Variance for 2-asset portfolio?
Variance for 3-asset portfolio?
-All investors are risk adverse.
-E(R), σ, σ² are known for all assets and know future values.
-Optimal portfolio constructed SOLEY by E(R), σ, σ².
-No tax / transaction costs
σ²p = w₁²σ₁²+w₂²σ₂²+2w₁w₂Cov₁₂
Cov₁₂ = Corr₁₂ * (σ₁*σ₂)
σ²p = w₁²σ₁²+w₂²σ₂²+w₃²σ₃²+2w₁w₂Cov₁₂+2w₂w₃Cov₂₃+2w₁w₃Cov₁₃
Cov₁₂ = Corr₁₂ * (σ₁*σ₂)…etc
Minimum variance frontier? Steps to construct this?
Global minimum variance portfolio?
Graph of the expected return/variance combinations for all minimum variance portfolios.
1. Estimation step = E(R) and σ² for each asset and Corr for each pair of assets. (HISTORICAL DATA is unbiased estimate of future, but NEED GOOD ESTIMATES)
2. Optimization step = for all target returns, find the weights that guarantee that variance is minimized.
3. Calculation step = figure out the target E(r)
Portfolio with the lowest variance of them all!
Maximum return for the same variance for portfolios on the minimum variance frontier.
Equally-weighted portfolio risk?
Correlation in equally weighted?
2. Number of assets in portfolio.
σ²p = (1/n * σ²i_) + ((n-1)/n * Cov_)
σ²i_ = average variance of all asset in portfolio
Cov_ = average covariance of all pairings in the portfolio.
as n gets large
=(1/n * σ²i_) approaches 0.
=((n-1)/n) approaches 1.
σ²p approaches Cov_ as n gets large in an equally weighted portfolio.
σ²p = σ²i_ * ((1 – Corr_ / n) + Corr_)
Cov_ = average correlation bw all pairs of 2 assets.
as n gets large
(1) σ²p ≈ σ²i_ (Corr_) = MAXIMUM risk reduction of individual asset variance is equal to the Corr_ * asset variance.
(2) level of Corr_ affects the number of assets needed to realize diversification benefits. (lower Corr_ — need more assets to achieve max diversification benefits).
X-axis = st dev. Y = E(R);
Slope _______ for risk-averse investors.
Standard deviation for a portfolio return when 1 asset is riskless?? (assume Corr with risky asset is 0). Difference with Efficient frontier?
Combining a risky portfolio with a risk-free asset is known as the ____________. The line representing the optimal risky asset portfolio is known as the ____________.
CAL answers 3 important question:
1. How should investor chose risky portfolio to combine w the risk free asset?
2. Given investor’s tolerance, what rate of return should be expected?
3. Given investor’s risk/return objective’s, % allocation to rfr and risky portfolio?
σportfolio = w₁σ₁
(1 is the risky asset).
Turns into a LINE when you combine a risky asset w a riskless asset, rather than a curve.
two fund separation theorem; capital allocation line.
1. Investor should choose risky portfolio that maximizes reward to risk tradeoff. Tangency porfolio (t) – efficient portfolio / CAL. Sharpe ratio = (E(Rt) – rfr) / σt.
2. CAL E(R). Can go on indefinitely if risk free borrowing is available. If not, the CAL ends at 100% risky portfolio.
CAL equation: (c = combination of rfr and optimal portfolio).
E(Rc) = rfr + ((E(Rt)-rfr)/ σt) * σc
3. See above:
σportfolio = w₁σ₁
CML equation of the line?
Rewrite CML? What does this mean?
Capital Market Line (CML)
E(Rp) = Rf + ((E(Rm) – Rf) / σm) * σp
E(Rp) = Rf + (E(Rm) – Rf) (σp/σm)
-For every unit of market risk, σm, the investor is willing to take on in the portfolio, σp, get an additional unit of MRP.
systematic risk ?
Give 2 examples.
How many securities need to own to diversify away unsystematic risk?
