Chapter 7: Performance Measures

Dollar Weighted Return

A dollar-weighted return provides a return between two points in time, incorporating all the cash in-flows, such as dividends, interest, and contributions, and all cash out-flows, such as withdrawals, from the portfolio. The dollar-weighted return differs from the simple arithmetic return and the time-weighted return because it gives more weight to periods where more funds are in the account or portfolio. For example, imagine a retiree who is withdrawing funds from his account, slowly depleting the account. The dollar-weighted return will be weighted more toward the earlier days of the account and less toward the latter days, because there were more funds in the account in the early days. If the performance in the account was worse at the earlier date, but then improved over time, the dollar-weighted return will weight the earlier performance greater because more money was in the account. But this does not accurately reflect the portfolio manager's actual performance. For this reason, portfolio managers typically do not report dollar-weighted average returns. They can be useful, however, in determining whether a customer will meet some financial goal by a specified time. Conceptually, a dollar-weighted return is the internal rate of return on an investment. It is calculated in the same manner as the internal rate of return on any investment, by figuring the return that is necessary to achieve a net present value of 0 given the increase in the investment, and all cash in-flows and out-flows.

Time-Weighted Returns

A time-weighted return is used to measure the compounded rate of growth in a portfolio. Conceptually, the time-weighted return is the compounded growth rate of the initial investment over a given period of time. This return is calculated using the geometric mean, rather than the arithmetic mean. Unlike the dollar-weighted average, as discussed below, the time-weighted return gives a return that does not incorporate deposits or withdrawals into the account. For this reason, it does not weight the return toward periods when the account has more money in it. Because the time-weighted return is free of the biases of dollar-weighted averages, it is the standard measure used by mutual fund portfolio managers to report performance in their fund prospectuses. It is also used by most investment advisers when reporting the returns of a client's portfolio.

After-Tax Return

An after-tax return removes the money that will be paid in taxes before the return is calculated.

Stan bought 1,000 shares in ABC Company at $50 per share. Two years later, he sold the shares at $40 per share. The company paid dividends of $2 per share per year. What was Stan's annualized total return on his investment?

Answer: -6% Explanation: Loss on XYZ = $40,000 - $50,000 = -$10,000 XYZ dividends = $2 x 2 years x 1,000 shares = $4,000 total return = (appreciation or loss in value + dividends and interest) / original investment total return = (-$10,000 + $4,000) / $50,000 = -12% annualized total return = -12% / 2 years = -6%

Jake bought 1,000 shares of XYZ Company at $25. Two years later, Jake sold his shares at $30. In that period, the company paid a $.25 quarterly, per share dividend. Calculate Jake's annualized total return, not taking into account compounded interest.

Answer: 14% Explanation: Appreciation on XYZ = $30,000 - $25,000 = $5,000 XYZ dividends = $.25 x 8 quarters x 1,000 shares = $2,000 total return = (appreciation or loss in value + dividends and interest) / original investment total return = ($5,000 + $2,000) / $25,000 = 28% annual total return = 28% / 2 years = 14%

Debby invested in Triple Threat Corporation last year. Her total return for the year was 10%. The country that Debby lives in experienced deflation of 4% for the year. What was Debby's real rate of return?

Answer: 14% Explanation: Deflation is a negative inflation rate, so in this question the inflation rate would be -4%. real rate of return = total return - inflation rate = 10% - (-4%) = 10% + 4% = 14%

Stan purchased 100 shares of a stock for $20 and sold them for $25. Stan's holding period was one year and one day. During the year, the stock paid $.50 in dividends per share. These are qualified gains. Stan is in a 30% tax bracket, and his long-term capital gains rate is 15%. What is Stan's after-tax return, ignoring any compounded interest?

