The basic quantitative building block for professional judgments about investment performance is the rate of return. How well do we really understand it? And how can we use past rates to assess the prospects for future performance? You may be surprised to learn that “expected return” may not be what you think.
Rates of return are always more complicated than we think. To understand them fully requires a complete mathematical model – and you need to understand that model, but the model can get complicated quickly.
Consider a basic application. If you have two investments combined 50-50 in your portfolio, and they realize 10% and 20% for the year, you can take their arithmetic average to get your portfolio’s return of 15%.
But if it’s only one investment for two years, and its rate of return is 10% in the first year and 20% in the second, you compound and annualize them to get the “average” annual return of 14.89% (that is, the geometric average). In this context, the arithmetic average (15%) of the first and second year’s returns has no real meaning.
How quickly it gets complicated
Next, suppose you have two investments combined 50-50 for two years. Then you must average their rates in each year, then compound and annualize the two annual results to get the portfolio rate of return over the two years.
But that’s only if you rebalance once, at the beginning of the second year, to a 50-50 mix.
If you don’t rebalance, then you must separately compound the annual returns for each investment for the two years, then take the arithmetic average of the results and annualize that number. If you do it in any other order, you get the wrong answer.
See how quickly it gets complicated? And this is almost the simplest example there is.
The nutty world of conversations about rates of return
Because most people who are not investment professionals have an extremely hazy grasp of compound return, statements and discussions about rates of return are frequently off-the-wall.
For example, I’ve heard someone discuss a friend who they’ve been told earns 10% a month as a day-trader. They don’t realize that 10% a month compounds to 214% a year, and to 9,600% in four years.
If the friend started with $11,000 and made 10% per month, he’d have more than a million dollars in four years – an implausible assertion.
Among more sophisticated discussants – who, nevertheless, still don’t have a very firm grip on a model – the confusion is at least a bit higher-level. For example, I have often heard people debate which is appropriate to use: the arithmetic average of a time series of historical returns or the geometric average. Since the arithmetic average has no discernible real-world interpretation, it’s hard to know what they’re arguing about. Rates of return in successive time periods don’t add or average any more than apples and oranges do – they compound.
Dollar-weighted vs. time-weighted rates
For so-called time-weighted rates of return, there is a formula, which, as mathematics goes, is relatively simple. A rate of return is calculated for each time interval between cash flows, then these rates are compounded to get the time-weighted return for the whole period.
Since 1968, this time-weighted rate of return calculation has been the measure of choice to evaluate investment managers, the standard method endorsed by the Bank Administration Institute. The BAI promoted this measure over an alternative, the dollar-weighted rate of return, because the time-weighted return is unaffected by how many dollars flow into and out of a fund, while the dollar-weighted return is not. The assumption was that the manager has no control over the inflows and outflows of funds, only over the success of the investments immediately under his or her control. (Of course, one can mount a strong argument that the latter assumption is false.)
It’s really the dollar-weighted rate of return that hits investors’ pocketbooks. As I pointed out in my recent article on Sebastian Mallaby’s book about hedge funds, More Money than God, the example that illustrates the flaw of time-weighting returns is a hedge fund that has great performance for a period of years, at which point investors get wind of it and pour in a huge quantity of new investment. If the fund then plummets, the time-weighted return over its whole life might be quite good, but the average dollar invested will have lost a bundle.
Continuous rates of return, or “log-returns”
To make things more confusing, rates of return can be defined over different intervals. The annual percentage rate (APR) was introduced to try to make these different rates commensurate and bring transparency, but it still doesn’t clear up the confusion. For example, a rate of 0.5% per month comes to an APR of 12 x 0.5% = 6%. But the annual percentage yield (APY) – which is the actual annualized rate of return – is 6.17%, because a monthly rate of 0.5% compounds to 6.17% in a year.
If you dice the periods into finer and finer intervals (monthly, daily, hourly, secondly, etc.), but still keep the 6% nominal annual rate, you get the continuous or continuously-compounded rate. This continuous rate of 6% is also called, by some people, the log-return, because it is the natural logarithm of one plus the actual annual rate of 6.18365%.
The nice thing about the log-return is that – while being an equivalent stand-in for the annual rate of return – instead of having to be compounded, it can be added to other log-returns to get the log-return for a cumulative time period. This can make the arithmetic easier, but you still won’t do it right if you don’t understand the model.
Looking backward
We can only calculate data about funds with historical data, because, of course, we don’t have data about the future. The problem is that there is no evidence to show that historical information about the performance of any particular investment has any statistical bearing on its performance in the future.
What, then, is the use of historical performance data? If it can’t be used to help decide the best action to take in the future, what can possibly be the purpose of compiling it?
Worse yet, many investment professionals test strategies they have conceptualized by “backtesting” them – that is, running them through the historical rate of return data for the various assets they plan to invest in, to see how the strategy would have performed historically. There’s virtually no evidence that this practice has any ability to predict how they will perform in the future.
Looking forward
A better way of testing a strategy is to simulate its results going forward into the future. This involves generating a number of possible alternative future series of rates of return, with a probability assigned to the occurrence of each series, then seeing how the strategy will perform in each alternative future. This procedure, called Monte Carlo simulation, generates a probability distribution of future results.
With this procedure, however, you have to make a number of decisions about how to generate those alternative future series of rates of return, and how to assign a probability to each one. That kicks up another hornets’ nest. The rest of this article will discuss how to do that, and the pitfalls you can run into.
