# Modern Portfolio Theory Statistics

### Benchmarks:

At Rootmont, we use a variety of benchmarks to measure the behavior of a coin in different contexts.
Given a set of coins X and a specific coin c in X, we define the benchmark for X to be the market-cap weighted sum of the performance of all coins in X.
Essentially, one can think of it as simulating the following investment strategy.

Suppose you have $6000 and your set X is comprised entirely of two coins, A and B. Suppose further that on day 1, A has a marketcap of$1m and B has a marketcap of $2m. Then we would put$2000 in A and $4000 in B. Suppose then at the end of the simulation that A’s marketcap had risen to$1.5m and B’s had increased to $4m. Now our portfolio is$3000 on A and \$8000 on B, giving us a return of 83%. This would be our benchmark.

### Simulations:

Please understand the following limitations to our investment simulations and take them into account when evaluating data on Rootmont.
1/ We do not simulate trading fees. Trading fees vary between exchanges and typically depend on volume.
2/ We do not account for price movement due to trading. This is primarily because historic data on “Level 2” of the exchange’s order books is hard to come by.
3/ Changing even one atom in the past would have immeasurable consequences moving forward. “No man steps in the same river twice, because it is not the same man and it is not the same river”.
4/ The performance of the strategy is dependent upon the sampling rate of the data. We use daily data, but a lot can happen in a day, so performance may vary significantly if the simulation is run at a different frequency.
5/ Each coin has a relatively short window of data available for it, so the portfolio statistics displayed on Rootmont will have a wider error window than their equivalents in equities.
6/ Coin prices are taken to be the last price the coin traded at on midnight UTC time. Due to the global reach of cryptocurrencies and 24/7 trading, we expect aliasing issues to be minimal.

### Covariance:

Covariance is a measure of how independent two variables are. When covariance appears on a coin report, we are using the coin variable and the corresponding benchmark variable, typically price.
A zero covariance means that the variables are entirely independent, knowing one tells you nothing about the other.
A positive or negative covariance means that the deviation from the mean in one variable makes deviation from the mean in the other variable more likely.
Positive covariance means that both variables are likely to be on the same side of their respective means, whereas negative covariance means they are likely to be on opposite sides.

$$cov(c,b) = \frac{1}{n} \sum_{i=0}^{n-1}(b_i – avg(b)) * (c_i – avg(c))$$

where $$avg(x) = \frac{1}{n} \sum_{i=0}^{n-1}$$ $$x_i$$, and $$c_i$$ is the i-th value of the coin time series, and $$b_i$$ is the i-th value of the benchmark time series.

### Beta:

We define the beta of a coin to be the amount of variance in the coin that is explainable by variance in the benchmark. More precisely,
$$\beta = \frac{cov(b, c)}{var(b)}$$

Where $$b$$ is the time series of percent changes in the value of the benchmark, $$c$$ is the time series of percent changes in the value of the coin and $$std(x) = \sqrt {\frac{1}{n}\sum_{i=0}^{n-1} (x_i – avg(x))^2}$$ and $$var(x) = std(x)^2$$

### Alpha:

There are many ways to define alpha. At Rootmont, we define the alpha of a coin as the risk-adjusted excess-performance of a coin’s price, relative to the benchmark.
$$\alpha = c_r – \beta * b_r$$
Where $$c_r$$ is the average of the daily percent return that the coin achieved in the given time window and $$b_r$$ is the average of the daily percent returns that the benchmark achieved in the given time window. More precisely, if $$c_i$$ is the coin value on the i-th day, and $$b_i$$ is the benchmark value on the i-th day, then

$$c_r = \frac{1}{n} \sum_{i=1}^{n-1} \frac {(c_i – c_{i-1})}{c_{i-1}}$$
$$b_r = \frac{1}{n} \sum_{i=1}^{n-1} \frac {(b_i – b_{i-1})}{b_{i-1}}$$

### Upside Capture:

We define the upside capture ratio to be the proportion of the upside of the benchmark that the coin captures. More precisely,
$$upside(c,b) = 100 * \frac{\sum_{i=0}^{n-1}max(0, c_i)}{\sum_{i=0}^{n-1}max(0, b_i)}$$

### Downside Capture:

We define the downside capture ratio to be the proportion of the downside of the benchmark that the coin captures. More precisely,
$$downside(c,b) = 100 * \frac{\sum_{i=0}^{n-1}min(0, c_i)}{\sum_{i=0}^{n-1}min(0, b_i)}$$

### Sharpe Ratio:

The sharpe ratio is a measure of risk-adjusted return, meaning that good returns are tempered by high risk (variance in the price) and amplified by low risk.
$$sharpe(c,b) = \frac{avg(c – b)}{std(c)}$$

### Sortino Ratio:

The sortino ratio is a different measure of risk-adjusted return, meaning that good returns are tempered by high downside and amplified by low downside.
$$sortino = \frac{avg(c – b)}{std(neg(c))}$$
where $$neg(x)$$ forms a new series by only selecting the negative values of the argument series.

### Treynor:

Treynor is another risk adjusted ratio that weights by beta, so being highly correlated with the benchmark reduces the impact of excess returns while being uncorrelated amplifies the effect of excess returns.
$$treynor = \frac{avg(c – b)}{beta}$$

### R-Squared:

$$r^2$$ is a normalized version of covariance that weights by the geometric mean of the variance of the coin and the variance of the benchmark.

$$r = \frac{cov(c,b)}{ \sqrt{(var(c) * var(b))}}$$