Answer: 23.4% Explanation: Original Investment = $2,000 value of investment at sale = $2,500 dividends = $.50 x 100 shares = $50 Stan's investment has appreciated by $500 before taxes. His tax rate is 30%, but long-term capital gains are taxed at 15%, so we multiply his gain by (1 - 0.15) to find his after-tax gain (500 x (1 - 0.15) = $425). His dividends are qualified, so they will be taxed at the same rate as his long-term capital gains rate, 15%. Stan's after-tax dividends will be $50 x (1 - 0.15) = $42.50 Thus, Stan's after-tax return = ($425 + $42.50) / $2,000 = 23.4%

Given the following assumptions for a particular portfolio, what would the portfolio's alpha be using the capital asset pricing model to calculate expected return? Actual return: 13%, riskless rate: 1%, expected return on general stock market: 7%, beta of portfolio: 1.5, Sharpe ratio: 2.

Answer: 3% Explanation: The formula for the capital asset pricing model is given by the following: return on stock = riskless rate + beta of stock x (return on market - riskless rate). Plugging in for the portfolio gives an expected return of 10%, (1% + 1.5 x (7% - 1%)). Alpha is calculated by subtracting the actual rate of return minus the expected return (13% - 10% = 3%). Note the Sharpe ratio is not used in the CAPM formula.

Stan Smith purchased a bond at par with a nominal yield of 5%. Stan sold the bond one year later at 102. During this time inflation was 3% and the value of the dollar fell by 2%. What was Stan's real rate of return on the bond?

Answer: 4% Explanation: Stan earns $20 in appreciation on the value of the bond ($1,020 - $1,000 = $20). Stan earns $50 of interest on the bond (0.05 x $1,000 = $50). His return on the bond is ($20 + $50) / $1,000 = 0.07 = 7%. Stan's real rate of return on the bond = 7% - 3% = 4%.

According to the capital asset pricing model, if the risk-free rate of return is 2%, the market rate of return is 6%, and the investment's beta is 1.5, what is the expected rate of return of the investment?

Answer: 8% Explanation: To calculate the expected rate of return according to the CAPM, subtract the risk-free rate from the market rate to get the market premium, then multiply the market premium by the beta and add that product to the risk-free rate to get the expected rate of return of the investment. In this case 0.02 + (1.5 x (0.06 - 0.02)) = 0.08, or 8%.

Gary Smith purchased a bond for 96 and sold it at 98 exactly one year later. The bond paid a nominal yield of 10%. Gary is in a federal tax bracket of 25% and he pays 5% in state taxes. What was Gary's after-tax return on the bond?

Answer: 8.75% Explanation: Gary purchased his bond for $960 and sold it for $980, so he earned a $20 gain in appreciation on the value of the bond. Gary also earns $100 in interest on the bond (0.10 x $1,000). Gary's $20 gain will be taxed at his ordinary tax rate, because he has only held it for one year, not over a year. Gary's interest on the bond will also be taxed at his ordinary income rate ($20 + $100) x (1 - 0.30) = $120 x 0.70 = 84. Gary's after tax return = $84 / $960 = 0.0875 = 8.75%.

Based on historical information, Big Brokerage has calculated that an investment in XYZ Corporation's common stock has a 40% chance of returning 10% and a 60% chance of returning 8%. What is the expected return on an investment in shares of XYZ Corporation?

Answer: 8.8% Explanation: expected return = (0.40 x 0.10) + (0.60 x 0.08) = 0.040 + 0.048 = 0.088 = 8.8%

At the beginning of the year, your client, Mrs. Meyer, had a portfolio with an expected return of 6% and a standard deviation of 2. At the end of the year, her actual return showed a positive alpha of 3%. What was her actual return on her portfolio?

Answer: 9% Explanation: The alpha level represents the difference between the expected return and the actual return. Alpha levels can be both positive and negative. In this example, expected return + alpha level = actual return (6% + 3% = 9%).

Bond X has a duration of 10. Bond Y has a duration of 8. Which of the following are true? I. If interest rates fall by 1%, Bond X's price will decrease by about 10%. II. If interest rates rise by 1%, Bond Y's price will fall by about 8%. III. Bond X is more sensitive to interest rate changes than Bond Y. A. I only B. I and III C. II and III D. I, II, and III

Answer: C Explanation: Duration measures the sensitivity of a bond's price to change in interest rates. A duration of 10 means that a 1% change in interest rates will result in about a 10% change in price. Since price and interest rates have an inverse relationship, for Bond X, a 1% decrease in interest rates will result in a 10% increase in price, not a 10% decrease in price. Statement I is therefore false. For Bond Y, with a duration of 8, if interest rates rise by 1%, the price will fall by about 8%, so Statement II is true. Statement III is also true; since Bond X has a higher duration than Bond Y, it is more sensitive to interest rate changes.

