# How to calculate the risk-free rate?

#### June 16, 2020

The risk-free rate is defined theoretically as the return that an investor would expect in return from an investment with zero risk (no default and reinvestment risk), over a defined period of time. The real risk-free rate can be calculated by subtracting inflation over the same period of time.

**Why is important?**

- It is the reference used to define the rate of return of any investment, as the risk-free rate is the minimum return an investor expects for any investment. Basically is the starting point of any return required calculation.
- It is an Indicator of monetary politics and the state of the economy.
- It is a key component of the calculation of the required rate of return, therefore, is important to calculate any rate of discount and in consequence, the subsequent valuation made with the cash flows used.

**Facts in consideration: **

- It is possible to have a negative risk free rate.
- The risk-free rate is currency-specific.
- Not all the securities issued by governments are risk free (as some governments face default risk).

**Recommendations:**

- In the same currency instrument for the same period of time, we must use the lowest rate.
- Instrument maturity must match (closest maturity) with the time evaluated (Over 10 year period is recommended use only 10 years, because of the characteristics in terms of efficiency and liquidity).

**Step#1: Find the Local Currency Government Bond Rate for the period required. (Example US 10Y Bond Rate, Mexican 10Y Bond Rate, etc.)**

**Step #2:**

**Calculate the Default spread in the instrument: We have 3 alternatives as following.-**

Find sovereign bonds denominated in US dollars (or Euros), issued by the country analyzed.

- Default spread = Country analyzed Bond Rate (in US/EUR $) - US/UE Treasury Bond rate (with the same maturity).

CDS spread: Obtain the traded value for the country analyzed CDS.

- Default spread = Sovereign CDS spread (Difference with USD).

Sovereign-rating based spread: estimate the spread based upon a sovereign rating.

- Find the sovereign rating provided by a credit rating agency and then convert rating to default spread.

**Step #3:Calculate the difference between the instrument and the default spread, as following: Instrument return - Default spread = Risk-free rate**

**Example: Mexican Risk-free rate for 10Y **

**1.- Find the local currency Government Bond Rate for a 10 year Bond in Mexican pesos (May 12, 2020). Rate = 5.91% Source: ****https://tradingeconomics.com/bonds**

**2.-Calculate the Sovereign Default spread**

**Approach #1: Default spread from Government Bonds**

First, we need to find a Mexican 10Y sovereign bond yield rate issued in US dollars (or the S&P/BMV Sovereign International UMS 5-10 Year Bond Index USD) and the US Risk-free rate 10Y Bond.

Results:

- México 2030, YTM: 3.58%
- US Risk-free rate 10Y Bond:

Then, we calculate the difference between them: 3.58% -0.66% = **2.92%**

**Approach #2**

- Find the US and México 5-Years Credit Default Swap quotation

- The US5-Years Credit Default Swap is 19.00 bp and Mexico 5-Years Credit Default Swap is 205.96bp
- Calculate the difference: 2.0596% - 0.19% =
**1.8696%**

**Source: **http://www.worldgovernmentbonds.com/sovereign-cds/

**Approach #3**

- Find the sovereign rating provided by a credit rating agency

- Moodys México sovereign rating: Baa1

Source:https://tradingeconomics.com/country-list/rating

- Look at the table provided by Dr. Damodaran
- Result for Baa1=
**1.50%**

3.-**Finally, we calculate the difference between the return of the 10Y Mexican bond and the default spread (Different approaches), having the following options to choose: **

#1 5.91% - 2.92% = 2.99%

#2 5.91% - 1.8696% = 4.04%

#3 5.91% - 1.50% = 4.41%

**Alternatives:** in case of not having trust in the government bond rate, we can:

1.calculate the risk-free rate by adding the expected inflation in currency with the expected real interest rate. Example: Inflation rate expected 5% and TIPs is 1%, therefore the risk-free rate is 6%.

2. Differential inflation: Using the dollar or euro rate and the expected inflation in the US/UE and the country analyzed, using the following formula:

It is clear that it is difficult to define a precise risk-free rate to use whenever it is needed, as there are multiple factors involved, but, it is important to take into account that there are several approaches and each one can be suitable for a specific need. As any financial theory, there are multiple opinions, approaches, and theories, therefore it is important to mention that this article was mainly inspired by Damodaran’s approach used on his books, articles, and classes.

*Credits: Marcos Abraham Rascón Corona, Luis Miguel Almanza Rueda, Itzel Pamela Becerril Tovar, Rodrigo Alejandro Ibarra Vizcarra, Sara Escamilla Enríquez, Ilian Murillo Palma, Gabriel García Nevárez, Priscila Elizabeth Terrazas Rico, Ivan Elias Castro Cordero.*