Introduction to Risk in Finance

Risk – in its essence, is present everywhere around us. We generally seek to mitigate or manage it, and the same applies in the field of finance. Whether the objective is to eliminate risk entirely or to control it so that our financial decisions do not turn against us, the underlying goal remains the effective management of uncertainty.

In finance, risk can be defined as the possibility that actual returns will deviate from the expected returns, $E[R]$. While this deviation can occur in either direction—and most investors would certainly welcome a few extra percentage points of profit—the main focus usually lies in managing and mitigating downside risk. Investors are typically more concerned with protecting their portfolios from losses than with limiting unexpected gains.

1. Financial Risk

In the world of finance, there are many types of risks lurking around the corner. Some are quite apparent, while others may not be as obvious. In this section, we will classify and discuss the major groups of risks that an investor may encounter in a financial context.

In general, financial risk can be divided into two main categories: systematic risk and specific risk.

systematic risk represents the portion of total risk that is always present and cannot be eliminated, as will later be seen in the definition of market risk. The terms systematic risk and market risk are often used interchangeably, as many authors define market risk as a form of systematic risk.

Specific risk, on the other hand, refers to the portion of risk that can be mitigated—or in some cases, even eliminated entirely. Depending on the type of risk, various strategies can be employed to reduce it, with hedging and diversification being among the most common approaches.

1.1. Market Risk

This type of risk usually arises from macroeconomic events, structural changes in the economy, political shifts, and similar factors. It represents the portion of total risk that while can be mitigated via hedging, or diversification, it cannot be eliminated by the investor entirely, which is why it is often also referred to as systematic risk. Two prominent examples of market risk are currency risk and interest rate risk.

Currency risk (often called foreign exchange or FX risk) is a classic example of systematic risk. While hedging strategies can help mitigate it, there will always remain an element of uncertainty that cannot be completely eliminated. Currency risk can be defined as the risk of a change in the value of an asset denominated in a currency different from the investor’s base currency, caused by fluctuations in exchange rates in the financial markets.

Interest rate risk is another important example, and it has become particularly relevant in recent years as central banks have adjusted their monetary policies. Although investors can mitigate this risk through hedging, it cannot be entirely removed, which again makes it a form of systematic risk. Interest rate risk can be defined as the risk of a change in an asset’s value resulting from shifts in interest rates driven by central bank policy decisions.

1.2. Credit Risk

Credit risk is a type of risk most commonly present in the capital markets, particularly when investing in money market instruments or debt securities. It represents the possibility that a borrower’s ability to repay the lender will deteriorate, or that the borrower may become unable to meet their obligations altogether. This type of risk depends on various factors, such as the borrower’s liquidity position, leverage, and overall financial health.

Credit risk can generally be assessed and partially mitigated by examining the borrower’s credit rating or by analyzing the credit spread—the difference between the yields of two bonds with similar characteristics (such as maturity and principal). The credit spread provides insight into how much additional risk the market attributes to a particular security relative to a risk-free benchmark.

1.3. Liquidity Risk

This type of risk represents the low ability—or even the inability—to liquidate a given asset. If an asset carries a high level of liquidity risk, it means the market for that asset is relatively illiquid, exposing the investor to potential complications when attempting to sell it. In such cases, one might have to wait for a buyer or sell the asset at a discount to its current market value in order to attract interest. This situation often arises when an investor needs to sell an asset abruptly or under unfavorable market conditions – which of itself is a very inconvenient time to sell anyway.

1.4. Operational Risk

Operational risk represents the possibility of incurring losses due to failures in internal processes, systems, or human error. It can also arise from indirect or external causes, such as natural disasters, cyberattacks, or other unforeseen events that disrupt normal business operations. Unlike market or credit risk, operational risk does not stem directly from market movements or counterparty default but rather from the way an organization conducts its daily activities.

Examples of operational risk include system outages that prevent transactions from being processed, accounting or data entry errors that lead to financial discrepancies, or breaches in internal controls that allow for fraud or unauthorized activity. It is important to note that well-established institutions might be exposed to operational risk, as complete prevention is virtually impossible. However, it can be effectively mitigated through strong internal controls, risk management frameworks and regular audits.