Alpha

As discussed above, alpha is a measure of how much better a portfolio of securities performs than what could be expected from the market. A portfolio's alpha is typically used to evaluate a fund manager's performance. An alpha of 2% means that the manager was able to add 2 percentage points more to the portfolio return than what would be expected if the fund was not actively managed. An alpha of -3% means that the manager actually cost the portfolio 3% compared to the benchmarks it is measured against. To calculate alpha, subtract the expected return of a portfolio from its actual return. When one is using alpha to evaluate a portfolio, expected returns for each stock are often calculated using the stock's beta or the capital asset pricing model. alpha = portfolio's actual return - portfolio's expected return

Beta

Beta coefficients are also used to assess the systematic risk of individual securities, meaning the beta coefficient will tell us how much the price of a security can be expected to be influenced by movement in the entire market. We will begin our discussion of beta coefficients by examining their usefulness when evaluating individual securities and then move from there to a discussion of how they can be applied to portfolios. A beta coefficient is a measure of the volatility of a security relative to the overall market. While betas can be used to assess the performance of any type of asset, they are most commonly used to study the performance of stocks. If a stock has a beta of 1.0, it means that the stock moves up and down in lockstep with the market. Thus, if there is a 5% increase in the market, an investor can expect to see around a 5% increase in the price of that security. If a stock has a beta of 1.5, it means that a stock moves up or down at a rate that is 50% greater than the market. If the market goes up 10%, this stock historically has gone up by 15%. If the market has declined by 10%, the stock has historically shown a decline of around 15%. Stocks that have betas that are greater than 1.0 are more volatile than the market and, hence, carry more systematic risk than the market.

Central Tendency

Central tendency is a single value that summarizes a set of scores. For investors, central tendency is often used to summarize a set of returns. For example, an investor may wish to know the most representative return from a set of annual returns. The mode, median, and mean are all measures of central tendency.

Correlation

Correlation refers to the amount of relationship between the two variables being measured. Correlation can be measured through a correlation coefficient, which ranges in value between -1 and 1. A coefficient of 1 indicates the two securities are perfectly correlated, meaning if the first security increases by a certain percentage, the other security increases by the exact same percentage. A coefficient of -1 means that if the first security increases by a certain percentage, the other will decrease by that same percentage. At 0, there is no correlation, meaning the securities move independent of each other.

Current Yield

Current yield is calculated by dividing the annual interest paid on the bond by the purchase price of the security. For example, a bond that pays a 5% coupon rate, or $50 per year, and was purchased for $900 would have a yield of 5.6% (5.6% = $50 / $900).

Duration

Duration is a measure in years of how long it takes for the price of a bond to be repaid by its cash flows. Bonds with lower coupon rates and longer maturities will have higher durations, because it takes more time for investors to earn back the money they invested. Zero coupon bonds don't pay any cash flows until maturity, so they have the highest durations, and their durations are always equal to the length of maturity. Conceptually, duration measures how sensitive a bond's price is to changes in interest rates. The prices of bonds with higher durations are more sensitive to interest rate changes than the prices of bonds with lower durations. To understand this, start by thinking about how much money a bond is paying out in total. Some of the money will be paid at maturity and some will be paid in interest along the bond's life. With respect to interest rate risk, the more money you receive in interest payments early on, the better, because once a payment is made, the money is no longer subject to interest rate risk. If most of your money is received at the end of the bond's life as in zero coupon bonds, it is subject to a lot of interest rate risk so the duration is higher. Duration can be used to calculate how a bond's price will change with a change in interest rates. A duration of 12 means that a 1% change in interest rates will result in about a 12% change in price. Since price and interest rates have an inverse relationship, a 1% decrease in interest rates will result in a 12% increase in price, and a 1% increase in interest rates will result in a 12% decrease in price.

Rick invested in Combo Corporation last year. His total return for the year was 8%. The CPI was 3%. What was Rick's real rate of return?