Is unsystematic risk compensated?
-Systematic + Unsystematic.
1. Harvey Davidsons have high systemic risk as its returns are highly correlated with that of the market (luxury good, thus at the whim of the market)
2. Utility companies have low systemic risk.
-One study 12-18 stock, another 30 stock takes away 90% of the diversification possible.
-No not in equilibrium, because it can be eliminated for free through diversification. SYSTEMATIC risk is the only risk priced by the market.
Estimate Betas by?
What is this line called? Slope of the line?
If we can plot the relationship between the market excess return and asset excess return, we can plot the E(Ri) and Systemic Risk (Cov(i,m)). This line is called the ___________ and its equation is basically __________
Main Difference between SML and CML?
βi = Cov(i,m) / σ²m
Substituting we get:
βi = Corr(i,m) (σi) (σm) / σ²m
βi = Corr(i,m) (σi) / σm
Regressing returns on the asset on those of the market index. Make a least squares regression line with market excess return as the x-axis and asset excess return (over rfr) as the y-axis.
Called the security characteristic line. Same slope as the Beta calculated.
security market line (SML); CAPM (ie E(Ri) = rfr + β(MRP))
-CML plots TOTAL RISK (σ) on the x-axis, hence only efficient portfolios (combining a RFR OR BORROWING IT) will plot on the CML (minimum unsystematic risk).
-SML plots SYSTEMATIC RISK (β) on the x-axis, all properly priced securities will place here regardless of how much unsystematic risk (only plots covariance with the market).
CML: Market portfolio Sharpe ratio; SML: Market risk premium
CML: Used to determine the appropriate asset allocation; SML: Used to determine appropriate expected (benchmark) returns for securities;
1. Historical means, variance, and correlations
2. Estimating betas using the market model. Main assumptions? Leads to? Forecasts can be derived from just? 3 predictions?
a. VERY arduous process. To develop efficient frontier, need ‘market’ of assets. For a portfolio of n assets, we need the following forecasts:
n individual asset expected returns
n individual asset st deviations
b. Large estimation error.
2. More practical than historical. Used to estimate Beta (systematic) and estimate a security’s abnormal returns (unsystematic). Ri = αi + βiRm + ei
Rm = market return
Bi = Slope coefficient
αi = Intercept
ei = Abnormal return on asset i
1. E(ei) = 0; 2. Errors are uncorrelated with the market; 3. Errors are uncorrelated across assets.
THIS LEADS to simplified mean, variance, and correlation.
E(Ri) = αi + βi x E(Rm)
(σ²i) = (β²i)(σ²m) + (σ²e)
Covij = (βi)(βj)(σ²m)
3n+2 parameters for mean, variance, and correlation.
3. Calculating adjusted betas
Beta instability problem?
Adjustments to historical Beta to improve its ability to forecast future Beta?
If α₁ = 1?
βi,t = α₀ + α₁(βi,t-1) + (ui,t)
u = error term with expected value of 0.
Future Beta (βi,t) is equal to historical beta plus a random walk. ie FORECASTED βi,t = βi,t-1
Beta instability problem addressed by adjusting the Beta to account for its tendency to gravitate to a value of 1 over time.
FORECASTED βi,t = α₀ + α₁(βi,t-1)
α₀ + α₁ = 1;
Most popular (Bayesian) are α₀ = 1/3; α₁ = 2/3;
Why is this a concern?
What can analyst do?
– Statistical inputs are unknown and must be forecast; greater uncertainty leads to less reliability on efficient frontier;
– Statistical input derived from historical sample estimates often change over time, causing efficient frontier to change (time instability);
– Small changes in inputs can lead to large changes to frontier, (“overfitting problem”), leading to unreasonably large SHORT positions and frequent rebalancing.
-Avoid rebalancing until significant changes
-Employing forecasting model that estimate better than historical
-Constraining weights (prohibiting short sales to avoid negative weights).
Fama and French multifactor model?
3 GENERAL classifications of multifactor models?
Assumes asset returns are driven by more than one factor.