2. Quantifying Financial Risk

In the financial setting, we can use the mean expected return $E[R]$ as an indicator of an expected return of our investment, and the standard deviation - also noted as sigma $\sigma$ to describe the risk of an asset or more often a portfolio of assets. We’ve discussed how to quantify the risk of a single asset in the previous article.

2.5. Measuring the Expected Return of a Portfolio

Before we can go and measure the risk of a portfolio, we first need to define a way to compute the expected return of that portfolio. Since risk is defined and measured as the standard deviation from this expected return, it represents a key metric we must know before doing anything else.

In the case of a single asset, we can compute the mean of the holding period returns simply by calculating its average return over time, as shown in the following equation:

$$\text{E[R]}=\frac{1}{N}\sum_{t=1}^N r_t\,\,\,\,\,\,\text{ where } \,\,\,\,\,\,r_t=\frac{P_{t+1}-P_t}{P_t}\approx \text{ln}\left(\frac{P_{t+1}}{P_t}\right)$$

To compute the expected return of a portfolio, we can use the expected returns of each asset and take their weighted sum, where the weights represent the percentage allocations of the assets within the portfolio. The following equation expresses how to calculate the expected return of a portfolio in mathematical terms:

$$\text{E[P]}=\sum_{i=1}^N E[R_i]\cdot w_i\,\,\,\text{where}\,\,\,w_i\text{ represents the weight of an asset}$$

2.6. Measuring the Risk of a Portfolio

When it comes to measuring the risk of a portfolio, at first glance it might seem as simple as computing the weighted sum of the standard deviations of the assets within it. However, that assumption—although it might sound quite reasonable—couldn’t be further from reality.

In fact, when assessing the risk of a portfolio, we must not only consider how individual assets fluctuate relative to their own means, but also how they move in relation to one another. This interaction between assets can either amplify the overall volatility of the portfolio—making it riskier—or, under certain conditions, reduce it through diversification effects. Based on this idea, we can begin to quantify the variance of a portfolio.

Before delving into the mathematics of modern portfolio theory, it is useful to visualize how assets move together using a two-dimensional matrix, which provides an intuitive view of how the components of a portfolio interact. This structure is known as the covariance matrix, and it plays a fundamental role in portfolio mathematics when evaluating overall portfolio risk.

$$\sigma^2[P]=\begin{bmatrix}
w_1,w_2
\end{bmatrix}

\cdot

\begin{bmatrix}

cov(r_1,r_1) & cov(r_1,r_2) \\
cov(r_2,r_1) & cov(r_2,r_2)

\end{bmatrix}

\cdot

\begin{bmatrix}
w_1 \\
w_2
\end{bmatrix}$$

If we look at the diagonal of the covariance matrix for two assets, we can see that it represents nothing other than the dispersion (or variance) of each asset. This is because the covariance of an asset with itself—essentially how it moves in relation to itself—is simply its own variance. The off-diagonal elements, meanwhile, represent how the assets move together. Naturally, all of these relationships must be weighted by the corresponding asset weights within the portfolio.

If we write this relationship as an explicit equation for a two-asset portfolio, it takes the following form:

$$\sigma^2[P]=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1 w_2 \cdot cov(r_1,r_2)$$

For a portfolio consisting of three assets, the covariance matrix becomes a $3\times3$ structure as follows:

$$\begin{bmatrix}

\sigma^2_1 & cov(r_1,r_2) & cov(r_1,r_3) \\
cov(r_2,r_1) & \sigma^2_2 & cov(r_2,r_3) \\
cov(r_3,r_1) & cov(r_3,r_2) & \sigma^2_3 \\
\end{bmatrix}$$

As the number of assets increases, the diversification effect becomes more pronounced. While the impact of diversification in a two-asset portfolio is limited, the number of covariance terms—and therefore the potential interactions—grows rapidly with the number of assets.

