Mutual funds and performance evaluation
Table of Contents
Quick Recap
Capital Allocation Line – Line created on a graph of all possible combination of riskfree and risky assets, and slope of which is known as rewardtorisk ratio.
 Used to choose how much to invest in a riskfree asset and one or more risky assets
 Based on the investor’s risk tolerance, these allocations can be made
Capital Asset Pricing Model
Model that describes the relationship between systematic risk and expected return for assets (stocks).

Security Market Line – Visualization of CAPM, showing the relationship between risk (measured by beta) and expected return.
 An investment evaluation tool derived from the CAPM
 If expected return of a stock is above the SML, then the stock is undervalued or underpriced; Buy the stock
 If expected return of a stock is below the SML, then the stock is overvalued or overpriced; Sell the stock

Capital Market Line – CAL where the risk portfolio is the market portfolio; Slope of the CML is the sharpe ratio of the market portfolio; Risk is measured by standard deviation
 Intercept point of CML and efficient frontier would result in the most efficient portfolio called the tangency portfolio.
A. Mutual funds
A mutual fund is a portfolio of financial securities. Many investors (typically investors) provide capital and a professional manager invests this ‘pool’ of capital in financial securities including stocks, bonds, money markets, etc.

Passive management – Invest in a welldiversified portfolio without searching for security mispricing.
 Examples include index funds, ETFs, etc.
 Assumes the efficient market hypothesis is true

Active management – Identifying the “mispriced” securities to beat the market
 Assumes the efficient market hypothesis is false
1. Net Asset Value (NAV)
NAV is the price per share of a mutual fund.$$\text{NAV} = \frac{\text{Market Value of Assets}  \text{Liabilities}}{\text{Number of Shares Outstanding}}$$
 Liabilities: Unpaid expenses, management fees, etc.
2. Mutual fund fees
 Frontend load – A fee charged when you buy the fund
 Frontend load does NOT affect NAV.
$$\text{Offer}_{t=0} = \frac{\text{NAV}_0}{1\text{frontend load}}$$
 Backend load – A fee charged when you sell the fund
 Backend load does NOT affect NAV.
$$\text{Redeem}_{t=1} = \frac{\text{NAV}_1}{1\text{backend load}}$$
 Expense ratio – % of NAV each year
 The expense is calculated on the increased NAV after the frontend load
Always note the following:
 Make sure to subtract the expense ratio from the return; this return the current NAV
Calculating Returns
$$\text{Return}_{fund}=\frac{\text{Redeem}  \text{Offer} + \text{Distributions}}{\text{Offer}}$$
 what is distribution??
 Income  if mutual fund includes stocks, then it includes dividends or bond can payout coupons
$$\text{Return_fund}=\frac{\text{NAV}(1+\text{cap gain})(1\text{exp ratio}) }{N}1$$
B. Portfolio Performance Evaluation
1. Risk Model: Jensen’s Alpha
$$\alpha_P = \bar{R}_P  \beta_P \bar{R}_M$$
 $\alpha_P$ Portfolio alpha
 $\bar{R}_P$ Average excess return on the portfolio
 $\beta_P$ Beta of the portfolio
 $\bar{R}_M$ Average excess return on the market
2. Mutual Fund Performance
If markets are efficient, then before expenses, an average mutual fund has $\alpha = 0$.
 Across all fund managers, the average $\beta$ is 0
C. Selecting Funds/Portfolios in Practice

Small investors select one portfolio (Entirewealth portfolio).
 Select portfolio with the highest sharpe ratio

Large investors hold many funds.
 Select funds using the
Treynor ratio
:
$$\text{Treynor ratio} = \frac{\bar{r_p}  \bar{r_f}}{\beta_p}$$
 $\bar{r_p}$ Average return on the portfolio
 $\bar{r_f}$ Average riskfree rate
 $\beta_p$ Beta of the portfolio
 Adding an actively managed portfolio: Information Ratio
 An actively managed portfolio delivers the benefit of $\alpha$, but adds idiosyncratic risk to our passive benchmark portfolio.
$$\text{Information ratio} = \frac{\alpha_p}{\sigma(e_p)}$$
 $\alpha_p$ per unit of unsystematic risk
 $\sigma(e_p)$ Standard deviation of the $e_p$ from an index model: $R_p = \alpha_p + \beta_p R_m + e_p$
Information Ratio is often used to evaluate hedge funds.
Hedge funds attempt to follow a market neutral strategy:
 Beta equals zero, so the fund is not exposed to market risk
 Alpha is positive
Performance Measure  Application 

Sharpe Ratio  To select one fund: for use as the optimal risky portfolio 
Treynor Ratio  Select fund of funds: for many portfolios 
Information Ratio  Add to benchmark: For adding an active fund to an existing passive benchmark 