Why Is Annualizing Risk Properly So Important exp12-30-16

Why Is Annualizing Risk Properly So Important? How to use the correct measure for each investor type

ISSUE 1 April 2013

Carl Moss

Introduction

Senior Managing Director cmoss@

One of the most common tasks faced in analyzing financial data is to summarize short-term data 011.44.20.7410.1529

into annual form. Often the data comes as a set of monthly returns that have to be turned into

meaningful annual statistics. These statistics can be used to compare competing investment opportunities, or may be fed into a long-term asset allocation model to decide the shape of a pension plan.

Vassilios Papathanakos, Ph.D. Executive Vice President Deputy Chief Investment Officer

Most of us are aware that we cannot simply multiply an average monthly return by 12 to get an annualized return. However, much less well-understood is the process of moving from monthly to

vpapathanakos@ (609) 497.9505

annual estimates of risk. Investors generally use a rule of thumb: to convert from monthly standard deviations to annual, just multiply by 12 3.46.

What is often overlooked is that the conversion from monthly to annual depends on the pattern of returns. The result of applying the square root rule is often wrong and can lead to misleading estimates; and if the estimates are wrong, the decisions made from them can go badly wrong, too.

Two hypothetical examples: the role of serial correlation

Here are two examples that illustrate the problem:

1) Stock A returns +25% in the first month, -20% in the second, +25% in the third, and so on. In other words, the pattern of returns is:

Year 1: +25%, -20%, +25%, ..., -20%; Year 2: +25%, -20%, +25%, ..., -20%; ...

In the long run this generates an average monthly return of 2.5% and a standard deviation of monthly returns of 22.5%.

However, the annualized return is 0%: $1 invested at the beginning grows by a factor 5/4 in the first month, falls by a factor of 4/5 in the second month, and so on. At the end of twelve months its value will still be $1; it generates 0% return each year, so its true annual standard deviation is 0%, not 22.5%*12.

2) Stock B has a different pattern of returns, depending on whether the year is even or odd, namely:

Year 1: +20%, +1%, +1%, ..., +1%; Year 2: -20%, -1%, -1%, ..., -1%; Year 3: +20%, +1%, +1%, ..., +1%; Year 4: -20%, -1%, -1%, ..., -1%; ...

The average monthly return on this stock, over the long term, is 0% with a standard deviation of about 5.85%. Using the 12 rule we conclude that the annualized standard deviation is about 20.3%. However, the annual return in Year 1 is about 33.9% and that in Year 2 is -28.4%. This leads to an annualized return of 2.8% and a true annual standard deviation of 31.2%.

Why are these annualized estimates so badly wrong? The key is that the monthly returns are not independent of each other. The returns on A reverse every month; those on B keep the same sign each month for a year. The simple 12 multiplication would be correct if monthly returns were independent of each other: that is if there were no relationship between the returns in one month and the next. But if there is a systematic tendency for returns to exhibit runs of good or bad performance, or if the returns reverse more often than not, then multiplying by 12 will give the wrong answer.

C-1215-1518 12-30-16

FOR INSTITUTIONAL INVESTOR USE

We can measure the tendency of a sequence of returns to either show runs or reversals by a statistic called autocorrelation1. In essence, this is the correlation of one month's returns with the previous month's. A positive number indicates a tendency toward runs of better or worse than average; a negative number indicates a tendency to reverse. Stock A, for example, goes from above to below average and back all the time; its autocorrelation is negative. Stock B has a positive autocorrelation because it has long periods when it is consistently either above or below the monthly average.

Because stock A's monthly returns reverse so frequently, a monthly estimate of standard deviation will overstate its annual standard deviation. From an annual viewpoint, stock A is much less risky than it appears when we look at monthly data. The opposite is true of B: runs of moderate monthly returns compound to large annual returns, so the annual returns on B are much more risky than its monthly data suggest.

What makes this more than a mathematical curiosity is that we see autocorrelation at work in the returns of many of the most common assets. Of course, it is seldom as obvious or extreme as the two examples above, but it can still have a material effect. As a result, annualizing monthly standard deviations will not generally give an accurate picture of the true annual risks for investors.

Equity volatility

Here is a comparison of the annualized monthly and annual standard deviations (SD) for a few common equity benchmarks,

measured between 1995 and 20122. Each of

these benchmarks shows a positive monthly

Benchmark Annual SD Annualized Monthly SD

Autocorrelation

autocorrelation of the order of 0.3?0.4, and its S&P 500 effect is to make the annualized numbers

20.1%

15.7%

0.32

materially understate the actual annual risks. Russell 1000

20.1%

15.9%

0.34

So far we have concentrated on the difference MSCI World

19.8%

between annual risks and annualized risks. But MSCI ACWI many investors consider risks over longer

20.4%

periods than a year, and here the picture

MSCI Emerging

35.3%

becomes more complicated. If monthly returns

were truly independent of each other then the standard deviation of returns over longer

and longer periods would follow the familiar square root rule. We do not see this

regularity in practice. The chart on the right compares the annualized standard deviation

of the MSCI All Country World Index returns over a number of sampling periods

(comparing estimates using overlapping vs non-overlapping time intervals). This plot

suggests that, over the short term, market returns tend to be positively correlated, while

over the long term, market returns tend to be negatively correlated.