Good multifactor model with
1. MRP; 2. firm size, 3. firm book value to market value ratio
Cahart, adds to above with
4. Price momentum using prior period returns.
1. Macroeconomic factor model (ASSET RETURNS EXPLAINED BY SURPRISES IN MACRO RISK FACTORS).
2. Fundamental factor model (ASSET RETURNS EXPLAINED BY RETURNS FROM MULTIPLE FIRM SPECIFIC FACTORS).
3. Statistical factor model (2 primary types: FACTOR ANALYSIS — covariance factors and PRINCIPAL COMPONENT MODELS – variance). Do not lend well to economic analysis. MYSTERY FACTORS.
BE CAREFUL ABOUT?
Priced risk factors?
Fundamental factor models:
A risk that doesn’t affect many assets (unsystematic risk) can usually not be diversified away. However, macroeconomic factors are systematic risk factors (priced), ie they can affect even well diversified portfolios.
Can be estimated by regressing historical asset returns (ie retail stocks) on corresponding historical macroeconomic factors (ie GDP growth).
Similar to z-statistics. Ex:
b1 (factor sensitivity for P/E) = (P/Ei – ^P/E) / σp/e
^P/E = average of P/E across all stocks
σp/e = standard deviation of P/E across all stocks
Rates of return associated with each factor (ie rate of return difference between low and high P/E stocks). Estimated as slopes of cross-sectional regressions (returns – dependent variable, standardized sensitivities – independent variables).
Fundamental factor models are not ‘surprise’ factors, therefore, no comparison to E(R), need return intercept.
Number of factors?
If given two stock macroeconomic models, How to calculate actual returns?
M: Surprises; F: Rates of return estimated from regression;
M: Small in number, represent systematic risk factors (parsimonious model); F: Large in number, more detailed;
M: Equals E(R); F: Simply the regression intercept;
Take the weighted average of the factors and expected returns to derive a ‘portfolio.’
Differences bw APT and Multifactor models?
– Returns are derived from a multifactor model
– Unsystematic risk can be completely diversified away.
– No arbitrage opps exist (investors will get to it and take it all away)
E(Rp) = rfr + β1(η1) + B2(η2)
η = expected risk premium for each risk factor.
β = factor sensitivity (loading) of p to each risk factor.
UNLIKE CAPM, APT does NOT require that one of the risk factors is the market portfolio. (TECHNICALLY CAPM can be considered a special case of the APT).
CAPM suggests that all investors should hold rf asset and the market. The APT states that asset returns follow a multifactor process… allowing investors to manage more than one risk factor (ie workers affected by recession, want to bid up non-cyclical. Higher risk premium for cyclical, lower for non-cyclical. This is why GDP is often a sign factor in multifactor models and wealthy investors can earn premium investing in cyclical stocks).
1. APT: Explains variation ACROSS ASSETS’ expected returns during a SINGLE time period. MF: Time series regression that explains variation OVER TIME in returns for ONE ASSET
2. APT: Assumes no arbitrage opps, derived from equilibrium theory; MF: Ad hoc (factors are searched for, rather than derived from theory).
3. Intercept Term: APT: rfr; MF: E(R);
Active risk can be separated into two components?
(ie tracking error or tracking risk)
st dev(Rp – Rb)
1. Active factor risk = attributable to portfolios factor sensitivities; Ex: Manager may decide to overweight certain industries relative to the benchmark;
2. Active specific risk = attributable to deviations of portfolio’s weightings vs benchmark’s weightings.
(Active risk)² = Active factor risk + Active specific risk;
Ex: Manager may decide to overweight stocks within specific industries relative to the benchmark;
(^Rp – ^Rb) / std dev (Rp-Rb)
Hedging or speculation. Ex: Manager thinks that GDP growth will be larger than expected, wishes to have a GDP factor ‘long portfolio’.
Manager wants to have multifactor model, but hedge against GDP. Then invested long in multifactor model and short the ‘GDP’ portfolio by the sensitivity weighting.
