Carl Mastrangelo

A programming and hobby blog.

Risk vs. Reward

In order to be a good investor, you need to be able to determine if an investment is a good idea. A good investment is one that has a good chance of having high returns, and a low chance of poor returns. This seems obvious, but I find myself having a hard time keeping this in mind when trying to decide where to put money.

Two subtleties are easily forgotten:

  1. Risk vs. reward really does mean both are necessary to consider. For example, the tax adjusted returns of S&P 500 Index funds are approximately 7% annually. This means that if you invest $1,000 at the beginning of the year, you can reasonably expect $1,070 by the end of the year. As of this writing, my bank is offering a savings rate of about 1.26% in a savings account. This means at the end of the year, that same $1,000 will be about $1,010 after taking into account taxes.

    The difference between the two investments is $60. That’s a fancy end of year dinner just for parking our money in the fund! So which one is the better choice? The answer is we don’t know, because we only calculated half of the equation. We only calculated the reward, but forgot about the risk. We got focused only on the positive outcomes, but forgot about the negative ones. Essentially our flawed math looks like:

    outcome = return / risk

    Wrongly, we compared the following two things:

    • S&P500 Index:
      • outcome = $70 / ?????
    • Savings Account
      • outcome = $10 / ?????

    Instead we should have used the following calculation:

    outcome = return / risk

    Unfortunately, I can really only estimate the risk for the bank. The most likely risk is inflation, which will erode the buying power of the savings account. We made $10 with our 1.26%, but the price of everything went up by 1.8% (the rate of inflation for 2017). Realistically, the bank won’t collapse (the events of 2008 not withstanding), the account won’t be drained by theft, and there won’t be hyperinflation (which did happen in the 1970s). These could (and do) happen, but affect many of the other investment options too, so it isn’t worth penalizing the savings account. Additionally, if these did happen, it would only give credence to the larger point that we don’t assess risk appropriately.

    The index fund on the other hand, is much riskier. The stock price could plummet, or could just be lukewarm. It’s like flipping a coin every day; we know the odds, but we don’t know what side will come up. We know that it generally does go up, and by what amount, but never what it’s going to do next. Short term holdings of stocks, funds and other securities is going to have higher risk. It’s only longer term where the odds actually play out, and we arrive at the expected value of 7% growth.

    To sum up, the risk needs to be taken into account to compare two opportunities.

  2. The chance of returns is not a fixed number. It’s a distribution. Continuing with the stock example, the return from investing $1,000 is more like the following:

    Chance Outcome
    2% $990
    5% $1,010
    11% $1,030
    17% $1,050
    23% $1,070
    19% $1,090
    13% $1,110
    7% $1,130
    3% $1,150

    I made up these numbers but you can see that it’s not as simple as an average. There’s a tiny possibility we lose $10 of our initial $1,000, but a corresponding chance that we make much more than 7%.

    Compare this to what the bank is offering us:

    Chance Outcome
    0.1% $1,007
    99.8% $1,010
    0.1% $1,013

    Obviously the savings account is much safer bet! The take away is that it isn’t helpful to compress the distribution of returns into a single number. As an extreme example, consider a gambler betting the entire $1,000:

    Chance Outcome
    99% $0
    1% $107,000

    This also has a 7% rate of return, but the number of times you can lose before you realize the gains is a lot less!

I struggle with remembering both of these points. I think both are true, but it’s difficult to keep them in the foreground while making a decision.

One thing I have noticed is that the risk is hard to express or evaluate, but how soon I need the money is easy. “I will need this money in 5 years” is a much easier goal to reason about. Thinking this way let’s me think choose an investment with an appropriate level of risk. Once that is sorted, I can pick the best returns for investments with equivalent risks.

The Vanguard Glide paths in their Target Retirement funds are a good rule of thumb. For money in 5 years, a 6040 stock-bond split is their recommendation. Since they have already gone through all the work of figuring out the risks and rewards of their funds, they make a good starting point.


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