## Bottom Line

The winner is the participant who posts the highest Sharpe Ratio by the end of the competition.

## Example: Wild Bill & Clever Susan

Let’s say that at the end of the competition we have two students: a risk-taker named Wild Bill and a shrewd trader named Clever Susan.

#### Wild Bill

Bill puts all of his money into a tiny, risky startup that has just been listed on the NASDAQ. He then forgets about it and lets it ride until the end of the competition, at which time he’s earned an amazing 250% return!!!! He clearly should win the competition, right?

Well… when you look at the value of Wild Bill’s performance, it was all over the map during the competition. As news was released about the startup, Bill was down as much as -300% on some days, and up as high as 375% because the tiny company’s price was fluctuating so much.

That means, that if Wild Bill had had a cash emergency or needed to close his position and withdraw his investment at any time, there’s no telling how much he’d have had available on a given day. Maybe he’d have a lot of cash, maybe not much at all.

In fact, the next day after the competition, Bill’s gains were wiped out and he was down a full -50% from where he started, meaning that in reality, Bill just got lucky that the competition ended when it did. Clearly, there’s more to the story here.

#### Clever Susan

Susan takes a different approach. She creates a well-balanced portfolio of different assets in different maket sectors, and at the end of the competition she posts a very healthy 12% return – much less than Wild Bill.

However, the value of Susan’s portfolio grew steadily and predictably through the competition. On her worst day, she had still earned a return of 8%, but upon the whole it never varied too much from 12%.

## So who should win– Wild Bill or Clever Susan?

In other words, how do we quantify the difference between these two students?

Investment performance is scored by calculating the Sharpe Ratio of your end-of-day account balance over time. The Sharpe Ratio is a measure of your risk-adjusted return; in other words, how much money you made vs. the risk that you took on.

In finance, risk = volatility and is defined as the standard deviation of the returns that a portfolio earns over time. You want as large a return and as small a volatility of returns as possible.

This is the entire game and the goal of every investor: high returns, low volatility. The Sharpe Ratio is a way to quantify this concept. Read on to learn more about the components of this very important metric and how it’s calculated.

## The Risk-Free Rate $$r_{f}$$

At any time, just about anybody can simply buy a T-Bill from the US Government or another large sovereign nation. The thinking is that if a big country like the US defaults (doesn’t pay) a T-Bill debt at maturity, then something is so terribly wrong with the economy, the world political situation, etc., that nobody is worried about finance anymore… they’re worried about surviving the space alien invasion, global war, or other major disaster that must have taken place to cause the country to default. Obviously this turn of events would be very bad, but fortunately, the prevailing financial thinking is that the likelihood of such an occurrence is so low that its probability can be assumed to be 0.

In other words, these investments are thought of as risk free, to which we assign the symbol $$r_{f}$$.

So just buy government bonds, earn a zero-risk return, and forget about it, right? Well… you could, but becuase the investment is risk-free and easily available, it doesn’t pay very much.

The US Department of the Treasury compiles daily risk-free rates and publishes them on their website, where you can go to view & download the rates and learn about the methodology the USDT uses to calculate them.

## Daily Log Returns

Your return is calculated each day by observing the end-of-day value of all your assets and comparing it to the end-of-day value on the day before. That means that over a period of $$N$$ days, we’ll have $$N - 1$$ total observations of a portfolio’s return because you can’t measure a return for the very first day – you have nothing to compare it to.

The formula used to calculate the return you earned on your portfolio is the log ratio of $$V_{N}$$ – the value of your portfolio on Day $$N$$ – divided by $$V_{N-1}$$ – the portfolio’s value on the previous day; i.e., $$R_{N} = \log(\frac{V_{N}}{V_{N-1}})$$.

In the competition, daily log returns are calculated for every student’s portfolio and for the risk-free rate.

## Average (“Expected”) Return

TLDR: We use the geometric mean of period-over-period log returns.

Details: Let’s say you invest $100k into an investment portfolio. Each quarter, you measure the return you earned at the quarter’s end with respect to the how much the portfolio was worth at the beginning of the portfolio. You observe the returns +25%, -25%, +25%, -25% for each quarter. In other words, an initial investment of$100k would be worth $125k,$100k, $125k, and$100k at the ends of quarters 1, 2, 3, and 4, respectively.

The question is, what’s the overall return that characterizes this investment’s performance over the year?

#### Arithmetic Mean

You might think: “just take the average” of the returns, which works out to 2.5%. But that’s not quite descriptive of the whole picture is it… after all, if you tell a client that you earned an average of 2.5% on their investment of $100k, then they’d naturally expect to see something around$102.5k when they look up the value of the account right?

…but the value of the account is $100k, meaning that you’re probably going to have a client banging your door down wondering where their$2,500 went. The “normal average”, more properly called the arithmetic mean of the returns is therefore misleading in this way!

#### Geometric Mean

A better way to assign an overall charactaristic (“expected”) return for a time series of returns is to use the geometric mean, calculated by multiplying all of the returns together and taking the $$n$$th root, where $$n$$ is the number of returns observed in the period. The geometric mean of a series of returns over $$n$$ periods can be expressed as two equivalent formulae $$\left(\prod_{i=1}^{n}R_{i}\right)^\frac{1}{n}$$ and $$\sqrt[n]{R_{1}R_{1}R_{1}...R_{n}}$$.

Note that, in the example above, the geometric mean rate of return for the investment is 0, just as it should be.

## Excess Return

You’re going to be managing a portfolio – some set of bonds, stocks, currency trades, etc, on which you earn a portfolio return $$R_{p}$$. As an investor, you only care about earning a return that is better than the risk-free rate $$r_{f}$$ because otherwise, why are you even bothering to invest? Better to just buy government debt.

In other words, what you really care about is your portfolio’s Excess Return over the risk free rate.

Excess Return is defined as the difference between the return your portfolio earned and the return that was available by buying sovereign debt during a certain time period; in other words, $$R_{p} - r_{f}$$.

## Volatility (Risk)

For a series of returns, the volatility (or “vol”, for short) is defined as the standard deviation of the observed returns. In finance, risk is thought to be defined as volatility. A portfolio’s vol is assigned the symbol $$\sigma_{p}$$.

## The Sharpe Ratio

Now we can put it all together and, as William Sharpe did in 1966, write down a metric that goes down when expected return goes down & vol goes up, and up when expected return goes up and vol goes down:

\begin{align*} {\tt Sharpe\ Ratio} = \frac{R_{p} - r_{f}}{\sigma^{2}_{r}} \end{align*}

#### Wherein:

\begin{align*} R_{p} & {\sf: Portfolio\ Return} \\ r_{f} & {\sf: Risk\ free\ rate\ of\ return} \\ \sigma^{2}_r & {\sf: Volatility\ of\ return} \end{align*}