To create a numerical representation of risk, we often use variability of returns as a proxy for measuring uncertainty. By using the variability in an investment product’s historical return stream, we can create a variety of simple statistical measures that will allow us to put performance in perspective.
Return Histogram
A simple way to measure the variance of returns is to create what is known as a histogram. When using this method, you select a series of return ranges, such as the ones set in Exhibit 5.1, and then count the number of returns that fall within each range.
Both the table and graph in Exhibit 5.1, which represent CAM’s his- torical performance, illustrate that 49% of the portfolio’s monthly returns have been greater than +5% or less than –5% on a monthly basis. The portfolio’s excellent long-term performance record can be attributed largely to the fact that its monthly return has exceeded 5% on 21 occasions (32% of the time).
Standard Deviation
Standard deviation is the most popular method of measuring an invest- ment’s variability in performance. This statistical measure is designed to first calculate the average return of an investment product over a period of time and then to calculate the typical (standard) difference (deviation) from the average. As a result, we can think of standard deviation as a mea- sure of an investment product’s historical return dispersion.
Formula 5.1 (Cumulative Standard Deviation)
Standard Deviation IR PA
=
∑
( − )2N
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EXHIBIT 5.1Return Histogram % of RangesCountTotal >5%2132% 4% to 5%58% 3% to 4%23% 2% to 3%12% 1% to 2%69% 0 to 1%35% –1% to 035% –2% to –1%58% –3% to –2%35% –4% to –3%58% –5% to –4%12% <–5%1117%
Positive Returns Negative Returns 05
10
15
20
25
> 5%
4% to 5%
3% to 4%
2% to 3%
1% to 2%
0 to 1%
–1% to 0 –2% to –1%
–3% to –2%
–4% to –3%
–5% to –4%
< –5%
80
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where IR = Portfolio’s individual monthly return. The returns can be based on daily, monthly, or quarterly performance periods or on any period you can consistently apply. It is important that the measurement period be consistently applied.
PA = Portfolio’s average return, based on the following formula:
N = the total number of return observations.
Formula 5.1 represents the formula for standard deviation based on what is known as a “population” as opposed to a “sample.” The only dif- ference between the two types of standard deviation is that the calculation based on the population uses the product’s entire performance history, whereas the sample calculation is based on some subset of the product’s history. The only technical adjustment in the formula is that the denomina- tor changes from “N” to “N– 1.”
As with many statistical measures, the longer the data set the more sta- tistically significant the result. If, for example, we were to calculate the standard deviation for a particular investment product that has a history of just one year (12 months), we would be much less confident about the re- sults. This is because over shorter time periods, one or more outlier returns might have a larger impact on the overall calculation than is appropriate.
As a general rule, a data set that consists of a minimum of 20 data points (period returns) is a good starting point. Anything less and you should take the results with a grain of salt. Once an investment product’s performance history goes past three years (36 months), the results become more meaningful.
In Exhibit 5.2, we have used the historical performance record of CAM’s small-cap value product and calculated its standard deviation. The actual monthly performance numbers are shown on the left of the table, while the calculation for standard deviation is broken out in to steps on the right side of the table. As the exhibit illustrates, the cumulative standard deviation for this product is 6.49. But what does this figure actually mean and how do we interpret it?
Before we can analyze the results we need to make sure that our results are comparable to the performance figures calculated in the previous chap- ter. As you recall, the most efficient and effective way to measure the return of an investment product with a history greater than one year is to annual- ize the performance. The same holds true when we look at the risk side of the equation. To properly assess the variability of historical returns, we first need to annualize the standard deviation figure. The formula used to annualize standard deviation is highlighted in formula 5.2.
