# Risk and Return: Standard Deviation and Sharpe Ratio

In last week’s Risk and Return article, we discussed Unsystematic Risk (Alpha) and Systematic Risk (Beta). In continuation of risk and return discussion, we will discuss Standard Deviation (SD) and Sharpe Ratio in this article. I suggest you read last week’s article “Risk and Return: Making Sense of Alpha and Beta” before reading this article.

## Standard Deviation

Standard deviation (SD) is one of the most common measures to gauge volatility of a mutual fund’s or Exchange Traded Fund (ETF). You may be familiar with a “bell curve” from your statistics class, which is a graphical representation of a normal distribution of the data. Assuming the data follows a normal distribution pattern, 50% of the data would be less than mean and the other 50% greater than the mean. Then Standard deviation is a statistical measure of how spread out the data is from its mean. In general,

• 68% of values falls within + / – 1 SD of the mean
• 95% of values falls within + / – 2 SD of the mean
• 99.7% of values falls within + / – 3 SD of mean

Figure Source: By M. W. Toews – Own work, based (in concept) on figure by Jeremy Kemp, on 2005-02-09, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1903871

In terms of a mutual fund or an ETF’s return, Standard Deviation measures how widely fund’s or ETF’s return are spread out from its average return over a period of time. So it is an indicator of volatility based on past performance. Although past performance is not a guarantee of future result, the historical data on volatility at least gives you a sense of volatility to be expected in the future.

The higher SD value indicates more volatility and lower SD value indicates lower volatility. In other words, the fund with widely varied returns would have higher standard deviation than the fund with narrow return range.

## Standard Deviation Example

Let’s look at a couple of examples.

The standard deviation of Vanguard Total Stock Market Index Fund (VTSAX) is 18.43 based on 3-year data per morningstar.com. The average return (mean) of VTSAX is 11.74 (3-year). So now let’s calculate the range based on the mean (average return) and standard deviation.

-/+ 1 Standard Deviation (SD) from Mean

• -1 SD = (11.74) – 1*(18.74) = -6.69
• +1 SD = (11.74) + 1*(18.74) = 30.17

-/+ 2 Standard Deviation (SD) from Mean

• -2 SD = (11.74) – 2*(18.74) = -25.12
• +2 SD = (11.74) + 2*(18.74) = 48.60

-/+ 3 Standard Deviation (SD) from Mean

• -3 SD = (11.74) – 3*(18.74) = -43.55
• +3 SD = (11.74) + 3*(18.74) = 67.03

What does this tell you about the fund’s future returns?

• 68% of the time return is expected between -6.69 and 30.17; within -/+ 1 SD of the mean
• 95% of the time return is expected between -25.12 and 48.60; within -/+ 2 SD of the mean
• 99.7% of the time return is expected between  -43.55 and 67.03; within -/+ 3 SD of the mean

So you would expect Vanguard Total Stock Market Index Fund to return between -25.12% to 48.6% almost all the time (95% of the time). If you compare VTSAX annual return from 2001 to 2019, 18 out of 19 times the return is between -25.12% and 48.60%. That is 94.7% of the time VTSAX annual return is within +/- 2-standard deviation of the mean!

## Standard Deviation Bond Fund

Now let’s look at one of the bond funds. The standard deviation of Vanguard Total Bond Market Index Fund (VBTLX) is 3.44 based on 3-year data per morningstar.com. The average return of VBTLX is 5.34 (3-year). So what does this tell you about the fund’s future returns?

• 68% of the time return is expected between 1.9 and 8.78; within -/+ 1 SD of the mean
• 95% of the time return is expected between -1.54 and 12.22; within -/+ 2 SD of the mean
• 99.7% of the time return is expected between  -4.98 and 15.66; within -/+ 3 SD of the mean

In comparison with Vanguard Total Stock Market Index Fund (VTSAX), the bond fund VBTLX is a lot less volatile with SD of 3.44 vs. VTSAX with SD of 18.43. At the same time, as expected, the upside and downside for the stock fund (VTSAX) is significantly higher than that of the bond fund (VBTLX).

## Caution with Standard Deviation

Standard deviation is not a measure of risk, rather it is the measure of volatility. For example, Fund A with following return over the past five years -12%, -5%, -10%, -15%, and -1% has the same SD (5.0) as Fund B with +12%, +5%, +1-%, +15%, and +1% return. The fund or ETF’s SD would increase with both above average return and below average return. In other words, the fund with consistent positive performance appears to be as volatile as the fund with consistent negative performance.

