The risk-free rate is the return an investor receives for investing in an instrument that is “risk-free.” In a completely rational world, it also represents the absolute minimum return an investor will accept for holding a risky portfolio. Theoretically, an investor should be compensated for bearing risk and should therefore earn a higher return than the risk-free rate. Otherwise, the investor could invest in a risk-free asset (like short-term U.S. government bonds) to earn a higher, more certain return. The return a risky investment delivers above the risk-free rate is known as “excess return.”

In this post, we’ll cover:

- How the risk-free rate factors into return decomposition generally
- The history of the risk-free rate in the U.S.
- How investors can incorporate the risk-free rate into investment and portfolio analysis

### Risk-Free Rate Overview

As shown in a recent Venn blog post, “One Investor’s Alpha is Another Investor’s Risk,” it is possible to break down a manager’s excess returns into alpha and beta components, where beta is the return coming from exposure to known risk factors, and alpha is the unique return of the manager. Where does the risk-free rate come in? It must be considered before calculating the excess return breakdown.

Let’s illustrate this with an example. Imagine a manager who returned 10%. Over the same period, the risk-free rate was 2%, the S&P 500 returned 6%, and the manager’s beta to the S&P 500 was 1.2.

First, an investor with exposure to this manager can attribute the manager’s 10% returns to its risk-free and risky components:

* Manager’s excess (or “risky”) return = Manager’s total return – Risk-free rate*

* Manager’s excess return = 10% – 2%*

* Manager’s excess return = 8%*

Next, the investor should do the same calculation for the S&P 500 to determine what the return premium was for investing in the equity market, which is a risky asset.

* S&P 500’s excess return = 6% – 2%*

* S&P 500’s excess return = 4%*

Now that we have the excess returns for both the manager and the S&P 500, we can decompose them into beta and alpha components:

* Manager’s excess return = Alpha **+ (Beta to the **S&P 500 *** S&P 500’s excess return)*

* 8% = Alpha **+ (1.2 * 4%)*

* Manager’s excess return attributable to beta exposure = 1.2 * 4% = 4.8%*

* Manager’s excess return attributable to alpha = 8% – 4.8% = 3.2%*

Putting this all together, we can now decompose the 10% total return of the manager by its risk-free return, its return coming from beta exposure to the S&P 500, and its alpha (or unique) return:

* Manager’s total return = Risk-free rate + Excess return attributable to beta exposure + Excess return **attributable to*

* alpha*

* 10% = 2% + 4.8% + 3.2%*

### Recent History of the Risk-Free Rate

In the previous example, we used a risk-free rate of 2%. How does one arrive at this number, and how does the risk-free rate change over time?

To answer the first question, Venn measures the risk-free rate using 3 month sovereign benchmark yields in the investor’s home currency. We believe this is a good proxy for a risk-free asset, as it is commonly assumed that it is unlikely for developed market governments to default on their short-term obligations.

To answer the second question, let’s look at the risk-free rate (as defined above) for a U.S. investor^{1} over the full history of the Two Sigma Factor Lens (December 26, 2002 to July 31, 2019).

**Exhibit 1: USD Risk-Free Rate **

Source: Exchange Data International. Time period: December 26, 2002 to July 31, 2019.

The risk-free rate decreased from its peak (since 2002) in the mid 2000s to near-zero levels in 2007/2008 in part due to dovish central bank policy in response to the economic deterioration leading up to and following the 2008 Global Financial Crisis. The risk-free rate stayed at these low levels until around 2017 when the Federal Reserve started to meaningfully raise rates.

As demonstrated in Exhibit 1, the risk-free rate has risen over the past few years to a higher level than we’ve seen for a decade. This implies that it will have a greater role in the total returns that investors earn, and it raises the return hurdle that rational investors will accept for investing in a risky portfolio.

