Sharpe Ratio with Julia

class: center, middle, inverse, title-slide

.title[
# Sharpe Ratio with Julia
]
.subtitle[
## Mathematical Programming with Julia
]

---

### Sharpe Ratio

**Here's the formula:**

**Sharpe Ratio = (Rp - Rf) / σ**

* **Rp:** Portfolio return
* **Rf:** Risk-free rate of return (e.g., return on a short-term government bond)
* **σ:** Standard deviation of portfolio returns

---

**Interpretation:**

* **Higher Sharpe Ratio:** Indicates better risk-adjusted performance. This means the investment generates higher returns for a given level of risk (volatility).
* **Negative Sharpe Ratio:** Suggests the investment's returns are below the risk-free rate after adjusting for risk. 
* **Sharpe Ratio of 1 or higher:** Generally considered good. 
* **Sharpe Ratio of 2 or higher:** Often considered very good.

---

**Key Points:**

* **Risk-Adjusted Return:** The Sharpe Ratio focuses on how much excess return you receive for each additional unit of risk you take.
* **Volatility:** Standard deviation measures the volatility or fluctuation of returns. Higher volatility implies higher risk.
* **Comparison:** The Sharpe Ratio helps compare the performance of different investments or portfolios, even if they have different levels of risk.

**Example:**

* If a portfolio has an annual return of 12%, the risk-free rate is 3%, and the standard deviation of returns is 8%, the Sharpe Ratio would be: 
   (12% - 3%) / 8% = 0.9

---

**Limitations:**

* **Assumes Normal Distribution:** The Sharpe Ratio assumes that returns are normally distributed, which may not always be the case in reality.
* **Focus on Total Volatility:** It considers both upside and downside volatility, while some investors may be more concerned with downside risk only.

**In Summary:**

The Sharpe Ratio is a valuable tool for evaluating the risk-adjusted performance of investments and portfolios. By considering both return and risk, it helps investors make more informed investment decisions.

---

```julia
function calculate_sharpe_ratio(returns, risk_free_rate)
    """
    Calculates the Sharpe Ratio for a given series of returns.

Args:
        returns: A vector of returns.
        risk_free_rate: The risk-free rate of return.

Returns:
        The Sharpe Ratio.
    """

avg_return = mean(returns)
    std_dev = std(returns)

sharpe_ratio = (avg_return - risk_free_rate) / std_dev

return sharpe_ratio
end

# Example usage:
returns = [0.02, -0.01, 0.03, -0.02, 0.01]
risk_free_rate = 0.01

sharpe_ratio = calculate_sharpe_ratio(returns, risk_free_rate)
println("Sharpe Ratio:", sharpe_ratio)
```

---

**Explanation:**

**1**. **Function Definition:**
   - `calculate_sharpe_ratio(returns, risk_free_rate)`: Defines the function with two arguments:
     - `returns`: A vector containing the series of returns.
     - `risk_free_rate`: The risk-free rate of return.

**2**. **Calculate Average Return:**
   - `avg_return = mean(returns)`: Calculates the average return of the given series of returns using the `mean()` function.

**3**. **Calculate Standard Deviation:**
   - `std_dev = std(returns)`: Calculates the standard deviation of the returns using the `std()` function.

**4**. **Calculate Sharpe Ratio:**
   - `sharpe_ratio = (avg_return - risk_free_rate) / std_dev`: Calculates the Sharpe Ratio by dividing the excess return (average return minus risk-free rate) by the standard deviation.

---

**5**. **Return Sharpe Ratio:**
   - `return sharpe_ratio`: Returns the calculated Sharpe Ratio.

**6**. **Example Usage:**
   - `returns = [0.02, -0.01, 0.03, -0.02, 0.01]`: Creates a vector containing sample return data.
   - `risk_free_rate = 0.01`: Sets the risk-free rate.
   - `sharpe_ratio = calculate_sharpe_ratio(returns, risk_free_rate)`: Calls the function to calculate the Sharpe Ratio.
   - `println("Sharpe Ratio:", sharpe_ratio)`: Prints the calculated Sharpe Ratio to the console.

This Julia script provides a concise and efficient way to calculate the Sharpe Ratio.