Reading 52: Portfolio Risk and Return P1

Major return measures

Holding Period Return (HPR)

Arithmetic Mean Return

Geometric Mean Return

Money-weighted Rate of Return

Gross return

Net return

Pretax Nominal Return

After-tax Nominal Return

Real return

Leveraged return

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 $S^2= \frac{\sum_{T=1}^{T}(R_{t}-\bar{R} )^2}{T}$

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}$

Correlation

Standard (unitless) measurement of co-movement that is bounded by (-1) to (1)

Formula: $\rho_{1,2}=\frac{Cov_{1,2}}{\sigma _{1}\times \sigma _{2}}$

Risk Aversion

Risk-aversion = prefer less risk to more risk

Given 2 equal expected return,

Financial assets are priced according to the preference of risk-averse investors.

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} $

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)

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

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

Capital allocation line (CAL)

The total return after deducting commissions and other costs necessary to generate the return, but before deducting management fees

Return after being deducted the management fees.

The numerical % return after considering tax but not inflation.

The numerical % return without considering tax and inflation.

the increase in investor's purchasing power

Gain/loss on an investment as % of investor's cash investment.

$HPR=\frac{P_{t}-P_{0}+Dividend_{t}}{P_{0}}$

to measure an investment's return over a specific period.

$\bar{R}= \frac{\sum_{t=1}^{n}R_{t}}{n}$

Simple average of a series of periodic return

$R_{GM}= \prod_{i=1}^{n}(1+HPR_{i})-1$

Compounded annual rate

is the IRR calculated using periodic cash flow into and out of an account,

Time-weighted Rate of Return

are effective annual compound returns

$ CF_0+\frac{CF_1}{1+MWR}+...+\frac{CF_N}{(1+MWR)^N}=0 $

$Annual.TWR=\prod_{i=1}^{n}\left [ \frac{End.Value_i}{Begin.Value_i} \right ] -1=\prod_{i=1}^{n}(1+HPR_{i})-1$

More appropriate to evaluate manager's performance, as manager cannot decide when the fund comes in or out

Mean: Centre of distribution

Tells about the volatility

Standard deviation = Square root of Variance

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}$

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.

Given 2 investment with same risk, investors prefer the one with higher return

Investors do not minimize risk. It's a trade-off

When $ \rho=1;\sigma _{portfolio}=\sigma_1\times w_1 + \sigma_2\times w_2 $

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

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

When the Indifference Curve just tangent to the CAL, that tangent point is the optimal portfolio

bias toward the return of the year where there are hugh cash inflow/ outflow

Annualized Return:

$R_{annual}=(1+R_{m})^{\frac{1}{m}}-1 $

m: Number of compound periods within 1 year

$ R_{real}=\frac{1+R_{nominal}}{1+inflation}-1$

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

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.