1. Single Index Model Lecture 4
FINC3017 INVESTMENTS AND PORTFOLIO MANAGEMENT
DR ANDREW AINSWORTH

2. Learning Objectives How are returns generated?
We can assume a single factor structure
How does this help us in terms of portfolio optimisation?
Reduces the number of inputs required for mean-variance optimisation
One factor drives the returns of securities
We can obtain estimates from regression analysis
Provides some economic intuition to portfolio construction
What are the benefits and limitations of mean variance optimisation?

3. Reading
BKM Ch. 8
Kritzman, M. (2011) “The graceful aging of mean-variance optimization”, Journal of Portfolio Management, Vol. 37 Issue 2, pp
Michaud, R.O. (1989) “The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?”, Financial Analysts Journal, Jan-Feb, pp

4. The Single Index Model (SIM)
Markowitz framework requires a significant number of inputs when dealing with a large number of securities
We can simplify the portfolio optimisation task by assuming that the risk of an individual security can be decomposed into a firm specific component and a market wide component
Sharpe (1963) overcame this problem using the Single Index Model (SIM)
The SIM uses this assumption to specify a particular form for security returns and simplify the portfolio selection problem
This reduces the number of required inputs

5. SIM
Returns can easily be expressed as an expected component and an unanticipated component in the following way:
where ui has a mean of zero and standard deviation si
Individual returns are generally correlated with each other
Common set of factors drive returns (e.g. common response to market changes)
Assume all stock returns are related to a single influence (market return)
Under the assumption of a single factor, security returns could be written as:
The macroeconomic factor (m) measures unanticipated macroeconomic surprises and therefore has a mean of 0 and a standard deviation of sm

6. SIM
SIM describes an asset return as made up of a constant and a sensitivity to a factor
This factor is often the ‘market’ index (e.g. ASX200, S&P500, FTSE100)
Where
Rit is the return on asset i in period t
Rmt is the return on the factor in period t
ai is the constant component of asset i’s return
bi is the sensitivity of asset i’s return with the factor
eit is residual component of asset i’s return that is not explained by the factor

7. SIM
The residual (eit) is assumed to be uncorrelated across assets and time, and is also uncorrelated to the factor return (Rmt)
The residual is assumed to have a mean of zero and a variance of
Sharpe assumes that the returns to the factor (Rmt) are generated as:
Where A is a constant and c is a random residual
This implies that the expected return on the factor is:
The standard deviation of the factor is simply:

8. SIM The expected return and variance for asset i are then:
The covariance between asset i and j is
The variance of the portfolio is then

9. SIM Let’s assume the factor is the return on the ASX300 index (Rm)
It is also common to work in ‘excess return’ form and deduct the risk-free rate from the both the asset’s returns and the market’s return:
If the excess return on the market is zero, then ai represents the expected excess return on the stock
The sensitivity of the securities return is given by bi (the larger bi is, the greater the swings in a stocks return when the market moves)
Given that E(ei) = 0, we can similarly express expected returns as
This is the expected return-beta relationship

10. SIM and Diversification
The SIM also provides intuition about the benefits of diversification
Let’s assume that portfolios are equally weighted (xi =1/N)
Alpha, beta and residual risk are therefore the averages:

11. SIM and Diversification
The return on a portfolio under a SIM is given by:

12. SIM and Diversification
Consider now the variance of the returns on this portfolio:
Since all the firm specific variance are assumed to be independent, we have that
Since the average variance is independent of N, as N grows (more firms are added) the portfolio’s idiosyncratic variance converges to zero
As an investor increases the number of assets in their portfolio, the proportion of total risk that is systematic risk will increase and in the limit, their portfolio variance approaches :

13. SIM and Diversification

14. Estimation
Now that we have developed our model, the question remains as to how it might be estimated?
The simplest (and best) way is to perform the regression model described earlier
We will consider 50 monthly returns from January 2013 to February 2017 on Commonwealth Bank (CBA), Ramsay Health Care (RHC) and Woodside (WPL)
The ASX200 is the single factor
I am not using a risk free rate so this is not excess returns
The 90-day Bank-Accepted Bill rate is a proxy for the risk-free rate

15. Monthly excess returns

16. Estimation: CBA

17. Estimation: RHC

18. Estimation: WPL

19. Estimating the SIM
Beta estimates can be supplied through commercial vendors or calculated directly
Require returns on the individual security
Require returns on a market index such as the S&P/ASX 200 accumulation index
Beta can be estimated using the following regression:
Beta estimates are imprecise:
They vary according to data used for estimation
Thin trading creates bias
They vary over time for same firm

20. Measurement error in beta
If a share is traded infrequently (thinly) then beta estimates using the above approach can lead to a bias in the measurement of beta
First order serial correlation between today and yesterday
Consider a case where the last trade in small and illiquid security, ARR happens at 1:20pm on a given day
If there is some news announcement between 1:20pm and the market close at 4:10pm that affects the entire market then the use of this return in a regression to estimate beta will be biased
When ARR next trades (e.g. the following day), its share price will adjust to this news
However, the market return the following day will be unrelated to the previous days news (on average)
We can correct for this bias using a Dimson (1979) adjustment