15.8% 16.2% 24.5%

0.38 0.37 0.39

Asset allocation

A typical investor might hold a 60%-40% mix of public equity and fixed income assets. In such a scenario, the risk of the allocation scheme depends crucially on the volatilities, as demonstrated in the tables below3. Keeping all of the various return assumptions constant, except for the volatility, we see that the risk at an annualized-monthly level equals 12.6%, while the annual risk is 15.9% -- an increase of more than a quarter!

Asset Public Equity

Weight 60%

Return 7.25%

Annualized monthly volatility

19.0%

Asset Public Equity

Weight 60%

Return 7.25%

Annualized volatility

24.7%

Fixed Income

40%

1.25%

6.5%

Fixed Income

40%

1.25%

6.5%

Portfolio

100%

4.85%

12.6%

Portfolio

100%

4.85%

15.9%

1 Strictly speaking, this is first-order serial autocorrelation. Other, more complicated, forms of autocorrelation can occur. 2 The annual standard deviation is computed by using the 18 calendar years in the period. 3 The return assumptions are taken from the "10-year Capital Market Return Assumptions" by BNY Mellon, and represent typical expectations. We use MSCI ACWI to represent the public

equity (PE), and the Barclays Global Aggregate Index to represent the fixed income (FI) asset classes. For simplicity, we assume that there are only two asset classes, and that the fixedincome volatility does not depend on the sampling time scale. Similarly, we assume that the correlation between PE and FI equals 0.35, and is independent of the measurement time scale.

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Low-volatility Investing

In the context of evaluating low-volatility alternatives to the various asset classes, especially equity, investors employ a variety of risk metrics, not the least of which is maximum drawdown. Nevertheless, the standard deviation of a low-volatility strategy can be used to quantify the volatility reduction as compared to a capitalization-weighted index; measuring the standard deviation correctly is vital. Furthermore, since low-volatility strategies are less tightly constrained to cap-weighted indices, they offer a substantial opportunity to reduce both the volatility and the autocorrelation of returns. Increasing the independence of returns serves investors by reducing the timing sensitivity for allocation into and out of asset classes, and simplifying risk budgeting.

Tailoring to clients' investment horizons

The choice of the time scale used to measure the standard deviation of a stock, or an asset class, should reflect the risk tolerance of each investor. Even though time is a continuous quantity, various practical considerations generally lead to the following discretization of time-scale sensitivity groups.

ETF insurance

mutual funds passive vehicles

pension plans endowments

daily/weekly

monthly/quarterly

annual/biennial

The first type of investors, often exemplified by exchange-traded-funds (which have daily disclosures) and insurance companies (that are exposed to strict daily accounting risk constraints), may well want to focus on the standard deviation samples at a frequency between daily and weekly.

The second group, mostly comprising mutual funds (exposed to monthly cash flows and quarterly disclosures) and passive vehicles (often attempting to control turnover by restricting rebalancing to no-higher than quarterly frequency), might concentrate on measuring and controlling monthly-to-quarterly sources of risk.

Finally, the third group, typically consisting of long-term institutional investors with a three-to-five year horizon, would be best served by choosing an annual-to-biannual sampling frequency, since this provides a good expectation of the maximum level of long-term volatility they will have to weather.

Conclusion

Properly understanding risk, and correcting for it when evaluating the value of expected future return, is a fundamental component of investing. The Global Financial Crisis of 2008?2009 has redrawn investors' attention to this basic principle. Recurring and continuing aftershocks since then have further demonstrated that risk has many sources, and that naive approaches to mitigating it (such as migrating from equities to `safer' assets) offer little more than psychological comfort. Indeed, the recent crisis has increased the urgency for investors to generate growth and retain or, more frequently, restore their funded status. In this environment, where investors need to carefully disburse the risk in their budget, they can ill afford to use approximate measures of risk. For this reason, tailoring the time scale for estimating the standard deviation to match an investor's risk profile is not only straightforward, but also very valuable.

MSCI makes no express or implied warranties or representations and shall have no liability whatsoever with respect to any MSCI data contained herein. The MSCI data may not be further redistributed or used as a basis for other indices or any securities or financial products. This report has not been approved, reviewed, or produced by MSCI. This material is for general informational purposes only. It is not intended as investment advice, as an offer or solicitation of an offer to sell or buy, or as an endorsement, recommendation, or sponsorship of any company, security, advisory service, or product. This information should not be used as the sole basis for investment decisions. Past performance does not guarantee future results. Investing involves risk, including the loss of principal and fluctuation of value.

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