# Get higher returns and less risk with science: Modern Portfolio Theory

I'm sure you've heard many times: "Stocks historically have the highest returns." Since you have decades left in the tank, you have put everything in stocks, because that's the strategy that yields the highest returns, right?

Actually, no.

Most investors don't know the scientifically correct way to maximize their returns. I'll tell you how it is.
You might have heard it before that risk and return go hand-in-hand. So if that's true, then by choosing one (e.g. returns) you always choose risk, too, right?

Almost.

## What is volatility?

If you own index ETFs, you'll see them go up 0.2% one day and down 0.1% the other. Sometimes they might go wild and go up even 1%. Very rarely you see them move more than 2% in a single day, but it happens, a few times a year.

You can calculate something called the standard deviation from this up-and-down movement, which mathematically describes how close to the "average movement" it stays on a daily basis.

With crypto currencies, it's very common to see +-10% daily movements. Needless to say that the standard deviation of crypto currency price changes is a gazillion times higher to index ETFs.

Both standard deviation and variance represent volatility. Variance is simply standard deviation to the second power.

No need to remember this, just keep in mind that when I talk about volatility, variance and standard deviation, it's all the same concept: how much an asset moves up and down.

## Volatility is married with expected returns

While the Capital Asset Pricing Model will definitely require its own post, the general gist is this:
• There's a risk free rate of interest, commonly thought to be arising from government bonds.
• Stock market returns have an excess to risk free rate, but come with volatility.
• With volatility comes always a risk to deprecate in value or a risk that you are forced to sell at a bad time.
• The excess return (compared to risk free rate) of an asset is equal to its volatility multiplier (aka. beta) times the excess return of the market. In other words, with double volatility comes double excess returns.
For more information on the CAPM, read Wikipedia. There are already more sophisticated asset pricing models available that suggest that volatility isn't the only thing to explain asset pricing. I'm also sure it's true, but the CAPM will do for now.

Clearly risk (i.e. volatility) goes hand-in-hand with returns. Ergo, since you've already decided to go to maximum returns, your risk level is determined as well, right?

Like I said, almost.

## Not all risk is equal

In horse racing, do you think your risk and returns follow the same formula? Or the lottery? Or the slot machine?

Indeed it's very easy to imagine situations where awful returns are accompanied with horrendous volatility, too. Russian roulette, anyone?

According to efficient markets however, in investing, smart money always finds a way to balance risk and return - at an individual asset level.

... at an individual asset level.

So, what happens when you combine multiple assets in your portfolio?

You can think of stocks as kind of random variables. When you have one stock in your portfolio, the variance (same as volatility) of your portfolio is the variance of the single stock. When you add another stock to your portfolio, what is the variance of your portfolio then?

Let's get a bit math-nerdy. Do you remember vectors?

Imagine your portfolio as a vector space with as many dimensions as you have stocks. In this example you only have two stocks, so your vector space is 2-dimensional: a plane.

Each of your stocks has a variance, represented by a vector on that plane. The length of the vector represents the amount of variance. The sum of those two vectors is the variance-vector of your portfolio. Wow!
So what if they point in a 90-degree angle? This is a case where the stocks are completely independent. If you remember the pythagoras' theorem, you'll realize that the resulting vector is the square root of the squares of each individual vector's lengths.

What if the vectors point exactly in the same direction? Well, then the two stocks are completely correlated and the resulting variance is the sum of the two variances.

For two stocks to be completely correlated it means when one moves up, the other one moves up too. When one moves down, the other one goes down as well.

"Wait what about when the vectors point opposite directions?! You didn't mention them at all!"

Good! Great that you noticed! Can you think of a situation where this might happen? What asset class often moves in the opposite direction to stock market?

Gold! Well, that's the answer I was looking for, although it's not exactly true. Granted, gold isn't very strongly correlated with stocks, but it's not quite inverse still. Unfortunately, gold isn't a great investment in terms of returns either.

In modern portfolio theory, the expected return of a portfolio is the weighed sum of the expected returns of each of the assets. Common sense dictates however, that this cannot always be completely true. It's still true enough that we can just believe it in a normal situation.

So if the expected returns are summed, but variances are only summed if the stocks are completely correlated, then how does that help?

## Efficient portfolios

Efficient portfolios are such where it is impossible to improve expected returns without increasing volatility. For any given expected return, the efficient portfolio is such where volatility is minimized.

When you own just one asset in your portfolio, it's rarely efficient. There's pretty much always a way to improve the efficiency (i.e. the returns-to-risk ratio aka. Sharpe ratio). The simplest way is to just add other assets with similar expected return and variance, but with less than perfect correlation.

However, another way is to reduce volatility with a small cost in expected returns.

The efficient frontier is the curve in the above picture that depicts the maximum possible return at any given volatility. Any point on that curve is considered efficient. However, the tangency portfolio is special. This is the most optimal point your portfolio that generates the highest marginal return for every unit of risk taken.

If you can borrow money to invest, the tangency portfolio is a gateway to all efficient portfolios at any return - even abofe the efficient frontier! Once you can get to the tangency portfolio, you can theoretically traverse the tangent by borrowing money at the risk free rate and thus leverage your portfolio.

Theoretically. Almost in practice as well.

## Practical applications

You might be one of those who only holds the Vanguard Total Stock Market Index (VTSAX). Or maybe you have two or handful of index ETFs in your portfolio. Chances are that these ETFs are very strongly correlated and therefore you don't get significant benefit from the diversification: one goes down, all go down.

Go ahead and try out a tool on Portfolio Visualizer, to which you can provide your tickers and it will produce the correlations. Won't cost you anything.

You might also find Portfolio Visualizer's Portfolio Optimization tool useful, which will calculate the optimal weights for your chosen assets. A tool like that is required since optimizing a portfolio manually is a fairly complicated task, involving calculating expected returns and correlations. (Would you be interested in the math behind that?)

As a final word: while a total market ETF such as VTSAX feels greatly diversified, you need to remember that it gives exposure only to the US market and in that, the large, usual suspects. It is likely that balancing mainly US stocks with small caps, other asset classes (a REIT?), and other indexes (Europe, Japan, China) should result in a more efficient portfolio.

And keep your mind open to leveraging.

Just make sure you know what you're doing, ok? 😉 I'm not a financial advisor.
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