Investment Analysis and Portfolio Management Lecture 6 Gareth

Investment Analysis and Portfolio Management Lecture 6 Gareth Myles

Announcement l There is no lecture next week (Thursday 27 th February)

The Single-Index Model l Efficient frontier Shows achievable risk/return combinations l Permits selection of assets l l Can be constructed for any number of assets l Given expected returns, variances and covariances l Calculation is demanding in the information required

The Single-Index Model l More useful if information demand can be reduced l The single-index model is one way to do this l Imposes a statistical model of returns Simplifies construction of frontier l The model may (or may not) be accurate l l The reduced information demand is traded against accuracy

Portfolio Variance l The variance of a portfolio is given by This requires the knowledge of N variances and N[N – 1] covariances l But symmetry ( ) reduces this to (1/2)N[N – 1] covariances l

Portfolio Variance l So N + (1/2)N[N – 1] = (1/2)N[N + 1] pieces of information are required to compute the variance l Example l l If a portfolio is composed of all FT 100 shares then (1/2)N[N + 1] = 5050 This is not even an especially large portfolio

Portfolio Variance l Where can the information come from? l l 1. Data on financial performance (estimation) 2. From analysts (whose job it is to understand assets) But brokerages are typically organized into market sectors such as oil, electronics, retailers l This structure can inform about variances but not covariances between sectors l So there is a problem of implementation l

Model A possible solution is to relate the returns on assets to some underlying variable l Let the return on asset i be modeled by l = return on asset i, = return on index, = random error l Return is linearly related to return on the index l This model is imposed and may not capture the data l

Model Three assumptions are placed on this model l The expected error is zero: l l The error and the return on the index are uncorrelated: l The errors are uncorrelated between assets:

Model The model is estimated using data l Observe the return on the market and the return on the asset l Carry out linear regression to find line of best fit l

Example 10 British American Tobacco Barclays 8 20 6 10 4 0 -10 2 -5 0 -2 0 5 -10 0 -10 -5 5 10 -20 -4 -30 -6 -40 Monthly data on stock return and FTSE 100 return l Observe different scales on vertical axis l 10

Example 20 10 0 -10 -8 -6 -4 -2 0 -10 -20 -30 -40 2 4 6 8 10 BAT Barclays

Model l The estimated values are l With l l The estimation process ensures the average error is zero The value of is the gradient of the fitted line

Example 10 British American Tobacco Barclays 8 30 6 20 4 10 2 0 -10 -5 0 5 10 -5 0 -2 -20 -4 -30 -6 5 -10 -40 R 2 = 0. 6304 10

Example 30 20 10 BAT 0 -10 -8 -6 -4 -2 0 2 4 6 8 Barclays 10 Linear(BAT) -10 -20 -30 -40 Linear(Barclays)

Model l If the model is applied to all assets it need not follow that If the covariance of errors are non-zero this indicates the index is not the only explanatory factor l Some other factor or factors is correlated with (or “explains”) the observed returns l

Model l Note: l And l These observations permits a characterization of assets

Assets Types If then the asset is more volatile (or risky) than the market l This is termed an “aggressive” asset l ri r. I

Assets Types If then the asset is less volatile than the market l This is termed a “defensive” asset l

Risk For an individual asset l If then l

Risk l This can be written l So risk is composed of two parts: 1. market (or systematic) risk 2. unique (or unsystematic) risk

Return l Portfolio return

Return l Hence The portfolio has a value of beta l This also determines its risk l

Risk l Portfolio variance is

Risk l The final expression can also be written Consequence: now need to only know and , i = 1, . . . , N l For example, for FT 100 need to know 101 variances (reduced from 5050) l

Diversified Portfolio l A large portfolio that is evenly held l The non-systematic variance is l This tends to 0 as N tends to infinity, so only market risk is left

Diversified Portfolio l That is l tends to l l is undiversifiable market risk is diversifiable risk

Market Model A special case of the single-index model l The index is the market l l l The market model has two additional properties l l l The set of all assets that can be purchased Weighted-average beta = 1 Weighted-average alpha = 0 Issue: how is the market defined? l This is discussed for CAPM

Adjusting Beta l The value of beta for an asset can be calculated from observed data l This is the historic beta l There are two reasons why this value might be adjusted before being used Sampling l Fundamentals l