Specifically, the number of covariances increases with $n(n-1)$ meaning that for a portfolio of 10 assets, there are $10\cdot10-10=90$ covariances influencing the total portfolio variance, which far exceeds the 10 individual asset variances.

$$\sigma^2[P]=\sum_{i=1}^n\sum_{j=1}^n w_i\cdot w_j\cdot cov(r_i,r_j)$$

We could go on and try to write down the equation for a three and more asset portfolios, but as the number of assets grows, and as already mentioned thus the number of covariance combinations grows too, it can become quite tough and unreadable.

That’s why in the above equation we instead have a look at the general formula for the portfolio’s variance $\sigma^2[P]$, this equation is also a representation of the covariation matrix we had a look at earlier.

2.7. How Diversification Can Mitigate a Part of the Risk in a Portfolio

Diversification as a way to mitigate risk is undoubtedly a great and popular concept among investors. We hear it everywhere—and it’s true: “distributing eggs between many baskets” can mitigate the risk of a portfolio. With this new knowledge, we can go further and actually look at how diversifying our portfolio mitigates part of the risk—the specific risk, to be precise.

$$\sigma^2[P]=\sum_{i=1}^n w_i^2\cdot\sigma^2_{r_i}+\sum_{i=1}^n\sum_{j\neq1}^n w_i\cdot w_j\cdot cov(r_i,r_j)$$

We can take the previous general formula for the portfolio’s risk $\sigma^2[P]$, split it into two parts as shown above, and observe how it behaves as we hypothetically increase the number of assets in the portfolio towards an infinite amount of assets. To simplify, we’ll assume the weights of the different assets in our hypothetical infinite portfolio are the same across all assets.

$$\sum_{i=1}^n w_i^2\cdot\sigma^2_{r_i}=\left(\frac{1}{n}\right)^2\sigma^2_{r_i}$$

If we assume that the risk of each asset is the maximum risk present among the assets in the portfolio, so that $\sigma_{r_i}^2 = \sigma_{max}^2$, we can prove that as the number of assets approaches infinity—by using the limit—no matter the risk of the asset, this part of the risk can be diversified to zero.

$$\lim_{n\to\infty}\,\frac{1}{n^2} \cdot n\cdot\sigma^2_{max}=0$$

For the second part of our equation, we’ll make a different assumption—that we have an average covariance $\overline{\sigma}$ of our portfolio, which is the sum of the covariances of the portfolio divided by the number of covariances $n(n-1)$, as discussed earlier.

$$\overline{\sigma}=\frac{\sum_{i=1}^n\sum_{j\neq1}^ncov(r_i,r_j)}{n(1-n)}$$

Our second part of the equation then becomes as follows. It’s worth noting that the weights stay the same under the assumption of equal asset weights in our hypothetical portfolio, and we must multiply by $n(n-1)$ to maintain equality.

$$\sum_{i=1}^n\sum_{j\neq1}^n w_i\cdot w_j\cdot cov(r_i,r_j)=\frac{1}{n^2}\cdot\overline{\sigma}\cdot n(n-1)=\left(1-\frac{1}{n}\right)\cdot\overline{\sigma}$$

Then, as we compute the limit of this function for $n \rightarrow \infty$, we find that while a certain part of the risk can be lowered by diversification as proven by the limit of the first part of our equation, there will still remain a part of the (systematic) risk that can’t be mitigated through diversification as proven by the following limit.

$$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)\cdot\overline{\sigma}=\overline{\sigma}$$

3. Conclusion

Risk will always be part of the financial world—it’s what makes investing both challenging and worthwhile. Every choice we take in the financial markets carries a degree of uncertainty, and that’s what keeps markets alive. The real goal isn’t to remove risk completely but to understand it well enough to keep it from taking control or end up going against us. By recognizing the difference between risks we can manage and those we can’t, we give ourselves the ability to act with clarity instead of fear.

Strategies like diversification and hedging don’t make risk disappear, but they help keep it in check. They give investors a sense of direction when the market feels unpredictable. Still, managing risk isn’t just about using financial tools; it’s about judgment, awareness, and learning from experience. Markets change, new risks appear, and what worked yesterday might not work tomorrow.

In the end, risk is what makes finance dynamic—it’s the price of opportunity. To manage it wisely is to accept that uncertainty is part of progress, and that smart decision-making isn’t about avoiding losses entirely, but navigating them with purpose and understanding.