4. Passive Risk & Return Flashcards
What is passive return?
Return of a market index (e.g. S&P500, NZX50, etc.)
Why is portfolio risk lower than the weighted average risks of assets included in the portfolio?
Diversification
Harry Markowitz is credited with…
developing the first modern portfolio theory (MPT)
Harry Markowitz model is the…
theoretical framework for the analysis of risk-return choices, with these choices made based on the concept of efficient portfolios
What is an efficient portfolio?
Portfolio’s expected return has the smallest variance for that particular level of expected return OR
a portfolio with the largest expected return for a specified amount of portfolio variance
Efficient Frontier
Set of optimal portfolios that offer the highest expected return for a given level of risk (no rational investor would choose to hold a portfolio not located on the efficient frontier)
Unsystematic risk
Can be reduced through diversification
Systematic risk
Risk inherent to the whole market
Three ingredients of optimisation
Objective function, choice variables and constraints
Return of portfolio
h (weights) vector transposed x R (return) vector
Expected return
h (weights) vector transposed x E(R) vector –> E(R) vector = h vector transposed x average return vector
Variance
= h vector transposed x VCV x h vector
Beta of a portfolio relative to benchmark
Bp,b = h vector transposed x VCV x hb vector/hb vector transposed x VCV x hb vector
Beta of a portfolio that’s not fully invested
delta x beta of a portfolio relative to benchmark where delta = proportion of portfolio invested in risky portfolio
Beta for individual stock, i, relative to a portfolio thats not fully invested
1/delta x beta of portfolio relative to benchmark
covariance
hp vector transposed x VCV x hb vector
Local extrema problem in numerical optimisation
Need to try different starting points to ensure that we are achieving the actual solution and not just local extrema.
Numerical vs. Analytical Optimisation
Numerical - software (complicated/more constraints)
Analytical - pencil and paper (faster and more accurate)
Why do practitioners care about MPT?
- Introduced the notation of unsystematic risk
- Passive MPT portfolios can be used in active portfolio management
Main consideration of Markowitz frontier
Considers expected risk and return of individual assets and their interrelationship as measured by correlation of VCV.
CAPM is simplified
Markowitz MPT
Tangent of the Tobin Frontier is
most optimal
Tangency portfolio is
market portfolio
SML depicts the relationship between
Systematic risk and return (beta and return)
CML depicts the relationship between
total risk and return
Systematic risk is measured with
beta
CAPM relates the returns on…
individual assets or entire portfolios to the return on the market as a whole
Investors are compensated for
systematic risk but not unsystematic (firm specific risk)
Unlike other points on Tobin frontier, the tangency portfolio is
fully invested (weights in risky assets = 1), the tangency portfolio is the optimal portfolio of risky assets (a.k.a market portfolio) and holds the highest sharpe ratio compared with any portfolio on the efficient frontier.
CAPM says that
the return on asset is the riskless rate plus a risk premium –> remember that investors are compensated only for systematic risk
Beta in CAPM =
Covar/var
What does CAPM say about alpha?
CAPM says that alpha in the long run has an expected value of zero, meaning that the returns investors get are due to to their exposure to the “systematic risk”
Ex Ante Meaning
If we plot the CAPM-derived expected return on stocks/portfolios against the betas of those stocks/portfolios, we find that they lie on a straight line with intercept equal to the risk-free rate and slope equal to the expected excess return on the market (SML)
Ex Post Meaning
“after the facts” - if we plot actual return against their betas, then we might find a scatter diagram
Can we test CAPM?
Roll (1977) - CAPM is untestable, the only economic prediction of CAPM is that the market portfolio is mean-variance efficient.
CAPM’s market portfolio cannot…
be observed - market portfolio includes all assets in the universe therefore, we use stock indexes as proxies for the return of a market portfolio
We can’t say what the true reason is for discrepancies between actual returns and returns forecasted by CAPM it could be…
the proxy for the market is not correct or CAPM is not valid for the chosen security
The linear return/beta relationship can be found in any sample irrespective of
how returns are determined in the market, we only need to identify an index that is ex-post efficient “after the facts”
All existing CAPM tests only tell us
whether the market proxy used by researches is efficient or not, nothing about the efficiency of market portfolio itself –>market proxies only contain information about the subsets that are equities, not the whole market –> testing only a fraction of the market
A true market portfolio should include
all assets in the universe!
