Risk-Adjusted Returns and Diversification
I’ll state the obvious. A return is a measure of how much you gain or lose from an investment relative to the amount you put in. Two common methods of measuring returns:
Absolute Return: If you invest $1,000 and end up with $1,200, you have an absolute return of $200, or 20%.
Annualized Return: In practice, we often want to compare multiple investments over different timespans. Annualized return scales your total return up (or down) to a yearly figure, which makes it easy to compare investment opportunities. For instance, if your 20% total gain occurred over two years, the annualized return is roughly 9.545% per year.
$1,000 * (1.09545) ~ $1,095
$1,095 * (1.09545) ~ $2,000
This compounded return, however, does not tell you how bumpy the journey was. You can think of it as a ‘smoothed’ return. For example, you could have invested $1,000 and returned:
· -10% in year 1, leaving you with $900
· 33.33% in year 2, leaving you with $1,200
The absolute return was still 20%, the compounded return is still 9.4545%, but you encountered much more risk along the way (your returns were more volatile).
In finance, risk typically reflects the uncertainty associated with an investment’s returns. More uncertainty (volatility) increases the likelihood of experiencing large swings – both up and down – in the investment’s value.
Standard deviation is the commonly used statistic to measure this volatility. It tells us how “spread out” an investment’s returns are from its average. A higher standard deviation means that returns can fluctuate widely around the average, a.k.a. higher volatility.
As investors, we don’t usually like volatility. But we certainly like high returns. Our goal? Maximize our expected return; minimize our risk. But we can’t just chase high returns – we need to be wary of the risk we take on to achieve them. If you look at an investment in isolation, a 9.545% annual return might seem impressive. But if that return was achieved by enduring high volatility (large swings in value), then the risk-adjusted performance might not be as attractive (the juice may not be worth the squeeze).
If we want to measure the tradeoff between risk and reward, we can’t look at each item in isolation – returns separated from risk are unhelpful. To combine these items, active managers look at something called the Sharpe Ratio.
· Rp = Portfolio return
· Rf = Risk-free (or benchmark) return
· Sigma = Standard deviation of the portfolio
The Sharpe Ratio quantifies how much extra return you earn for the additional risk you take on. A higher Sharpe indicates that an investment’s returns are more attractive given the amount of risk incurred.
The Sharpe Ratio is particularly useful for comparing different portfolios. If two portfolios deliver the same returns but one incurs lower volatility, the portfolio with lower volatility (hence a higher Sharpe Ratio) is more appealing.
“But why are we subtracting the risk-free return from the portfolio return?”
Great question.
Subtracting the risk-free rate from the portfolio return isolates the excess return – the additional reward you receive for taking on extra risk. The risk-free rate represents the return you could earn without any risk, so by subtracting it, you're essentially asking:
"What extra return am I getting for the extra risk I'm taking on compared to a completely safe investment?"
This excess return, when divided by the portfolio's volatility (its standard deviation), gives you the Sharpe Ratio, which measures the efficiency of the return relative to risk. Without subtracting the risk-free rate, you wouldn't know how much of your return is just compensation for bearing risk, as opposed to what you could have earned risk-free.
Other Ratios
The process of standardizing returns per unit of risk is very common when measuring investment performance, and there are plenty of variants to the Sharpe Ratio that measure investment performance standardized by some kind of risk statistic.
For example, the Sortino Ratio is a metric that does exactly what the Sharpe Ratio does, but focuses only on downside volatility.
By focusing only on downside risk, the Sortino Ratio provides a more tailored view of how well a portfolio performs in avoiding losses.
The Treynor Ratio evaluates portfolio performance by comparing excess returns to systematic risk, measured by the portfolio's beta (β). Beta is simply how much the portfolio's returns move in relation to the overall market.
Unlike the Sharpe Ratio, which uses total volatility (standard deviation), the Treynor Ratio assumes that diversified portfolios have minimized unsystematic (idiosyncratic) risk, so only market-related (systematic) risk is relevant.
The Treynor Ratio is useful for comparing well-diversified portfolios, where non-market risk has been largely eliminated. It tells you how much return you're generating per unit of market risk, allowing you to gauge whether you’re being adequately rewarded for the risk that truly matters in a diversified setting.
Putting it all together
So now that we have an understanding of returns, risk, and risk-adjusted returns, and now that we know how to measure and benchmark those risk-adjusted returns, let’s take it one step further. I’m going to demonstrate how and why combining assets into a portfolio (diversifying a portfolio) can increase risk-adjusted returns for an investor.
In modern portfolio theory, there is a concept called the “efficient frontier.” The concept of efficiency here is very simple – it means for every level of risk we take on for a particular portfolio, we maximize our level of portfolio return.