The other drawback of standard deviation measure is that it is not a relative measure. Meaning fund’s SD is not compared to its relative benchmark or a similar fund in the same category. So the Fund’s SD is more meaningful when comparing it with a benchmark or a similar fund in the same category.

Again, let’s use Vanguard Total Stock Market Index Fund (VTSAX) as an example. Based on 3-years data, VTSAX standard deviation is 18.43 while the standard deviation of the fund’s relative benchmark is 18.19. This indicates that the fund’s volatility is similar to that of its benchmark.

## Sharpe Ratio

The Sharpe Ratio was developed by William Sharpe, Noble Laureate Professor of Finance at Stanford University.

Here is the mathematical formula for Sharpe Ratio.

```                     Return of Investment - Risk Free Return*
Sharpe Ratio =     -------------------------------------------
Standard Deviation of the Investment

*Risk-Free Return: i.e., 90-days US Treasury```

Mathematically, higher investment return (in numerator) and/or less volatility (standard deviation in denominator) would generate higher Sharpe Ratio.

In other words, Sharpe Ratio measures risk-adjusted return of an investment. The higher the Sharpe Ratio, the better the investment’s risk-adjusted performance.

## Making Sense of Sharpe Ratio

Sharpe Ratio is useful when comparing two funds or ETFs. For example, if you are comparing two funds with similar return, which one is a better investment considering risk and return?

Following is the comparison of Sharpe Ratio between the two large blend funds: Vanguard Dividend Appreciation Index Fund (VDADX) and Strategic Advisers Core Fund (FCSAX).

• Vanguard Dividend Appreciation Index Fund (VDADX): 0.76
• Strategic Advisers Core Fund (FCSAX): 0.61
• Category (Large Blend ): 0.52
• Benchmark Index: 0.65

Although the Sharpe Ratio for both funds, VDADX and FCSAX, are higher than the large blend category, VDADX has better risk-adjusted performance than FCSAX. Because VDADX Sharpe Ratio is higher than FCSAX.

## Caution with Sharpe Ratio

We know that higher Sharpe Ratio is better but Sharpe Ratio number by itself is meaningless. For example, Fund A with a Sharpe Ratio of 1.3 seems high but by itself does not tell you whether it is a good investment or bad investment. You have to compare the Sharpe Ratio with a similar fund and/or benchmark to make sense of the fund’s risk-adjusted return.

## R-Squared

R-Squared is another statistical measure of Modern Portfolio Theory (MPT). It indicates investment’s similarity to its relevant benchmark. The value of R-Squared is between 1 and 100 where R-Squared of 100 indicates a fund or ETF moving in lockstep with the benchmark.

In general, the statistical measures such as alpha and beta are meaningless unless the R-Squared value of the investment is 75 or higher against the index.

## Risk and Return: Putting All Together

Let’s use Vanguard Total Stock Market Index Fund (VTSAX) risk and volatility measures to make sense of risk and return of the fund. Following is the snapshot of the fund’s risk and volatility measures over the last 10-years period from morningstar.com.

Alpha (-0.63): The negative alpha indicates under performance compared to its benchmark index. However, the fund is performing better than the category as the category alpha is -1.66 compared with fund’s alpha of -0.63.

Beta (1.04): Since the beta value is close to 1.0 it indicates that the fund moves in tandem with the benchmark index.

R-Square (99.35): Since R-Squared is close to 100, indicating the fund’s movement is in lockstep with its benchmark index. Hence, alpha and beta of this fund are meaningful measurements.

Sharpe Ratio (0.95): The higher Sharpe Ratio of the fund compared to its category (0.86) indicates that the fund’s risk-adjusted performance is better than the category over the last ten-years period.

Standard Deviation (13.76): The fund’s volatility is similar to that of the category with SD of 13.74 and the index with SD of 13.54.

In nutshell, you would expect VTSAX to generate above average risk-adjusted return compared to its category based on historical measures.

## Risk and Return: Fund vs. Fund Comparison

Following is the comparison of two funds in the large blend category: Vanguard Total Stock Market Index Fund (VTSAX) and Nuveen Large Cap Select A Fund (FLRAX).

So between VTSAX and FLRAX, VTSAX is better performing fund with higher risk-adjusted return based on last 10-years data.

In conclusion, all these statistical measures are based on the past performance. Nobody can predict funds volatility and its performance in the future with certainty. However, past performance at least provides some indication of what to expect in the future. So use the statistical measures as a rough guide in evaluating funds and ETFs.

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