Introduction

On part one of this portfolio performance series: Into The Rabbit Hole, we came up with important questions that come up when evaluating portfolio performance. We also went over some of the key metrics needed to answer these questions. On part two, we’ll expand our discussion on important measures of performance and describe in detail each metric calculation. As a heads up, the math might feel a bit harder, but don’t worry about understanding all the equations. Focus more on understanding the concept. On part three: Help Along The Way, we’ll show you a helping hand to make it all simple and easy!

The Elephant In The Room

There is an aspect of portfolio performance that we still haven’t addressed. It is the elephant in the room, that we’ve been avoiding: risk! Everyone knows investing can be risky. The more risk you take, the higher the potential reward, so our performance metrics are intrinsically connected to risk.

As investors, we value predictability. A startup stock, with a price that roller-coasters every month is riskier than cash in the bank. Investors willing to take more risk, holding the startup stock, should be rewarded accordingly. Is 30% return per year better than 20%? The answer is: it depends! Which investment is generating more return per level of risk? Any performance evaluation of our investment would be incomplete without assessing risk. Measuring volatility of returns is one way to capture the level of up and down swings in performance. Variance and standard deviation of returns are volatility metrics we can use to measure the total risk of an asset or portfolio.

Total risk can be split in two parts: one part relates to overall market risk, known as systematic risk; another part is connected to specific asset risk, also known as idiosyncratic risk.

We briefly touched on this topic, when we talked about the CAPM and the beta metric. Beta is a metric that represents overall market risk. Dividing the covariance of the asset returns and the market returns, by the variance of the market returns, gives us the beta of an asset. We can evaluate the market risk of an asset or portfolio in relation to the overall market. The market has a beta equal to one, beta greater than one means the asset swings are higher than the market, implying more systematic risk, and vice-versa.

As for idiosyncratic risk, it is said that the only free lunch in finance is diversifying specific asset risk. Rational investors can easily eliminate specific risk by adding more assets to their portfolio. When you hear someone talk about diversifying a portfolio, this is what they are after. And since everyone can do this easily, investors should not be rewarded for taking specific asset risk. Any expected reward should be based on the systematic risk of our portfolio. That's why, when answering our third important question and using the CAPM model, our expected return depended only on market factors and the beta of the asset.

Standard deviation gives us an idea of total risk. Beta gives us the market risk. But how do we know the amount of specific asset risk of our portfolio? We want to know how diversified our portfolio is. To answer this question, we go back to the CAPM. The CAPM model is a simple linear regression, where excess return of the asset is the dependent variable and excess returns of the market is the independent variable. Beta is the slope of the line obtained from the regression. Idiosyncratic risk is captured on the error component of the regression, the unobserved deviations that are not dependent on the market.

The R-squared of the regression is the metric we are looking for. It can tell us the proportion of the portfolio return that can be determined by the market return. Let's say we get a 0.9 R-squared from the linear regression. We can then say that 90% of the portfolio return can be explained by the market, the other 10% is related to specific factors, such as company or industry specific risk. In other words, the portfolio is 90% diversified.

Digging your way out

Are you in the mood for crunching some numbers? Warning: this is not for the faint-hearted! If you want the easy way out, go straight to part three of this portfolio performance series: Help Along The Way.

Now that we defined the metrics needed to manage our portfolio like a pro, let's go over the process and formulas needed to calculate everything.

Time Weighted Rate of Return (TWRR)

Start by defining a period of time, for which you want to calculate TWRR. For example, this year. Get all the end of day prices for the assets in your portfolio. From your bank or brokerage, get the initial positions for stock ownership at the beginning of the period, as well as all trades and distributions. Don't forget the fees!

For each day, compute the total value of the portfolio, based on the number of shares owned and end of day prices. Then, calculate the return for each day, discounting any cash flows. The overall period TWRR is the product of all the daily returns plus one. Here are the formulas:

Money Weighted Rate of Return (MWRR)

Define a period of time, for which you want to calculate MWRR, for example, this year. From your bank or brokerage get the initial and ending balances, as well as all the cash flows that happened during the period.

MWRR is essentially and IRR calculation, so we need to discount all cash flows by an unknown rate, which would set the Net Present Value to zero. Remember to set opposite signs (+ or -) for deposits and withdrawals. Assume the initial balance date as period t = 0, and count the number of days from cash flows to the initial balance to calculate period (t). MWRR will be a daily return, you can then convert it to monthly, yearly or overall period return. Here is the formula:

We can only calculate MWRR on this formula by trial and error. It is best to use a software for this calculation. Excel has the XIRR formula that can easily handle the task. Add dates in one column. On the other column, add values for initial balance, cash flows and ending balance. Use XIRR formula to run the trial and error solver.

Expected Return (Historical Data)

Select a reliable period for data, for example 5 years, and get weekly or monthly adjusted closing prices for each asset. Calculate weekly or monthly returns. The average return is the expected return for the asset. For the portfolio expected return, simply calculate a weighted average, where the weights are the percentage allocation of each asset in the portfolio.

Where E(R_i) is the expected return of the asset.

Where E(R_p) is the expected return of the portfolio and ω_i is the percentage allocation of the asset.

Expected Return (CAPM)

We need to know expected market return, expected risk free rate and the beta of the asset to calculate the assets and portfolio expected returns. There are different ways to obtain these values. A straight forward approach is to use the previous historical data method to calculate expected market return. For the risk free rate, we can use a short-term treasury yield.

The beta for each asset can be obtained by running a simple linear regression, where excess asset return is the dependent variable and excess market return is the independent variable. Excess returns are calculated deducting the corresponding period risk-free rate. The regression coefficient is the beta of the asset. Excel has the LINEST formula which helps, or you can use the regression coefficient formula below.

It's also possible to calculate expected portfolio return with the portfolio beta, the result is the same:

Expected Risk

When we talked about risk, we mentioned measuring volatility. We can simply calculate the variance and standard deviation of returns for each asset during a specific period, for example: monthly returns for the past 5 years. Portfolio variance and standard deviation is a bit trickier, since it's not just a percentage allocation weighted average of the asset's returns. At the heart of this formula, lies the secret sauce of portfolio diversification. If you are interested in understanding why, take a look at this article: Demystifying The Magic of Modern Portfolio Theory.

We talked about volatility being a measure of total risk. It has two parts: market related risk, which cannot be diversified; and asset specific risk, which can be diversified. The R-squared from our CAPM regression, measures the proportion of each asset or portfolio return, which is explained by the market. We can think of the R-squared as the percentage of market risk. Accordingly, one minus R-squared represents the percentage of asset specific risk.

Where ω_i and ω_j are asset allocation weights and σ_ij is the covariance of asset returns.

Conclusion

In part two of this series, we introduce the concept of risk and how it is a key component for evaluating portfolio performance. We also went over all the formulas and processes for calculating each metric. If you love number crunching, I am sure you enjoyed it. If not, don’t worry, it is more important to understand the principles behind all the formulas. On part three of this series: Help Along The Way, we’ll see how all this process can be automated and the data insightfully displayed, so we can focus our energy on the portfolio analyzes.