# 9. Mutual funds and performance evaluation

## Quick Recap

Capital Allocation Line – Line created on a graph of all possible combination of risk-free and risky assets, and slope of which is known as reward-to-risk ratio.

• Used to choose how much to invest in a risk-free 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 well-diversified 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

1. Front-end load – A fee charged when you buy the fund
• Front-end load does NOT affect NAV.

$$\text{Offer}_{t=0} = \frac{\text{NAV}_0}{1-\text{front-end load}}$$

1. Back-end load – A fee charged when you sell the fund
• Back-end load does NOT affect NAV.

$$\text{Redeem}_{t=1} = \frac{\text{NAV}_1}{1-\text{back-end load}}$$

1. Expense ratio – % of NAV each year
• The expense is calculated on the increased NAV after the front-end 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

1. Small investors select one portfolio (Entire-wealth portfolio).

• Select portfolio with the highest sharpe ratio
2. 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 risk-free rate
• $\beta_p$ Beta of the portfolio
1. 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:

1. Beta equals zero, so the fund is not exposed to market risk
2. 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
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