---
title: Expected risk and return
slug: expected-risk-and-return
docTags: 
createdAt: 2023-05-26T14:18:54.000Z
---

# Expected return

The expected monthly return of a portfolio is given by the following formula:

```tex
\mathbb{E}(r) = \mathbb{E}(r)_1 \times w_1 + \mathbb{E}(r)_2 \times w_2 + \dots + \mathbb{E}(r)_n \times w_n

```

where

```tex
\begin{align*}
E(r)_i & : \text{expected monthly return of instrument in the portfolio}, \\
w_i & : \text{weight of instrument in the portfolio}, \\
n & : \text{total number of instruments in the portfolio},
\end{align*}

```

and

```tex
w_1 + \dots + w_n = 1.

```

The expected annual return of the portfolio is calculated by   

```tex
return_{annual} = (1+E(r))^{12}-1.

```

Expected return for the instruments in the portfolio will vary based on the configuration. It can be one out of the three below: 

1. Historical return of the category where instrument is connected 
2. Houseview of the category
3. Houseview for instrument

If method one is applied, the expected monthly return of an instrument is calculated by   

```tex
E(r)_i = E\{log(\frac{pf}{po})\}

```

where *po* and *pf* are the instrument values at the beginning and the end of month, respectively.

# Expected risk

The calculation of the expected volatility of a portfolio is done through the following two steps. First we calculate the monthly variance of the portfolio by using the following formula: 

```tex
\sigma^2 = w^T \Sigma w

```

where:

```tex
\begin{align*}
\sigma^2 & : \text{variance}, \\
w & : \text{(column) vector of weights for each instrument in the portfolio}, \\
\Sigma & : \text{covariance matrix of the expected monthly returns of the instruments in the portfolio}, \\
T & : \text{transpose operator of vector/matrix}.
\end{align*}

```

Here,

```tex
w = [w_1, \dots, w_n]^T.

```

Second, we calculate the expected annual risk (in %) by using this formula: 

```tex
risk_{annual} = \sigma \times \sqrt{12} \times 100.

```

Expected risk for the instruments in the portfolio will vary based on the configuration. It can be one out of the three below: 

1. Historical risk measured on a monthly data  
2. Houseview risk of the category
3. Houseview risk for the instrument