Two basic methods
There are two basic methods to generate future rate of return scenarios. The first assumes that rates of return in successive time periods are independent. The second assumes rates of return in successive future time periods are interrelated in whatever messy way they were in the past.
The overwhelming choice is the independence assumption. This is because it is relatively easy to implement, there is substantial evidence that it is correct for securities prices, at least to a very good approximation, and it can be underpinned with a logical, consistent and robust mathematical model. Other methodologies, by comparison, are prone to anomalies – I’ll give an example of one of these anomalies later.
The independence assumption means that the rate of return in one time period says nothing about the return in the next time period. Hence, there is no such thing as a trend, there is no such thing as momentum, and there is no such thing as reversion to the mean. All of these phenomena are ruled out by the independence assumption.
The one-period distribution under the independence assumption
When the independence assumption is used, it is usually also assumed that the probability distribution of rates of return is the same in every time period. (This need not necessarily be the case; a method that I often use, for example, called stochastic volatility, varies the standard deviation of the probability distribution of rates of return from one period to the next.1)
The mathematical model under these assumptions – when returns in successive time periods are assumed to be independent and to have the same probability distribution – is sometimes called a stationary stochastic process, or a stochastic process with independent, identically-distributed increments. (“Stochastic” is a technical word for things that pertain to chance or probability.)
Two methods are commonly used to determine the probability distribution. One is to simply use the set of past returns. Hence, for example, there are 85 annual returns on U.S. large stocks over the period 1926-2010 in the Stocks, Bonds, Bills and Inflation database. The assumption could be that each one has a one-in-85 chance of being the rate of return on large U.S. stocks in any given year in the future.
The second method is to assume a “parametrized” probability distribution of future returns, usually the lognormal2. One then needs two parameters, the mean or “expected value” of the distribution, and its standard deviation, to determine the probability of any given rate of return. In generating a future rate-of-return scenario, one invokes an algorithm that generates a “pseudo-random”3selection from this probability distribution in each future year.
1. The stochastic volatility method has the virtue of fitting historical returns distributions better by exhibiting fat tails and persistence of periods of high volatility.
2. The lognormal assumption is rightly criticized for failing to model the “fat tails” observed in real-world returns distributions; this assumption continues, however, to be embedded in much of mainstream financial theory including option pricing and variance-at-risk models.
3. The number is pseudo-random because it is created by a computer, which can’t create “real” random numbers because they are generated by a programmed algorithm. Pseudo-random numbers are created by a program to look random and to fit the probability distribution.
A linguistic paradox
I was prompted to write this article because of my realization that there is a linguistic paradox relating to the expected rate of return.4 This is important, because the choices of the expected return and standard deviation make a huge difference in the results of a parametric simulation.
The expected return of a probability distribution is defined in probability theory as the probability-weighted average of the possible values of the random variable.
So let’s suppose, for simplicity, that we assume the probability distribution of U.S. large-stock returns has the same expected return and standard deviation as the 85 historical annual returns.
Since each of the 85 returns in this probability distribution is equally likely, their probability-weighted average is their simple arithmetic average, which for S&P 500 annual returns is about 11.9%.
But if we run a simulation for a very long period of future years using these assumptions, what do we expect to be the long-run annualized rate of return? The answer is that over a long time period, the rate of return will converge to (that is, it is eventually almost certain5 to be) the geometric average of the 85 returns – about 9.8%, two percent less than the “expected return” as defined in probability theory.
Thus, what we would normally mean by the expected return – namely, the rate of return we will almost certainly get if we wait long enough – is different from the definition of “expected return” in probability theory. It’s easy to get confused by this – in fact I’ve done it myself.
The perils of ad hoc simulation
Often people perform simulations not by reference to a mathematical model, but by taking a variety of sequences of rates of return from the historical data. This is an example of a method that does not assume independence of successive returns, and if it is not employed in the context of a sound model, things can go awry.
A method I have frequently seen used is to pluck a number of series of rates of return from the past. For example, one might simulate 30 years into the future by using as the scenarios all the 30-year historical rate-of-return sequences. If this is done by using the 30-year historical annual periods, they might be the periods 1926-1955, 1927-1956, and so on, ending with the period 1981-2010, for a total of 56 30-year return scenarios.
Unfortunately, if each scenario is given equal probability weight, the result is a probability distribution with too high an expected return and much too low a standard deviation. The problem is that the periods all overlap – which causes their standard deviation to be too low – and they under-represent the earliest years and the latest years. As a result, the expected 30-year annualized rate of return for the 1926-2010 time period is about 11.2% when it should be 9.9%, and the standard deviation is 1.4% when it should be 3.7%. That results in much too low a probability estimate for outliers.
To do it right you need a model
Alan Greenspan once said, “To exist, you need an ideology.” To paraphrase the former Fed chairman: To have a coherent worldview in rate-of-return space, you need a model. Just combining numbers in an ad hoc manner will not do. Without a consistent and correct mathematical model to refer to, you’ll get results that are wrong, misleading, or simply don’t make sense. It’s not easy – it would require a whole book to explain in complete detail all the uses and meanings of rates of return, how to use them right, and how they are often used wrong. But if someone writes that book, a wise advisor would do well to buy a copy.
4. I am grateful to Dave Loeper for an e-mail exchange that triggered my awareness of this fact.
5. This can be given a very precise interpretation in mathematics: the long-run annualized return converges to r (or is “almost certain” to eventually be r) if for any numbers e and p, however small, there exists an integer N such that for any number of years n>N, the probability that the annualized return will depart from r by more than e is less than p.
Read more articles by Michael Edesess
