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Reading 52: Portfolio Risk and Return P1 (Major return measures (Money…
Reading 52: Portfolio Risk and Return P1
Major return measures
Holding Period Return (HPR)
\(HPR=\frac{P_{t}-P_{0}+Dividend_{t}}{P_{0}}\)
to measure an investment's return over a specific period.
Arithmetic Mean Return
\(\bar{R}= \frac{\sum_{t=1}^{n}R_{t}}{n}\)
Simple average of a series of periodic return
Geometric Mean Return
\(R_{GM}= \prod_{i=1}^{n}(1+HPR_{i})-1\)
Compounded annual rate
Money-weighted Rate of Return
is the IRR calculated using periodic cash flow into and out of an account,
\( CF_0+\frac{CF_1}{1+MWR}+...+\frac{CF_N}{(1+MWR)^N}=0 \)
bias toward the return of the year where there are hugh cash inflow/ outflow
Gross return
The total return after deducting commissions and other costs necessary to generate the return, but before deducting management fees
Net return
Return after being deducted the management fees.
Pretax Nominal Return
The numerical % return without considering tax and inflation.
After-tax Nominal Return
The numerical % return after considering tax but not inflation.
Real return
the increase in investor's purchasing power
\( R_{real}=\frac{1+R_{nominal}}{1+inflation}-1\)
Leveraged return
Gain/loss on an investment as % of investor's cash investment.
Time-weighted Rate of Return
are effective annual compound returns
\(Annual.TWR=\prod_{i=1}^{n}\left [ \frac{End.Value_i}{Begin.Value_i} \right ] -1=\prod_{i=1}^{n}(1+HPR_{i})-1\)
More appropriate to evaluate manager's performance, as manager cannot decide when the fund comes in or out
Annualized Return:
\(R_{annual}=(1+R_{m})^{\frac{1}{m}}-1 \)
m: Number of compound periods within 1 year
Major asset classes for creating a portfolio
Small-capitalization stocks
Large-capitalization stocks
Long-term corporate bonds
Long-term Treasury bonds
Treasury bills
Mean, Variance, Covariance, Correlation of Asset Return
Variance \(\sigma ^2\)
Calculable when knowing the Period returns, Total number of periods, and Population Mean
Population Variance
\(\sigma ^2= \frac{\sum_{t=1}^{t}(R_{t}-\mu )^2}{t}\)
Sample Variance
\(S^2= \frac{\sum_{T=1}^{T}(R_{t}-\bar{R} )^2}{T}\)
Tells about the volatility
Standard deviation = Square root of Variance
Covariance
Say how 2 variables move together overtime
Positive covariance means variables (ex: rates of return) tend to move together. Negative covariance means variables tend to move in opposite direction. Zero covariance means no linear relationship
Sample Covariance: \( Cov_{1,2}=\frac{\sum_{t=1}^{n}\left [ \left ( R_{t,1}-\bar{R_{1}} \right )\times\left ( R_{t,2}-\bar{R_{2}} \right )\right ]}{n-1}\)
Population Covariance: \( Cov_{1,2}=\frac{\sum_{t=1}^{n}\left [ \left ( R_{t,1}-\bar{R_{1}} \right )\times\left ( R_{t,2}-\bar{R_{2}} \right )\right ]}{n}\)
Correlation
Standard (unitless) measurement of co-movement that is bounded by (-1) to (1)
Formula: \(\rho_{1,2}=\frac{Cov_{1,2}}{\sigma _{1}\times \sigma _{2}}\)
Mean: Centre of distribution
Risk Aversion
Risk-aversion = prefer less risk to more risk
Given 2 equal expected return,
the risk-averse investor will choose the less-risky investment,
risk-seeking investor will choose the more-risky one
risk-neutral investor will not care.
Financial assets are priced according to the preference of risk-averse investors.
Given 2 investment with same risk, investors prefer the one with higher return
Investors do not minimize risk. It's a trade-off
Utility function
\( U=E(r)-\frac{1}{2}\times A \times \sigma^2\)
U: Utility
E(r): Expected return
A: Risk-aversion measure, aka. marginal return for investor to accept more risk
\( \sigma^2\): Variance of investment
to decide which investment is best for who
Choose investment with highest Utility point
Portfolio standard deviation
\( \sigma _{portfolio} = \sqrt{(W_{1}^{2}\times \sigma_{1}^{2} ) + (W_{2}^{2}\times \sigma_{2}^{2} ) + 2\times W_{1}\times W_{2}\times \sigma _{1} \times \sigma _{2}\times \rho_{1,2} }=\sqrt{(W_{1}^{2}\times \sigma_{1}^{2}) + (W_{2}^{2}\times \sigma_{2}^{2}) + 2\times W_{1}\times W_{2}\times Cov_12} \)
When \( \rho=1;\sigma _{portfolio}=\sigma_1\times w_1 + \sigma_2\times w_2 \)
Investing in less-than-perfectly-correlated assets
Assets' returns are perfectly correlated (correlation = 1) → Greatest portfolios risk
The lower the correlation of asset return, the greater risk reduction (benefit of diversification)
Theoretically if we can find 2 assets with correlation = -1, we can create a risk-free portfolio with positive return. that is why alternative investment exist in portfolio
Minimum Variance, Efficient Frontiers of Risky Assets & Global Minimum Variance Porforlio
Minimum Variance Portfolio (MVP):
the portfolio with the least risk, for each level of expected return.
Minimum-variance Frontier
A line form by all the minimum-variance portfolios.
Global minimum-variance portfolio: is the portfolio on Minimum-variance frontier with the lowest standard deviation. It has the least risk (furthest to the left) on a risk-return graph.
Markowitz Efficient Frontier
A line formed by portfolios with highest expected return for each level of risk.
Efficient Frontier coincides with the top portion of the minimum-variance frontier.
Risk-averse investors would only choose a portfolio lying on the efficient frontier.
Optimal Portfolio Selection & Capital Allocation Line
Indifference curve
A curve plots combinations of risk and expected return that an investor finds equally acceptable.
Generally slopes upward (as investor will only take on more risk if being compensated with greater expected return)
Optimal Portfolio
When the Indifference Curve just tangent to the CAL, that tangent point is the optimal portfolio
Capital allocation line (CAL)
Risk averse investor will invest more on risk-free asset, and less on risky asset in the portfolio
when we put one risk-free asset and optimal risky asset into a portfolio, the risk/return of each possible weight combination is referred to as CAL