### Using the Risk-Free Rate in Investment and Portfolio Analysis

Investors can use the risk-free rate in both historical and forward-looking analyses. In a historical context, an investor can analyze the attribution of returns by breaking them down into risk-free and risky components, similar to the exercise above. Venn’s Factor Contribution to Return chart^{2} performs these calculations using the factors in the Two Sigma Factor Lens. These charts have been enhanced to include a row for the risk-free rate so that users can conduct a more complete return decomposition.^{ 3}

Return contributions from the factors represent the beta component of the excess returns, while the residual return contribution is Venn’s best approximation of the manager’s unique return (or the excess return that is not captured by the Two Sigma Factor Lens).

Investors can also incorporate the risk-free rate into forward-looking analysis, such as portfolio optimization and forecasting total return performance for an investment or portfolio. Venn’s optimizer, for example, considers a user’s factor return forecasts in combination with investments’ factor exposures to determine how the portfolio should be optimally allocated across various investments. The resulting optimal portfolio’s expected performance has been updated to consider the user’s preferred return value for the risk-free rate.^{4} A user can set his or her preferred return value in the Forecasts modal. As with the factors, Venn will provide a default return value for the risk-free rate if a user does not provide one.^{5}

We think clarifying our use of the risk-free rate throughout Venn and incorporating the risk-free rate in features that did not previously consider it will enhance the analysis provided on Venn and help users better understand the impact of the risk-free rate on investments and portfolios. Over the course of the next couple of weeks, we will be updating various features on Venn (including Factor Analysis, Optimization, Drawdown Analysis, and Venncast) to incorporate the risk-free rate.

Contact the Venn team to learn more about applying your custom factor and risk-free rate assumptions to forward-looking investment and portfolio analysis.

### Appendix: Recent History of the Risk-Free Rate in EUR

Let’s look at the risk-free rate (as defined above) for a European investor over the full history of the Two Sigma Factor Lens (December 27, 2002 to July 31, 2019).

**Exhibit 2: EUR Risk-Free Rate **

Source: Exchange Data International. Time period: December 27, 2002 to July 31, 2019.

Similar to the USD example, the euro’s risk-free rate is much lower today than it’s been historically (since 2002). In 2008, the European Central Bank (ECB) cut short-term interest rates to low levels and later launched a quantitative easing program^{6} in response to economic deterioration in the aftermath of the 2008 Global Financial Crisis.

Unlike the USD example, the euro’s risk-free rate has continued falling to unprecedented negative territory, meaning that investors in euro-denominated high-quality sovereign bonds pay interest (rather than receive it) for the certainty of their capital’s return after three months. This negative risk-free rate implies that risky assets denominated in EUR should require a much lower total return to justify taking on the investment risk. This effect of negative interest rates is the main reason for the ECB’s dovish policy – to provide greater motivation for banks and investors to hold relatively risky assets.^{7}

##### REFERENCES

^{1} See the appendix for an analysis of another currency supported on Venn.

^{2} Factor Contribution to Return charts are one of the outputs of Venn’s “Factor Analysis.” Users can run “Factor Analysis” on an investment or portfolio by selecting Factor Analysis on the Analysis page and entering the desired investment or portfolio name.

^{3} The risk-free rate has always been removed from the calculations shown in the Factor Contribution to Return charts. Venn has enhanced the display by adding a new row displaying the risk-free rate so that the sum of the contributions to return now equal the total return of the investment or portfolio.

^{4} Venn’s optimizer will not allocate to the risk-free rate unless it is directly added as an investment within the portfolio being optimized.

^{5} Venn’s default return value for the risk-free rate is the average 3 month sovereign benchmark yield in the investor’s home currency over the last month. Whereas, the default return forecasts for the factors use the geometric average calculated over the full period of historical returns available on Venn for each factor.

^{6} Duncan, Gary (8 May 2009). “European Central Bank opts for quantitative easing to lift the eurozone”. The Times. London.

^{7} Cœuré, Benoît. “Life below zero: Learning about negative interest rates.” European Central Bank. Frankfurt. 9 September 2014.

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