PA IR to
=
∑
1 NN
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EXHIBIT 5.2Example of Standard Deviation Using CAM’s Performance CAM’s Small-Cap Value Portfolio MonthlyDeviationAbsolute Value Returnsfrom Averageof DFA Squared Date(MR)(DFA)(SDFA)To calculate the cumulative standard deviation Jan-98–1.23%–2.22%4.94Step One:Determine average return=0.99 Feb-988.34%7.35%53.98 Mar-985.67%4.68%21.87Step Two:Calculate DFA for each monthSee table Apr-981.21%0.22%0.05 May-98–5.21%–6.20%38.48Step Three:Calculate SDFA for each monthSee table Jun-981.89%0.90%0.80 Jul-98–8.34%–9.33%87.11Step Four:Sum all the SDFAs=2,779.64 Aug-98–19.45%–20.44%417.92 Sep-986.45%5.46%29.78Step Five:Divide figure in step four Oct-984.89%3.90%15.19by total number of Nov-985.42%4.43%19.60observations (66) (Jan. 1998 Dec-984.99%4.00%15.98to June 2003)=42.12 Jan-99–0.70%–1.69%2.87Step Six:Take the square root Feb-99–7.21%–8.20%67.29of 42.12=6.49 Mar-99–0.20%–1.19%1.42 Apr-997.21%6.22%38.65To annualize the standard deviation May-994.51%3.52%12.37Multiply 6.49 by the square root of the Jun-995.82%4.83%23.30return frequency used (in this case, the frequency is monthly). 12 = 3.46=6.49 ×3.46 =22.48 Apr-039.17%8.18%66.86 May-038.76%7.77%60.33 Jun-034.21%3.22%10.35 Average0.99%Sum2,779.64
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Formula 5.2 (Annualized Standard Deviation ASD)
where CSD = Cumulative standard deviation.
RPF = Return period frequency (monthly = 12, quarterly = 4, etc.).
The result of the annualized standard deviation calculation for CAM’s small-cap value portfolio is 22.48.
The graph in Exhibit 5.3 simply shows the monthly returns (repre- sented by the bars) in the context of the maximum/minimum standard de- viation bands we calculated. The black bars in the chart represent months that fell outside of the +/–1 standard deviation band. A total of 16 monthly returns fell outside the band (8 positive and 8 negative). This downside de- viation will be discussed at length later in the chapter. However, at this point in our analysis, we can quickly determine if these outlier return peri- ods are really outliers by simply comparing the portfolio’s return in each of these months to the appropriate index return.
As Exhibit 5.4 illustrates, the returns that we have labeled as outlier periods (8/98, 9/01, and 7/02) are not outlier periods when compared to the small-cap benchmarks.
This leads us to conclude that these numbers in isolation tell us little about the portfolio’s level of risk. The numbers alone do not tell us if a small-cap value portfolio with a historical standard deviation of 22.48%
can be considered to be high-risk or low-risk. To understand the portfolio’s ASD=CSD× RFP
Risk Analysis 83
EXHIBIT 5.3 Historical Monthly Returns
1998 1999 2000 2001 2002 2003
+1 Standard Deviation (0.99 + 6.49 = 7.48) 0.99% Avergage Monthly Return
–1 Standard Deviation (0.99 – 6.49 = –5.50)
–20%
–15%
–10%
–5%
0 % 5 % 10%
15%
20%
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true level of risk, we must compare the standard deviation to some bench- mark or other representation of the asset class. If, for example, I told you that the small-cap value benchmark has a standard deviation over the same time period of 35%, only then could we conclude that the CAM portfolio has exhibited less risk, as defined by standard deviation, over time.
Exhibit 5.5 illustrates the annualized performance and standard devia- tion of CAM’s small-cap value portfolio compared to several benchmarks over the time period from January 1998 to June 2003.
Based on the return and risk statistics presented in Exhibit 5.5, we can conclude that the CAM small-cap value portfolio had much better long- term performance than did any of the benchmarks or the return of the me- dian small-cap value manager. Now the question is whether the portfolio manager or team in charge of the CAM product took on extra risk to achieve those wonderful returns. Since the CAM portfolio’s standard devi- ation of 22.48% is roughly in line with the standard deviation of the three small-cap indexes and the small-cap value manager median, we can hy- pothesize that CAM did not take any significant risks beyond those taken by the indexes over the period. We will be able to verify the hypothesis when we conduct a thorough analysis of CAM’s portfolio holdings in Chapter 6.
84 EQUITY AND FIXED INCOME MANAGER ANALYSIS
EXHIBIT 5.4 Comparison of Outlier Period Returns
Outlier Months
8/98 9/01 7/02
CAM small-cap value portfolio –19.5 –15.6 –15.9
S&P SmallCap Value Index –18.0 –14.3 –16.3
S&P SmallCap Growth Index –20.5 –17.7 –12.0
S&P SmallCap Index –19.3 –13.5 –14.0
EXHIBIT 5.5 CAM’s Annualized Performance and Standard Deviation Standard Performance Deviation CAM small-cap value portfolio 9.73% 22.48%
S&P SmallCap Value Index 4.77% 20.95%
S&P SmallCap Growth Index 2.73% 24.67%
S&P SmallCap Index 4.50% 21.83%
Median small-cap value fund 5.01% 23.55%
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Return/Risk Graph
Now that we have established a form of return variance that is generally viewed as a strong measure of portfolio risk (standard deviation), we can apply the two sides of the coin in a simple, yet effective analytical measure.
