Key statistics

It is very important to consider important indicators before investing. Backtesting is a useful tool for checking these things and can help you to modify and improve your strategy based on the results.

While most investors are only interested in returns, it is also important to reduce risk, as reducing risk is more effective in maximising compounding returns than increasing returns. We should therefore look at the Sharpe and Sortino indices, which measure risk-adjusted returns. A high risk-adjusted return indicates a relatively very efficient strategy.

Although these metrics are absolute, they are also useful for the relative evaluation of strategies, i.e. we can analyse their relative strengths and weaknesses on the same basis.

• Compound Annual Growth Rate (CAGR)

  Compound Annual Growth Rate, which is the annualised return of the total cumulative return in one year.

• Maximum Drawdown (MDD)

  Short for Max Drawdown, it refers to the decline from the maximum high to the maximum low up to that point in time.

• Volativity

  We define risk as the amount of variation relative to the average of an investment and a portfolio. We define volatility as the amount of variation from the average return over a given period.

• Sharpe Ratio

  The return of a portfolio divided by its risk (volatility) as a relative measure of investment performance. It refers to the return you get for each unit of risk. In principle, the risk-free rate should be subtracted from the portfolio return.

• Sortino Ratio

  Similar to the Sharpe Ratio, but calculates volatility with negative volatility by including only negative returns in the calculation of risk (volatility). It therefore represents the return that can be earned per unit of negative volatility (risk).

• UPI (Ulcer Performance Index)

  The Ulcer Performance Index (UPI) shows the excess performance per unit of risk by dividing the CAGR (compound annualised growth rate) over the risk-free return by the Ulcer Index (UI). A higher value indicates relatively better risk-adjusted performance.

• Annualized Turnover (Turnover Ratio)

  The annualised turnover ratio shows how much a strategy has bought and sold over its lifetime. It is calculated by taking into account both 'buy' and 'sell'.

Annual Turnover = ( Average Monthly Buy Percentage + Average Monthly Sell Percentage ) x 12

For example, an annualised turnover of 100% indicates that, on average, 50% of the total assets were switched (selling one asset and buying another of the same size) during the year. Assuming that all assets were switched in a year, the annualised turnover would be 200%.

This turnover is an important metric to consider in an asset allocation strategy. Since there is an absolute cost to changing positions in financial markets, it stands to reason that the strategy that can maintain the lowest turnover will be relatively superior.

One of the reasons we tend to use passive ETFs among our ETFs is that passive ETFs (index ETFs), which track an index, have almost no transaction costs of their own (other than moving in and out of the index).

Glossary

CAGR

It is the average annual percentage gain of the portfolio, the cumulative return (how much the backtest or strategy has made cumulatively over X number of years) is calculated and annualised. CAGR can be used as an easy way to compare the returns of different strategies. You should also use CAGR when backtesting, as a 30 year strategy and a 10 year strategy will have different cumulative returns over the period.

$$CAGR= ({EV\over BV}) ^{1/n} ​ −1×100$$

EV = Ending Value

BV = Beginning Value

n= Number of years

Annulized Volatility

The standard deviation of a portfolio's returns over a one-year period. Volatility is used as a measure of risk, so higher trading volumes indicate a riskier model.

$$\sigma_T = \sigma \sqrt{T}$$

σT = volatility in a given period

σ = standard deviation of returns

T = number of periods in a given period

$$\sigma{\displaystyle _{}= {\sqrt {{1\over n -1} \sum_{i=1}^{n} (R_i ​− \bar{R})^2}}} ​$$

Standard deviation is the square root of variance. Variance is used to measure the dispersion of returns around the mean and is calculated as the square of the distance from the mean. Standard deviation is the most commonly used volatility and risk measure for portfolio returns.

σ = standard deviation of return

n = number of periods used

Ri - R = Distance from the mean

Max. Drawdown (MDD)

The maximum decline in the portfolio or the maximum negative difference in the value of the portfolio. It is calculated as the difference between the peak and the trough before a new high is reached.

MDD is important because it shows you the biggest swings in your portfolio's returns, and you can see how long your returns have been negative and whether you are experiencing more losses and volatility compared to your benchmark. The MDD is always negative, so less is more (the closer to zero, the better).

$${\displaystyle \operatorname {MDD} (T)=\max _{\tau \in (0,T)}D(\tau )=\max _{\tau \in (0,T)}\left[\max _{t\in (0,\tau )}X(t)-X(\tau )\right]}$$

Sharpe ratio

The Sharpe ratio is named after its invention by William Sharpe. It is calculated as a strategy's risk/reward ratio, or risk-adjusted return, calculated as annualised return/annualised volatility.

The Sharpe ratio is one of the most famous ratios because it expresses the return an investor earns relative to the risk they take - in other words, it shows how much risk they take for each unit of return (the risk/reward ratio). The return and the risk use the average return and the standard deviation of the return. Therefore, a Sharpe ratio of 1 or higher indicates a good portfolio and more reward than risk.

$$Sharpe =\frac{\left[R_{p}-R_{f}\right]}{\sigma_{p}}$$

Rp = Portfolio Return

Rf = risk-free rate of return

σp = standard deviation of returns

The standard deviation measures how spread out the data is in a statistic. In the case of returns, how spread out the returns are means how volatile they are. The reason we use standard deviation is that the more spread out your returns are, the more likely you are to experience losses, and therefore the more volatile they are.

The risk-free rate is usually the yield on a government bond, or it could be the rate you would expect to earn if you were to buy and hold.

Sortino ratio

Return adjusted for downside risk, calculated as annualised return/annualised volatility of negative returns. Volatility is a measure of risk. It is almost identical to the Sharpe ratio with one difference. The difference is that Sharpe looks at all volatility, while Sortino looks at downside volatility.

In general, you want an asset allocation strategy that delivers a high Sharpe and a high Sortino. If you're interested in reducing overall volatility, use Sharpe, and if you're interested in reducing negative drawdowns in your portfolio, look at Sortino.

$$Sortino=\frac{\left[R_{p}-R_{f}\right]}{\sigma_{p}}$$

​​Rp = Return on portfolio

​​Rf = risk-free rate of return

​σp = standard deviation of the asset's negative return

Ulcer Index

The Ulcer Index (UI) is a measure of the magnitude and duration of price declines from previous highs. Like the Sortino index, it captures volatility only as a downward change in price. While standard deviation does not tell us how much risk a portfolio or strategy has reduced by avoiding market downturns, the UI avoids the problems that standard deviation has in measuring the risk of market timing strategies.

The greater the fall and the longer it takes to recover to the previous peak, the higher the Ulcer Index. Strictly speaking, it penalises large drawdowns more than small ones because it is calculated using the square of the decline in value. In short, the UI measures the severity of drawdowns.

$${\displaystyle R_{i}=100\times {price_{i}-maxprice \over maxprice}}$$

$${\displaystyle Ulcer={\sqrt {R_{1}^{2}+R_{2}^{2}+\cdots R_{N}^{2} \over N}}}$$