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Reading 53: Portfolio Risk and Return P2 (Beta (Formula: \(\beta…
Reading 53: Portfolio Risk and Return P2
Risk-free & Risky Assets Combination
Such combination allows investors to build portfolios with superior risk-return properties.
Portfolio expected return \(E(R_p)=W_{risky}R_p + W_{Rf}R_f\)
Portfolio standard deviation = \(W_{risky}\sigma_p \)
Capital Allocation Line (CAL) & Capital Market Line (CML)
Capital Allocation Line (CAL):
Formed by various combination of risky and risk-free assets on the risk-return graph.
Superior portfolio has steeper slope CAL when combining with a risk-free asset
Capital Market Line (CML):
When the linear CAL just tangent with the Efficient Frontier, the tangent point is the Market Portfolio
The CAL that connect Risk-free asset and the Market Portfolio (Best possible risky asset) is the Capital Market Line
Investor can adjust portfolio along the CML based on their risk preference.
Scare of risk → Put 100% of investment on Risk-free asset
Can accept more risk → increase the % of risky asset so long as you stay in the CML
If can bear all risk → put 100% of investment on Risky asset
If want more, borrow to invest in more risky asset so long as you stay in the CML
Systematic vs Non-systematic risks
Systematic Risk:
is due to factors that affect the value of all risky securities (Ex: GDP Growth)
Cannot be reduced by diversification
Measured with covariance of returns with returns on the market portfolio
Only systematic risk is rewarded with higher expected returns
Only systematic risk is priced
Unsystematic Risk
a.k.a. idiosyncratic risk
Firm-specific risk, can be reduced by portfolio diversification.
Return-generating Models
is an equation that estimates the expected return of an investment, based on a security's exposure to one or more macroeconomic, fundamental or statistical factor
Simplest Return-generating model
Has a single risk factor, the return on the market
\(R_{i}= \alpha _{i}+\beta _{i}\times R_{m}+e_{i}\)
\( \alpha _{i} \): is the intercept
\( \beta _{i}\): is the slope coefficient, measuring the asset's systematic risk (market risk)
The asset's beta ( ) is the sensitivity of its return to this risk factor
Asset returns are a linear function of market returns
Multi-factor models
\( E[R_i] - R_f = \beta_{i,1}E[F_1] + \beta_{i,2}E[F_2]+...+\beta_{i,K}E[F_K]\)
The factors (F's) are the expected value of each risk factor
The betas (\(\beta_{i,K}\)) are the asset's factor sensitivity or factor loadings for each risk factor
\( E[R_i] - R_f \) is the excess return on the asset
Risk factors (F) of three types
Macroeconomic factors (e.g., GDP growth. inflation, consumer confidence)
Fundamental factors (e.g., earnings, earnings growth, firm size, research expenditure)
Statistical factors, no basis in finance theory
Fama and French 3-factor Model
Risk factors are:
Firm size
Book-to-market ratio
Excess return on the market portfolio
Carhart added a 4th factor, momentum
These models explain US Equity returns better than the market (single-index) model
Beta
Formula: \(\beta {a}=\frac{Cov(R_{i},R_{market})}{\sigma _{market}^{2}}=\frac{ \rho_{i,market}\times \sigma _{i}}{\sigma _{market}}\)
\( \beta_{market}= \rho_{market,market(\frac{\sigma_{market}}{\sigma{market}})}=1 \)
Beta = 0 means the security's return is uncorrelated with market return.
\( \beta_{portfolio}=\sum_{i=1}^{n}w_i\beta_i\)
average Beta of all assets in the market = 1
Capital Asset Pricing Model (CAPM)
Assumptions
Investors are risk-averse, utility maximizing, and rational
Markets are free of friction like costs and taxes
All investors plan to use the same 1-period time horizon
All investors have the same expectations of security returns
Investment are infinitely divisible
Prices are unaffected by an investor's trades
No inflation and unchanging interest rates
Capital markets are in equilibrium
Investors are price takers
Security Market Line (SML)
is a graphical representation of the CAPM that plots the expected return for any security.
Anything that is not on the SML is currently mispriced
Buy stock that is staying above the SML line(since it has lower level of risk relative to the amount of expected return), and sell those are below the SML line
SML vs. CML
CML
CML is based on total market risk (horizontal axis is \( \sigma\)
only 2 assets are plotted on CML
slope of CML is Sharp ratio
SML
all assets are plotted on SML
SML is based on systematic risk (horizontal axis is \( \beta\)
Expected return
\(E(R_{i})= R_{f}+\beta_i[E(R_{m})-R_{f}] \)
Beta is a measure of systematic risk
In equilibrium, required return = expected return on an asset
\([E(R_{m})-R_{f}] \) is market risk premium
Applications
CAPM
An analyst can compare his expected rate of return on a security to the required rate of return indicated by the SML to determine whether the security is over-, under- or properly valued
Performance evaluation: Analyze risk and return of an active manager's portfolio
Attribution analysis: Analyze sources of the difference between an active manager's portfolio return and the benchmark portfolio's returns
CAPM and SML indicate what a security's equilibrium required rate of return should be, based on the security's exposure to market risk.
Sharpe Ratio
Measures the excess return per unit of total risk
is useful for comparing portfolios on a risk-adjusted basis
Formula: Sharpe Ratio = \(\frac{R_{p}-R_{f}}{\sigma _{p}}\)
Higher is better; must be higher than Market's Sharpe Ratio in order to "beat the market"
\(M^2\) (M-two) Ratio
Provides the same portfolio ranking as Sharpe, but is stated in percentage terms
Formula:M-squared Ratio = \( (R_{p}-R_{f})\times \frac{\sigma _{M}}{\sigma _{p}}-\left ( R_{M}-R_{f} \right )\)
\(M^2\) is the extra % of return for a portfolio with the same risk as the market portfolio
used to measure Risk-adjusted Performance (RAP)
if \(M^2\) > 0, you are beating the market
Treynor Measure
Measures a portfolio's excess return per unit of systematic risk.
Formula: Treynor Measure = \(\frac{R_{P}-R_{f}}{\beta _{P}}\)
if your Treynor Measure > Market's Treynor Measure (\(R_{P}-R_{f}\)) → You are beating the market
Jensen's alpha
is the % return above the equilibrium return for a portfolio with beta = \(\beta_p\)
Formula: Jensen's alpha: \(\alpha _{p}= R_{p}-\left [ R_{f}+ \beta _{p}\times \left ( R_{m}-R_{f} \right ) \right ]\)