Mutual funds and performance evaluation

Table of Contents

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
Open-end fund: Assume frontend load

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