use total variance to capture the systematic risk  How to capture systematic risk: covariance (ri, rm)  Systematic risk-return ratio = E(Ri)/cov(ri, rm)  Equilibrium: E(Rm)/σM2 = E(Ri)/cov(Ri,Rm) E(Ri) = cov(ri,rm)/σm2  this is beta E(Rm) B = cov(ri,rm)/σm2 E(Ri) – rf = Bi[E(Rm – rf)]  CAPM For any individual risky assets, risk can be split into time return and risk return (risk return can be further decomposed into price of risk and quantity of risk How to measure risk-return ratio  Risk: systematic risk  Return: risk premium 

On equilibrium:

Risk Premium j Risk Premium i = Systematic Risk i Systematic Risk j

σ Risk-return relationship of market portfolio, M  n → ∞:  The firm-specific risk is reduced to zero.  The total risk is due to systematic risk.  Risk-return compensation is measured by

E (R M ) σ

=

2 M

E (r M )−r f 2

σM

Risk-return relationship for individual security, i  For individual securities, total risk o Systematic risk, which is measured by the covariance. o Firm-specific risk.  Only systematic risk receives compensation on equilibrium. 

Risk-return relationship of security i is measured by

E( R i) E( Ri ) E( Ri ) = = systematic risk covariance cov ( Ri , R M )

Between M and i  On equilibrium, o Risk-return of all risky assets should be equivalent to each other.

E (R M ) E (Ri ) = cov(Ri , R M ) σ M2 cov(Ri , R M ) E( RM) E(Ri ) = 2 σM E ( r M )−r f E ( r i )− r f = cov ( Ri , R M ) ] ¿ σ M2 cov(Ri , R M ) = β i , we have CAPM Define 2 σM E (r M ) −r f ]. ** this formula can work for any portfolio E ( r i )− r f = βi ¿

o

o

o

o

o

CAPM: Summary  Security Expected Return= Time return + Risk return  Time return =risk free rate  Risk return= quantity of risk X price of risk (B)  Risk is measured by beta.  Price of risk is measured by market excess return. Extension to portfolio  CAPM holds for the overall portfolio because:

E (rP) =

å

w kE (rk) a n d

k

P

=å w

k

k

k

E(rp) = wk [rf + Bp[E(rm – rf)]]

 formula works for individual asset but can also be applied for any portfolio 

This also holds for the market portfolio (

E ( rM ) = r f +

M

βM=

éë E ( r M ) - r f ùû

Figure 9.2 The Security Market Line

cov(R M , R M ) 2

σM

=1 ):

σ SML = E(ri) = rf + Bi[E(rm) – rf] Only difference: slope here is price of risk (CAL – slope is sharpe ratio)

   

Total risk includes both firm-specific risk and systematic risk. Investor chooses the optimal risky portfolio by maximizing Sharpe ratio. This process is purely technical and independent of risk attitude. (Separation property) Systematic risk is measured by the covariance. Under the assumption of investor competing for better risk-return trade off, on equilibrium, o Only systematic risk is compensated by return. o The compensation for each unit of risk is the same among all securities. CAPM model:

E ( r i )− r f =

o

E (r M ) −r f ]. βi ¿

3. INDEX MODEL   

Markowitz procedure and single-factor model Risk decomposition o Systematic vs. firm-specific Single-index model and its estimation

Risky portfolio choice under N Assets (from lecture 2)  From the portfolios that have the same expected return, choose the one that has minimum variance.  Min n

s

2 p

=å

n

å

i =1 j =1



j

)

S. t. E (r p ) =



w i w jC o v ( r i , r

n

å

w iE (ri )

=a. By changing a, we could have optimal risky portfolio for different expected return, which could then be plotted. i= 1

If N = 2 s.t. 2 x 1 (need 3 inputs) If N = 3 S.t. 3 x 1 (need 6 inputs) N = 4 (need 10 inputs) N = N  I from 1 to N  need (N x (n + 1))/2 parameters Input of Markowitz Portfolio  N risky asset. o Expected return. Vector NX1. N parameters. o Covariance matrix. Matrix NXN. K=N+NX(N-1)/2 parameters. o Example.



❑

o

N=2.

σ1,

2

σ22 , σ 12 . K =2+

o

N=3.

σ21 ,

❑ σ22 , σ 32 , σ 12 ,

o

N=50.

K=50+

2 X (2−1) =3. 2 ❑

❑

σ 23 , σ 13 . K =3+

3 X2 =6 2

50 X 49 =1275. 2

Problem to estimate the covariance matrix when N is huge.

A single-factor market  Advantages o Reduces the number of inputs for diversification o Easier for security analysts to specialize

 

σ Model

r i=E (r i )+ unexpected suprise=E ( r i ) + β i m + e i o β i = response of an individual security’s return to the common factor, m; measure of systematic risk

m = a common macroeconomic factor e i = firm-specific surprises; independent tween asset i and j.

o o

*relationship between CAPM and index model: CAPM  E(r) = rf + Bi(E(rM) – rf)  E(rm) - non-diversifiable, optimal risky portfolio Factor model  ri = E(ri) + Bim + ei (1 asset)  M doesn’t need to be estimated – can be directly observed in market  Observable – index return  B and residual – can significantly reduce the number of parameters you need Risk under single-index model  Variance = Systematic risk + Firm-specific risk:

=

2 i o o o

2

2 M

i

+

2

(ei )

LHS = total risk Bi2σ2M = systematic risk σ2(ei) = firm-specific risk

* Variance  Can delete the E(rm) as its constant Risk decomposition under Index Model  Variance = systematic risk + firm-speciifc risk: 2  σ2i =β2i σ 2M +σ ❑ (ei ) 

The contribution of market risk to total risk is

β i2 σ 2M 2 2 β2i σ M + σ ❑ (e i )

Regression equation of index model  Regression equation:

R i (t ) =



 

+

R

( t ) + e ( t)

i i M i Expected return-beta relationship:

E (R i) =

i

+

E (R

)

i M α M =0, β M =1.

For the market index, The contribution of market risk to total risk is measured by the R square of the regression model.

How to run regression in excel  regression excel in week 3 folder Figure 8.2 Excess returns on HP and S&P 500 Figure 8.3 Scatter Diagram

R

H P

(t ) =

H P

+

H P

R

S& P 5 0 0

(t ) + e H P ( t )

σ Table 8.1 Excel Output: Regression statistic for the SCL of Hewlett-Packard Interpreting the output:  Correlation of HP with the S&P 500 is 0.7238  HP’s alpha is 0.86% per month (10.32% annually) but it is not statistically significant  HP’s beta is 2.0348 and is highly significant.  52% of return variance is due to market risk.

Portfolio construction and the single-index model  Single-Index Model Input List: n assets. o 1. Risk premium on the S&P 500 portfolio o 2. Estimate of the SD of the S&P 500 portfolio o 3. n sets of estimates of  Beta coefficient  Stock residual variances  Alpha values o 4. Total number of parameters: K=3n+2. This is much smaller than n+n*(n-1)/2 of Markowitz portfolio construction when n is large.  n=50, 3n+2=152, n+n*(n-1)/2 =275. Covariance determines effect of diversification (key parameter coming from index model)  covariance is output not input  Cov (E(ri) + Bim + ei, E(rj) + BjM + ej)  Cov (x1 + x2, y1 + y2) = cov (Bim, Bjm) + cov (Bim, ej) + cov(ei, Bjm) + cov(ei, ej) (all equal zero) = BiBjσm2 Cov(ri, rm) = BiBm...