Objective Function, What’s Your Function?

Objective Function, What’s Your Function?
March 9, 2012

Conjunction Junction

In between cartoons on Saturday mornings in the 1980s the networks would run educational vignettes about grammar and math topics set to catchy music. One such video has stuck with me for 30 years: Conjunction Junction. The jingle for the vignette went, “Conjunction Junction, What’s Your Function?” To this day, I remember the function of a conjunction because of this video. (For those who don’t remember, a conjunction is a word that joins other words in a sentence, like ‘and’, ‘or’ or ‘but’.)

Objective Function
In the domain of quantitative finance, we spend a great deal of time thinking about something called an ‘Objective Function’, which always reminds me of the above vignette. The objective function is the goal that we try to optimize with our investment strategy.
For example, most mutual fund managers are compensated on their ability to maximize an objective function called the ‘Information Ratio’. The Information Ratio quantifies the extra return that a manager has achieved over a certain benchmark, like the S&P TSX Composite index of Canadian stocks, without straying too far from the benchmark. By implication, mutual fund managers are compensated for delivering slightly better returns than their benchmark, not for delivering superb absolute returns. That’s why a Canadian equity mutual fund manager can receive full compensation in a year when his fund loses 40%.
Unfortunately, this doesn’t serve most investors very well. Most investors are concerned about achieving the highest return possible at a given level of risk. Taking this a step further, most individual investors want the highest, most consistent returns for a certain level of risk.
Now let’s disentangle the three concepts: returns, risk, and consistency.
Consistent Growth with Minimal Volatility
Everyone is familiar with returns; this just refers to the growth in your portfolio. Returns can come from capital gains, dividends or interest, with preferential tax treatment for capital gains and dividends.
Risk is a little more fuzzy. Most institutional investors think of risk as portfolio volatility. Volatility captures both upside and downside gyrations though, and most people like the upside gyrations, so maybe we should just concern ourselves with downside volatility.
However, we know that in markets downside volatility tends to cluster around bear markets, and bear markets often take a few years to recover from, so a more prudent measure of risk might focus on maximum potential loss, and the duration of that loss. This is especially relevant for retirees, who from a pure mathematical standpoint cannot afford to endure large sustained losses while continuing to also withdraw income.
This brings us to the idea of return consistency. For most people, it isn’t enough to earn an 8% average return over 5 years if the returns arrive inconsistently, for example as 0%, 0%, 0%, 0%, 40% sequentially. Most people would prefer a return sequence closer to 8%, 8%, 8%, 8%, 8%, and in fact this more consistent return stream also helps to maximize the retirement equation.
Bringing it all together, investors want strong returns, with tax treatment favouring capital gains over dividends, and dividends over interest. Furthermore, they want the lowest risk of large sustained losses, which means maximizing the consistency of returns over time. So how can we capture this in an objective function?
Our preferred objective function is the DVR, which measures the amount of return per unit of risk (sometimes called the Sharpe ratio), and multiplies this ratio by a measure of the linearity of the return stream, called the R-Squared. An investment strategy which maximizes the DVR delivers strong returns with minimal volatility where returns occur in a predictable sequence over time.
Oh, The Humanity
Purely from a mathematical standpoint, the DVR is the most rational objective function for non-pensioned investors to pursue in order to optimize their retirement sustainability and withdrawal rates. However, while many investors claim that all they want from their Advisor is strong, consistent returns with minimal risk, their behaviour is often inconsistent with this goal.
In our experience, many individual investors obsess over business news or the business section of newspaper, where they are inundated with performance metrics and exuberant market forecasts for their local stock market. Investors can’t help but anchor their expectations to the optimistic return prospects trumpeted by analysts through thick and thin, which contaminates their emotional objective function.
Rather than focusing on the most consistent long-term returns at the lowest possible risk, investors focus on capturing as much of the upside returns as possible from their local stock market index. Only when the index is dropping precipitously do investors begin to ignore the index and focus on those strong, persistent returns that will get them to retirement.
This presents an interesting challenge for managers: most investors have one objective function in rising markets and another objective function in falling markets. Unfortunately, this is impossible to facilitate.
So I would urge you to do some soul-searching about what your objective function really is. Are you really trying to optimize your retirement nest egg, sustainability and retirement income? Or are you determined to watch the business news and follow the investment media?
This is not a trivial matter. If you come to the conclusion that you are wedded to market outcomes, and like to feel the excitement of rising stocks, then we would urge you to look elsewhere for an Advisor. There are plenty to choose from, because most Advisors adhere to this objective function as well, either by choice or default because they have no alternative.
If, after careful contemplation, you discover that you would rather maximize your chance of retirement success rather than chase short-term market returns, or engage in the excitement of daily stock market drama, then our Core Diversified Program may be right for you.