Explanation: Real rate of return = total return - inflation rate = 8% - 3% = 5%

What are the main points you need to remember about time-weighted returns?

For the exam, remember the following points about time-weighted returns: • Mutual fund portfolio managers will usually report the time-weighted return in their fund prospectuses over the dollar-weighted return. • Investment advisers typically report the time-weighted return to their clients, because it is a better reflection of the adviser's true performance. • Time-weighted returns, because they incorporate compounding, rely on the geometric mean. • Time-weighted returns exclude the effects of cash in-flows (contributions) and cash out-flows (withdrawals).

Expected Return Using Probability Theory

In lieu of a crystal ball, an investment adviser who wants to try to predict future returns may use probability theory to calculate an expected return. Expected returns differ from the returns that we have discussed thus far, because they are forward-looking and, hence, unknown, as opposed to historical returns, which are known. One common method for determining an expected return is to use historical data to develop probabilities of different returns. These probabilities are combined to come up with a composite expected return. expected return = (probability x return) + (probability x return)

Nominal Yield

Nominal yield is the percentage of the face value of a security (the amount the bond will be redeemed at) that is paid out on the security. The nominal yield is also called the coupon rate. The nominal yield and face value of the bond do not change over time. For example, a bond that pays $70 per year and has a face value of $1,000 would have a nominal yield of 7% (7% = $70 / $1,000). Relatedly, a bond with a coupon rate of 5% would pay annual interest of $50 (0.05 x $1,000 = $50). Bonds typically have a face value of $1,000, meaning they pay out a principal of $1,000 at maturity. But a bond is often traded at either a premium (e.g., $1,100) or a discount (e.g., $980) to its face value. A bond's current yield incorporates whether the bond was purchased at either a premium or discount and reflects this additional gain or loss. A bond that is purchased at a discount will have a higher current yield than a bond purchased at a premium.

Holding Period Return

Perhaps the simplest type of return is a holding period return. A holding period return measures the return on an investment over the time period that the investment was held. The time period may be less than a year or more than a year. The important fact to remember is that a holding period return is not annualized, so the investor must be careful when comparing holding period returns.

What are the portfolio benchmarks PMs use to best represent their portfolio?

Portfolio managers try to choose benchmarks that best represent the investments in a portfolio. • The smallest broad-based index is the Dow Jones Industrial Average. It is the best known and oldest equity index in the U.S., but it tracks just 30 stocks. This is the best benchmark for a portfolio filled with blue chip stocks. • The S&P 500 is an index of 500 representative large-cap, publicly traded companies drawn from the primary sectors of the U.S. economy. The S&P 500 is considered the best gauge of large cap stock performance in the United States. It is also believed to be the best representation of the overall market. • The largest broad-based index is the Wilshire 5000. This is a super-composite index that measures the performance of all U.S. headquartered companies whose stock is traded on the New York Stock Exchange, American Stock Exchange, and NASDAQ. Despite the name, the index currently tracks substantially more than 5,000 stocks. • The Russell 2000 tracks 2,000 small-cap companies, and is commonly used as a guide to small cap performance. • Treasuries are often the bond portfolio benchmark.

Risk-Adjusted Return

Risk is another critical factor when assessing the performance of a security. In the context of investing, risk is defined as the volatility in the returns of a security. When volatility is high, the returns vary from high to low, so investors open themselves up to the possibility of bigger losses and bigger gains. Imagine two bonds that have equivalent returns at the end of the same time period, but one bond is a high-yield junk bond and the other is an AAA bond. Most investors would judge the AAA bond to be the better investment. In fact, investors require compensation for risk, which is why junk bonds offer higher yields than ultra-safe Treasury bonds. When one is choosing from a myriad of possible investments it can be important to determine a risk-adjusted return. The various risk-adjusted returns attempt to adjust for risk by mathematically normalizing it. Thus, the goal of calculating risk-adjusted returns is to again be able to compare investments on an "apples to apples" basis. Several key measures are used to calculate risk-adjusted returns.

