Estimating Sharpe Ratios from Portfolio Returns with Irregular Contributions

# Calculate Sharpe Ratio, Annualized Return, and Volatility for Uneven Cashflows and Mixed Asset Classes?


# Calculate Sharpe Ratio, Annualized Return, and Volatility for Uneven Cashflows and Mixed Asset Classes?












I am working on a portfolio problem and encountered some challenges related to calculating key performance metrics. I would greatly appreciate any guidance on the following:

- Say, I started with an initial investment of $50k in a portfolio consisting of x different stocks/ETFs. Following this, I have been making systematic investments (SIPs) of 1k dollar each month (or on random months) for N years. Now, I would like to compute the Sharpe ratio for this scenario, especially when considering different combinations of x stocks/ETFs. Given that the cashflows are irregular, I’m unsure how to compute the annualized return and annualized volatility in a mathematically robust manner. While I am familiar with the XIRR method for calculating annualized returns in case of uneven cash flows, I am uncertain how to approach the calculation of annualized volatility in this context.

- Additionally, I am curious about how to compute the Sharpe ratio for a combination of multiple asset classes. Specifically, I have investments in ETFs alongside trades in options. How should I compute the Sharpe ratio, annualized return, and annualized volatility for such a portfolio? Moreover, if I am using leverage by pledging the ETFs/stocks for options trading, how would the computation of these metrics—especially Sharpe ratio, annualized return, and annualized volatility—be affected?

Any insights, particularly with implementation examples in Python, would be greatly appreciated.

Thank you in advance!

## Answer by Con Fluentsy (score 1)

https://quant.stackexchange.com/a/81001

The sharpe ratio has nothing to do with the cashflows you are talking about; estimate the daily log price changes, this gives you the daily return sum; and you have the return over the period annualise this return, use the daily price changes to estimate the standard deviation. The sharpe ratio is return/ standard deviation. Thats it, you can make it a tougher measure by deducting risk free return from annual return and then dividing by standard deviation(volatility).XIRR has nothing to do with calculating sharpe ratio? If your assets are all market securities, just take the close of trade prices , to estimate returns:

### Portfolio Sharpe Ratio, Daily Log Price Change Returns, and Daily Standard Deviation

- Daily Log Price Change Returns (Log Returns)

Logarithmic returns (log returns) provide a continuous compounding view of returns and are calculated as:

r_t = ln(P_t / P_{t-1})

Where: r_t is the log return on day t, P_t is the price at the end of day t, P_{t-1} is the price at the end of day t-1, ln denotes the natural logarithm

Log returns are often preferred in financial analysis because they are time-additive, meaning returns over multiple periods can be summed up.

##### Example in R:

```
log_returns <- diff(log(price_data))
```

#### 2. Daily Standard Deviation from Log Price Changes (Volatility)

The daily standard deviation, or volatility, measures the amount of variation or dispersion in log returns. It is a key risk metric. The standard deviation is calculated as:

```
σ = sqrt( (1 / (n - 1)) * Σ (r_t - μ)^2 )
```

Where: σ is the daily standard deviation (volatility), r_t are the log returns, μ is the mean of the log returns, n is the number of periods.

##### Example in R:

```
daily_volatility <- sd(log_returns)
```

#### 3. Portfolio Sharpe Ratio

The Sharpe ratio measures the risk-adjusted return of a portfolio, calculated as the excess return per unit of risk. The formula for the Sharpe ratio is:

```
S = ( R̄ - R_f ) / σ
```

Where: S is the Sharpe ratio, R̄ is the average portfolio return (mean log returns), R_f is the risk-free rate of return, σ is the portfolio standard deviation (volatility).

##### Example in R:

```
mean_log_return <- mean(log_returns)
risk_free_rate <- 0.01 / 252  # Assuming 1% annual risk-free rate
sharpe_ratio <- (mean_log_return - risk_free_rate) / daily_volatility*sqrt(n)
```

By combining log price change returns, daily volatility, and the Sharpe ratio, we can analyze a portfolio's performance in terms of return per unit of risk. This helps investors understand whether they are being adequately compensated for the risks they are taking.

Python code as follows:

import numpy as np

## Example price data (replace with actual data)

price_data = np.array([100, 102, 101, 105, 107])

## Calculate daily log price change returns

log_returns = np.diff(np.log(price_data))

## Calculate daily standard deviation (volatility)

daily_volatility = np.std(log_returns, ddof=1)

## Calculate mean log return

mean_log_return = np.mean(log_returns)

## Assume 1% annual risk-free rate, converted to daily (252 trading days in a year)

risk_free_rate = 0.01 / 252

## Calculate Sharpe ratio

sharpe_ratio = (mean_log_return - risk_free_rate) / daily_volatility*(sqrt(n))

print(f"Log Returns: {log_returns}") print(f"Daily Volatility (Standard Deviation): {daily_volatility}") print(f"Mean Log Return: {mean_log_return}") print(f"Sharpe Ratio: {sharpe_ratio}")