Portfolio risk calculator and formula

We often say that risk and return are two sides of the same coin. So, when assessing the performance of a portfolio, we need to consider its risk as well as its return. In the previous lesson, we focused on portfolio return. Now, we turn our attention to portfolio risk.

Portfolio risk calculator

You can use the portfolio risk calculator below for portfolios containing up to three assets. Please note the following instructions:

For Asset 1, enter its variance (σ₁²) and its covariances with Asset 2 (σ₁₂) and Asset 3 (σ₁₃).
For Asset 2, enter its variance (σ₂²) and its covariance with Asset 3 (σ₂₃).
Finally, for Asset 3, enter its variance (σ₃²) only.
If your portfolio consists of 2 assets only, just leave 0s in the fields that you don’t need.
Make sure that investment weights add up to 100% (see the “sum of weights (%)” in the last row).
The calculator allows for both positive investment weights and negative investment weights (i.e., short selling).

Portfolio risk formula

In the previous lesson, we explained that a portfolio’s return is simply a weighted average of the returns of assets that constitute the portfolio. And, the weights are determined by the amount invested in each asset. This led us to the formula for portfolio return R_P:

where w_i is the investment weight for asset i and R_i is the realized return for asset i. In an earlier lesson, we introduced the variance of returns (let’s denote that as σ_i² for asset i) as a measure of risk. Could we then calculate the variance of returns on a portfolio σ_P² as a weighted average of the variance of assets in that portfolio? In other words, would the following equation capture portfolio risk?

The answer is no. When calculating σ_P², we need to consider not only the variance of each asset but also the covariance between each pair of assets. To understand why, imagine a portfolio with two assets only. Suppose that when the price of one asset goes up, the price of the other tends to go down. These opposite movements would partially cancel out each other, reducing the volatility of portfolio returns. So, the portfolio risk formula is:

where w_i is the investment weight for asset i, w_j is the investment weight for asset j, and σ_ij is the covariance between assets i and j.

Portfolio variance with two assets

For a portfolio containing two assets (A and B), the general portfolio risk formula simplifies into:

So, in a two-asset portfolio, σ_P² depends on:

The variance of returns for each asset (σ_A² and σ_B²).
The covariance between the returns of asset A and asset B (σ_AB).
And, the investment weights (w_A and w_B).

Portfolio variance with three assets

If a third asset (C) is added to the portfolio, we have:

Then, in a three-asset portfolio, σ_P² depends on:

The variance of returns for each asset (σ_A², σ_B², and σ_C²).
The covariance between each pair of assets (σ_AB, σ_AC, and σ_BC).
And, the investment weights (w_A, w_B, and w_C).

Portfolio variance with N assets

If we kept adding assets to the portfolio, we’d end up with the following formula:

This means that for a portfolio of any size, portfolio risk depends on the individual risk of each asset in the portfolio (σ₁², σ₂², …, σ_N²), the covariance terms (σ₁₂, σ₁₃, …, σ_{N-1 N}) and investment weights (w₁, w₂, …, w_N).

Summary

In this lesson, we have explained the portfolio risk formula, highlighting the importance of covariances between asset pairs. We have also provided a handy portfolio risk calculator.

What is next?

This lesson is part of our free course on investments.

Next lesson: We’ll investigate how investors can conduct mean-variance optimization to construct portfolios that are efficient in terms of risk and return.
Previous lesson: We explained how to calculate portfolio returns.

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