The efficient frontier helps investors visualize the best possible risk-return tradeoffs for a given level of risk. The efficient frontier is a line chart measured by:
· Risk (standard deviation) on the x-axis
· Expected Return on the y-axis
Note: Image above pulled from AnalystPrep.com
I am going to demonstrate how to create an optimized portfolio.
To start, we create a table of three assets. All data you see below is dummy data – only used for demonstration purposes.
The table below represents the universe of our investible assets. In the table below, you can see:
· Assets: These identify our investment options.
· Expected Returns: Forecasted annual returns for each asset (which can be based on historical data or analyst estimates). In this example, I use 3% for Asset A (conservative risk investment), 7% for Asset B (moderate risk investment), and 12% for Asset C (aggressive risk investment).
· Initial Weights: These are the percentage of the portfolio that each asset takes up.
To start this exercise, I simply put 1/3 of the portfolio into each asset.
Step 2 is building something called a covariance matrix. This sounds complex, but if you don’t know what that is, don’t be intimidated. A covariance matrix quantifies the relationship between the returns of different assets. Basically, you can think of covariance as the relationship between two assets – how they move in relation to one another. When an asset in the portfolio has a positive return, another asset in the portfolio might also have a positive return. When an asset in the portfolio has a negative return, another asset in the portfolio might have a positive return.
The covariance matrix tells you how much each asset’s returns deviate from its own average (variance) and how they interact with each other (covariance). If assets move independently or in opposite directions, combining them can reduce overall portfolio risk (diversification).
Again, don’t think too hard about the actual values shown. This is all dummy data only used as an example.
· Diagonal Cells (Variance): Represent each asset’s variance, showing how their returns vary on their own (e.g. 0.0016 is Asset A’s variance)
· Off-Diagonal Cells (Covariance): Measure how the returns of two different assets move together. A positive value means they tend to move in the same direction; a negative value indicates an inverse relationship. In this example, we assume a positive relationship between all assets.
With these assumptions, we can now calculate our portfolio’s expected return and risk. I’m abstracting some of the math involved for simplicity. Using the data above, we arrive at the following.
· Portfolio expected return: 7.26% (take the sumproduct of each asset’s expected return and weight)
· Portfolio variance: 0.0043 (use matrix multiplication)
· Portfolio standard deviation: 0.0656 (square root of portfolio variance)
But we can do better! Our portfolio weights aren’t optimized – all we’ve done so far is assume we put a third of our portfolio into each asset. That may help with diversification, but how do we know it’s the right way to weight our portfolio? I’ll answer this question after a quick recap.
Quick recap
Asset Table Setup: We’ve built a table with 3 assets, including expected returns and initial weights. This serves as the foundation for our portfolio analysis.
Covariance Matrix Construction: We’ve created a covariance matrix. This matrix helps us understand how assets move in relation to one another.
Portfolio Metrics Calculation:
Expected Return: We calculated the unoptimized expected return given our initial weighting assumptions.
Portfolio Variance: We used matrix multiplication to calculate the variance of our portfolio.
Standard Deviation: We took the square root of our calculated portfolio variance to give us a clear measure of the portfolio’s risk.
Now we need to optimize this portfolio. To do this, we adjust the asset weights so that we can maximize our expected return for a given level of risk. This will allow us to map out an efficient frontier. To do this, I’m going to use Excel’s Solver add-in.
My assumptions for Solver are as follows:
Goal: Minimize variance for a target level of return.
Constraints:
Sum of portfolio weights must be 1 (we must be fully invested in the market).
Prevent negative weights (no short selling).
Expected return constrained to our target (we optimize for a target level of return).
Here’s what that looks like in the Solver Window.
Again, Solver’s objective is to minimize variance for a target level of return. I iteratively pass into Solver a target return (say, 3%, 5%, 10%, etc.), and Solver finds, based on the expected returns and covariance matrix we built earlier, the standard deviation and expected return for the portfolio given that target return. Here is a table showing the output:
When we plot this table as a scatterplot, we get the following output:
This is the efficient frontier for our 3 investable assets. It is a visual representation of the best (lowest risk) portfolio for each level of return.
Here is what this looks like using Python. In this version, it’s easier to visualize how the efficient frontier outputs the best portfolio for each level of risk.
It’s important to note that while the efficient frontier provides a powerful quantitative framework for optimizing portfolios, qualitative analysis is absolutely crucial for portfolio management. Industry trends, economic forecasts, and other data play a crucial role in decision-making. This is just one piece of the puzzle.
While these metrics and visualizations are helpful for theoretical understanding, the underlying principles behind risk-adjusted returns are more important. Fundamentally, every investment has its limitations, whether it be opportunity cost, liquidity constraints, or something else. Recognizing and accounting for risk/reward trade-offs ultimately leads to a more resilient investment approach - one that prioritizes a risk-first mentality over returns-chasing.