The return/risk graphic is one of the most popular methods of displaying the relationship between a portfolio’s historical performance and the risk it assumed to achieve those returns.
Like they say: A picture speaks a thousand words. The graphic in Ex- hibit 5.6 clearly delineates the relationship between reward and risk for CAM’s small-cap value portfolio compared to the relevant indexes. As in- vestment manager analysts, we would like the managers we hire to appear in the upper left-hand corner of the graph because that area represents higher performance and lower risk relative to the other alternatives dis- played in the graphic. The risk/return graph is very durable, as it can be used to compare a specific product to any number of benchmarks or to a subset of its appropriate peer group of investment managers or the entire peer group (although in the latter case it becomes impossible to include product labels due to the sheer volume of points that tend to be included).
The risk/return graph does not rely exclusively on annualized perfor- mance and standard deviation as the return and risk variables. For exam- ple, we can substitute any of the risk measures that follow in this chapter for standard deviation, or we can substitute upside or downside returns in place of total annualized returns. The bottom line is that this graphical for- mat is as simple to calculate and interpret as it is flexible.
Looking at the graph in Exhibit 5.6, we can quickly see that the CAM small-cap value product achieved superior performance and did so
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EXHIBIT 5.6 Return/Risk Graph for CAM’s Small-Cap Value Portfolio
20% 21% 22% 23% 24% 25%
Annualized Risk (Standard Deviation)
Annualized Return
CAM SCV
Median SCV Fund S&P SC Growth Index S&P SC Index
S&P SC Value Index
0%
2%
4%
6%
8%
10%
12%
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with a level of risk that was less than the median small-cap value fund and the small-cap growth index, but a shade higher than the overall small cap index and a bit more than the small-cap value index since its incep- tion in 1998.
Sharpe Ratio
Thus far we have established that return and risk go hand in hand and can be calculated separately and displayed side by side. The Sharpe ratio, named for its creator, Nobel laureate William Sharpe, was one of the first statistical measures that factored both return and risk into a single for- mula, thus giving us a single statistical measure of risk-adjusted return.
Formula 5.3 (Sharpe Ratio)
where APR = Annualized portfolio return.
RFR = Annualized risk-free rate (90-day T-bills are typically used as a proxy).
StdDevAPR = Annualized standard deviation of the portfolio’s returns.
We can use formula 5.3 to calculate the Sharpe ratio for the CAM small-cap value portfolio. To simplify the calculation, we will assume that risk-free rate’s annualized rate of return was an even 2.00% over the time period being used (January 1998 to June 2003).
Sharpe ratio calculation for CAM’s small-cap value portfolio:
However, the Sharpe ratio for the CAM SCV portfolio in isolation does not provide us with any great insight. The 0.344 result simply states the incremental return per unit of total risk taken by the portfolio. This translates as follows: For every 1 percent of risk taken by the portfolio, it achieved a 0.34% rate of return. But it does not tell us whether this figure is good or bad. The only way of effectively gauging CAM’s Sharpe ratio is to compare it to a benchmark or to a peer group. Using the same method- ology for the small-cap indexes, we can calculate the Sharpe ratio for the small-cap indexes and peer group.
SR(CAM SCV) APR RFR StdDevAPR
= − = 9 73 2 00− = =
22 48
7 73
22 48 0 344
. .
.
.
. .
SR APR RFR
StdDevAPR
= −
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Sharpe Ratio Comparison (Sorted from High to Low) CAM small-cap value portfolio 0.344
S&P SmallCap Value Index 0.132 Median small-cap value fund 0.128
S&P SmallCap index 0.115
S&P SmallCap Growth Index 0.030
The results clearly indicate that CAM’s portfolio had achieved risk- adjusted returns far in excess of any of the benchmarks listed. In fact, CAM’s Sharpe ratio is more than twice that of the most appropriate benchmark (S&P small-cap value benchmark) and the median small-cap value fund.
M2Ratio
Another Nobel Prize–winning economist, Franco Modigliani, and his granddaughter, Leah Modigliani, a portfolio strategist at Morgan Stanley, created the Modigliani-Modigliani ratio. It is referred to as the M-squared ratio or the M2ratio.