Tracks the same factor exposures as the index. Manager usually tries to have some active specific risk (ie use the same weightings, but make superior asset selection)
Another version of the information ratio?
ex-post information ratio?
alpha measured after actual results become available.
Rpt = α + βRbt + εt
Rpt = excess return on portfolio p (return less rfr);
α = ex-post alpha on portfolio
Rbt = excess return on benchmark b (return less rfr);
forecasted alpha of residual returns;
NOTE: Alpha of a portfolio is the weighted average of the asset assets;
α / w
α = annualized residual return
w = annualized residual risk
NOTE: Managers LEVEL of aggressiveness (higher active risk), does not change the information ratio
tα / √n
tα = t-statistic of α in the regression model
n = number of years in the regression model
For a given information ratio, plot of residual return and residual risk.
Formula for VA?
Optimal level of residual risk to assume for given level of manager ability and investor risk aversion?
Active strategy vs. risk aversion?
VA = α – (λ * w²)
α = annualized residual return
w = annualized residual risk
λ = risk aversion parameter (↑ λ indicates higher risk aversion).
NOTE: use percentage, not decimal form.
i.e. α = 3%; λ = 0.05; w = 6.5%
VA = 3 – (0.05 * 6.5²) = 0.89(%);
VA = IR * w – (λ * w²)
(NOTE α = IR * w)
This graph, for a given level of λ, increases initially with risk, then decreases.
Optimal levels of:
w* = IR / 2λ
λ = IR / 2w*
VA = IR * w – (λ * w²); λ = IR / 2w*
VA* = IR² / 4λ
VA* = IR * w* / 2
SHOWS THAT INVESTORS CHOOSE MANAGER W HIGHEST INFORMATION RATIO, INDEPENDENT OF RISK AVERSION; Investor would choose to implement managers strategy based on risk aversion;
Breadth (BR) ?
Relation to information ratio?
Optimal level of residual risk / VA?
If IC forecasts are not independent (ie 2 asset forecasts)?
NOTE: Quantifying IC = 2* (Nc/N) – 1;
N = # of bets
Nc = # of correct bets; NOTE if Nc/N = 0.5, the IC will be 0;
Number of independent forecasts of exceptional return per year that the manager makes. INDEPENDENT means that forecasts should not be based on highly correlated information sets; ie forecasting outperform for high dividend AND value stocks are not independent. A lot of overlap.
NOTE: Quantifying BR = forecasts/month * 12
IR = IC * √BR
If you have 2 analysts (or want to expand coverage), here is the way to calculate.
IR² = IRold² + IRnew²
w* = (IC * √BR) / 2λ
VA* = (IC² * BR) / 4λ
ICcom = IC * (√(2/(1+r)))
IC = original IC
r = correlation between the two sources of info.
2. Sources of information are independent
3. IC is constant across bets. (Otherwise would use the additivity principle).
Steps of the portfolio management process?
Final step in planning stage?
3 common approaches used to implement.
Planning, Execution, Feedback;
Planning: Analyzing objectives and constraints, developing an IPS, selecting an appropriate asset allocation
-Creation of strategic asset allocation; (also consider the need for flexibility);
1. Passive Investment strategies (not responsive to changes in expectations)
2. Active investment strategies (much more responsive to changing expectations)
3. Semi-active, risk-controlled active, or enhanced index strategies. (INDEX TILTING — attempt to match index risk characteristics of a portfolio, but deviate to earn higher returns).
When investor’s ability and willingness to assume risk are in conflict?
Absolute vs. relative risk objectives?
Required vs. Desired returns?
Long time horizon?
Willingness and ability to take risk = risk tolerance
Unwillingness and Inability to take risk = risk aversion
Resolution is required.
Abs: st dev; Relative: tracking risk
Required — stricter benchmark – MUST be achieved.
1. Liquidity constraints
2. Time horizon
3. Tax constraints
4. Legal / Regulatory
5. Unique considerations.