21. Measurement error in beta
Dimson (1979) adjustment :
Where
Ri,t is excess return on stock i at time t
Rm,t is excess return on the market portfolio at time t
ei,t is the residual from the regression
k is the number of lagged market returns
j is the number of leading market returns
The stock’s beta is

22. Alpha and Security Analysis

23. Alpha and Security Analysis
The Treynor-Black Model
Use macroeconomic analysis to estimate risk premium and risk of market index
Use statistical analysis to estimate the beta coefficients of all securities and their residual variances, σ2(ei)
Establish the expected return of each security absent any contribution from security analysis
Use security analysis to develop private forecasts of the expected returns for each security
You can invest some portion in the market portfolio and some portion in an ‘active’ portfolio
If you only care about diversification then you hold the market portfolio
If you find alpha through security analysis you will not hold the market
Model trades off alpha against the cost of not diversifying efficiently

24. Single-Index Model Input List
Risk premium on the passive market portfolio (Rm-Rf)
Estimate of the standard deviation of the passive market portfolio
n sets of estimates for each stock
Beta coefficient
Stock residual variances
Alpha values
There are n+1 assets to invest in with portfolio mean and variance:

25. Optimal Risky Portfolio of the Single Index Model
Let all returns be in excess return form
The investor wants to maximize the Sharpe Ratio:
With two assets, it can be shown that the optimal weights for the two asset case is (see p. 217 of BKM):
Let’s use our knowledge of the single-index model to solve for the weights

26. Optimal Risky Portfolio of the Single Index Model
Substitute for expected returns and covariance
Divide all terms by the market variance
Cancel and collect terms

27. Optimal Risky Portfolio of the Single Index Model
Substitute in for the variance of the active portfolio
Multiply and divide the last term in the denominator by the market variance
Cancel terms

28. Optimal Risky Portfolio of the Single Index Model
Assume beta of the active portfolio = 1
But beta isn’t always equal to 1
If beta of the active portfolio is greater than 1 the diversification value of the passive portfolio is lower
The weight in the active portfolio (x*a) increases
And the weight in passive portfolio decreases

29. Optimal Risky Portfolio of the Single Index Model
That’s great, but what is the weight of each stock in the active portfolio?
This can be determined by examining each stock’s alpha divided by residual variance
The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):
The contribution of the active portfolio depends on the ratio of its alpha to its residual standard deviation – the information ratio
This measures the extra return we can obtain from security analysis

30. Markowitz vs SIM
Markowitz involves much more data and computationally difficult
SIM approach easier than Markowitz framework
Only need to estimate sensitivity to common factor
Markowitz makes no assumptions about return generating process
SIM makes assumption about return generating process
One common source of risk
SIM omits industry risk
SIM ignores correlations among firm specific risks and thus may not provide the best risk minimising weights in portfolio theory
In practice, the result of both approaches are similar

31. Index Model versus Full-Covariance Matrix
Full Markowitz model may be better in principle, but
Using the full-covariance matrix invokes estimation risk of thousands of terms
Cumulative errors may result in a portfolio that is actually inferior to that derived from the single-index model
The single-index model is practical and decentralizes macro and security analysis

32. Mean-Variance Optimisation Issues

33. Is optimized optimal?
Michaud, R.O. (1989) “The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?”, Financial Analysts Journal, Jan-Feb, pp
Benefits of MV optimisers
Incorporate constraints and objectives imposed by clients into the portfolio construction process
Control risk exposure of the portfolio
Is able to implement style objectives and philosophy of the investment manager
Avoid ad hoc weighting of information
Can adjust to changes in market conditions quickly by processing information quickly

34. Is optimized optimal? Reasons not to use MV optimisers
Political: The investment committee loses control of the investment decisions
‘Black box’ – too quantitative
Error maximisation
Overweight securities with higher expected returns, negative correlations and lower variance
These stocks are likely to contain the most error
Varying confidence in forecasts
Missing factors
Is one factor in the SIM enough?
No, we will talk about multi-factor models later in a few weeks
Unstable optimal solution

35. Like a fine bottle of wine?
Kritzman (2011), “The Graceful Aging of Mean-Variance Optimization”, Journal of Portfolio Management, Winter, pp. 3-5
“(Mean-variance optimization)... has survived and prospered, because it works.”
Kritzman rebuts criticisms
Garbage in, garbage out
Is this that surprising? Why use historical estimates of returns?
Optimisers are error maximisers
This criticism is true, but this does not have as big an impact on outcomes as the critics suggest
Mean-variance optimisation depends on false assumptions
Again, fair criticism, but does not have a substantial adverse impact in practice

36. What you should know
How does the SIM approach compare to the full covariance approach?
What are the benefits and limitations to the SIM approach?
How do we characterise the statistical properties of the SIM?
What are the benefits and limitations of mean-variance optimisation?
Next week: Capital Asset Pricing Model