Like the Sharpe Ratio, the Modiglianis’ statistic seeks to measure how well portfolios/funds perform after adjusting for risk. To make this adjustment, the M’s delever a portfolio until its volatility (as measured by standard deviation) matches that of its benchmark. Put differently, for a portfolio whose historical volatility has been less than its benchmark’s, they expand the portfolio by leveraging it at an assumed borrowing rate;
and for a portfolio whose volatility has been greater than its bench- mark’s, they contract it and invest the hypothetical proceeds at an as- sumed yield. The assumed interest rate for both borrowing and lending is typically the yield on short-term Treasury bills. This adjustment produces a portfolio-specific “equity share” or a leverage ratio that equates the portfolio’s risk to that of its benchmark. The portfolio’s actual return is then multiplied by its equity share, and the product of this calculation is compared to the benchmark’s actual return to determine whether the portfolio had outperformed or underperformed the benchmark on a risk- adjusted basis.
So what is the difference between the Sharpe and M2 ratios? In my view, the M’s created not a better mousetrap, but a different one. The M2 ratio is, perhaps, more user-friendly in that it is stated in actual perfor- mance terms as opposed to the Sharpe ratio, which is stated in somewhat more abstract terms.
Whatever method you prefer, just keep in mind that the individual ra- tios are meaningless in absolute terms. These figures are meaningful only when they can be compared to ratios of similar products or benchmarks.
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Information Ratio
The information ratio is used to measure a manager’s performance against its appropriate benchmark. This measure explicitly relates the degree by which the portfolio/fund has beaten its benchmark to the consistency by which the portfolio/fund has beaten the benchmark. It is basically a mea- sure of efficiency (or consistency) calculated by dividing the excess rate of return (alpha) by the standard deviation of the excess rate of return stream (tracking error).
Formula 5.4 (Information Ratio)
where Alpha = Average of the portfolio’s excess monthly returns over a specific benchmark.
Tracking Error = Standard deviation of the alpha.
We can use formula 5.4 to calculate the information ratio for the CAM small-cap value portfolio:
Like so many of these statistical measures, the 0.33 figure we calcu- lated takes on more meaning when we compare the results to other prod- ucts with similar mandates. For example, we could compare the information ratio we calculated for CAM against other managers we might have under consideration.
Treynor Ratio
Just as the Sharpe ratio was named for William Sharpe, the Treynor ratio was named after its creator, Jack Treynor. The numerators of the Sharpe and Treynor ratios are identical; it is in the denominator where the two risk measures differ. While the Sharpe ratio is concerned with total risk (standard deviation), the Treynor ratio is concerned only with systematic or market risk (as measured by beta).
Formula 5.5 (Treynor Ratio)
TR APR RFR
= Beta−
IR(CAM SCV) Alpha
Tracking Error
= =0 42 =
1 30. 0 33
. .
IR Alpha
Tracking Error
=
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To fully understand the Treynor ratio and its application as a risk-ad- justed performance measure, we must understand what beta is and how it is calculated. The first point that needs to be made is that beta is a compar- ative measure, meaning it needs to be calculated in relation to some stated benchmark. It is critically important that the benchmark being used be ap- propriate to the portfolio or fund under review. The beta can be inter- preted as the slope of the line in a regression equation or in a capital asset pricing model (CAPM) context. The only difference is that excess portfolio and index returns are used in the CAPM calculation whereas total portfo- lio and index returns are used in the regression format.
Because this book is focused more on the practical application of in- vestment methodologies and less so on formal statistical derivations, I will avoid a full discourse on beta. Beta can be easily calculated using any sta- tistical software package or simply by using the beta function in Microsoft Excel or some other spreadsheet package.
Beta can be interpreted in the following way:
■ Beta > 1:The portfolio or fund under review is more volatile than the index being used. For example, it can be interpreted that if the market goes up 10%, a portfolio with a beta of 1.5 would be expected to go up 15% (1.5 times the market).
■ Beta < 1:The portfolio or fund under review is less volatile than the in- dex being used. For example, it can be interpreted that if the market goes up 10%, a portfolio with a beta of 0.5 would be expected to go up 5% (0.5 times the market).
■ Beta = 1:The portfolio or fund under review exhibits volatility that is equal to the index being used. For example, it can be interpreted that if the market goes up 10%, a portfolio with a beta of 1.0 would be ex- pected to go up 10% (equal to the market return).
To calculate a portfolio’s beta using Microsoft Excel, use the following format:
=SLOPE (A1:A30,B1:B30)
Statistically, beta is the slope of the regression equation. The Excel for- mula can be interpreted as follows: The term “A1:A30” represents the his- torical portfolio returns in column format. The term “B1:B30” represents the historical index returns in column format.
So the Treynor ratio can best be described as a measure of a given portfolio’s historical returns in excess of those that could have been re- turned on a riskless investment per unit of market risk assumed. Because the denominator of this ratio is wholly dependent on the benchmark
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