Stock A has a return of 10% and a historical SD of 2, whereas Stock B has a return of 15% and a historical SD of 4. Three-month Treasury bills are currently returning 3%, and the S&P returned 10% last year. Using the Sharpe ratio, which stock has the best risk-adjusted return?

Sharpe ratio of Stock A = (10% - 3%) / 2 = 3.5% Sharpe ratio of Stock B = (15% - 3%) / 4 = 3.0% Hence, Stock A has the higher Sharpe ratio, and therefore is the better investment after adjusting for risk.

Sharpe Ratio

The Sharpe ratio is one of the most well-known measures of risk-adjusted return. The Sharpe ratio rewards investments that show superior returns without taking excessive risk. Thus, high Sharpe ratios suggest high performance investments with low risk, while low Sharpe ratios suggest that performance may have been due to higher risk. In the numerator of the formula above, the riskless rate of return is subtracted from the investment's actual return. The riskless rate of return is the return on a kind of investment with, essentially, no risk. This is usually the rate of three-month Treasury bills, or 10-year Treasury notes. Also, note that standard deviation is in the denominator of the formula, indicating that when an investment has a higher SD, it will have a lower Sharpe ratio. In contrast, investments that have lower SDs will have higher Sharpe ratios.

William purchased 300 shares of Regal Robots Corporation at $100 per share. Five years later, he sold the shares at $200 per share. What was William's holding period return? What is his annualized return?

The answer is: 100%. Following the formula above we get: original investment = 300 x $100 = $30,000 investment's value at sale = 300 shares x $200 = $60,000 William's holding period return = ($60,000 - $30,000) / $30,000 = 100% William held his investment for five years, so we must divide his return by five to get the annual return (100% / 5 years = 20% per year).

Sharon purchased 300 shares of Rubber Robots Corporation at $100 per share. Six months later, she sold the shares at $125 per share. What was Sharon's holding period return? What is her annualized return?

The answer is: 25%. Following the formula above we get: original investment = 300 x $100 = $30,000 investment's value at sale = 300 shares x $125 = $37,500 Sharon's holding period return = ($37,500 - $30,000) / $30,000 = 25% • Remember: For holding period return, the time period that the investment is held is irrelevant and, therefore, is not included in the formula for calculating it. Sharon held her investment for half a year, so we must divide her return by 0.5 to get her annual return (25% / 0.5 years = 50% per year).

Inflation-Adjusted Rate of Return (Real Rate of Return)

The inflation-adjusted return, also known as the real rate of return, measures the difference between investment return and inflation. This type of return assesses an investment's purchasing power, meaning whether the investment is able to keep up with the change in the cost of goods and services. real rate of return = total return - inflation rate

R-Squared

This measure represents an attempt to determine how much of a portfolio's performance could be explained by a portfolio's beta as described above. For a stock portfolio, R-squared would assess how much of the portfolio's performance could be explained by the performance of the stock market. R-squared is measured from 0 to 1.00, with a score of 1.00 meaning that 100% of the movement of the portfolio can be attributed to a movement in an index or benchmark and the securities' betas relative to the benchmark. A typical R-squared may be 0.60, which would mean that 60% of the portfolio's performance could be explained by the performance of the market and the specific stocks' risk exposure. The remaining 40% may be due to a portfolio manager's skill at picking stocks that will outperform expectations (or lack of skill) or simply luck. Note that R-squared does not tell us whether the portfolio manager performed well or not; it simply tells us how much of the performance of the portfolio could be explained by the performance of the broader market.

Yield to Maturity

Yield to maturity measures the annualized yield of a bond assuming that the bondholder will hold the bond to maturity. Thus, while current yield compares bonds based on their coupon rates and their purchase prices, yield to maturity compares bonds on these measures, but also takes into consideration how long the bonds will be held by the investor. In fact, the yield to maturity involves four variables in its calculation: the face value of the bond, the purchase price of the bond, the coupon rate, and the number of years to maturity. Because yield to maturity incorporates the lifetime of the investment, it gives a more useful assessment of the yield than either current yield or nominal yield. Bonds are typically quoted in yields to maturity.

Geometric Mean

geometric mean for a set of numbers = ((1 + annual return) x (1 + annual return))1/t - 1