**While there are several competing risk and return models in finance, most of them require three inputs to come up with an expected return. The first is a riskless rate [commonly called **“* Free-risk Rate*“

**], which acts as a floor on your required return and measures what you would make on a guaranteed investment. The second is a risk premium, which looks at the extra return you would require as an investor for investing in the average risk investment. The third is a risk parameter or parameters (depending on the model you use) that captures the relative risk of the specific investment that you are evaluating. Most risk and return models in finance start off with an asset that is defined as risk-free and use the expected return on that asset as the risk-free rate. The expected returns on risky investments are then measured relative to the risk-free rate, with the risk creating an expected risk premium that is added on to the risk-free rate. But, what makes an asset risk-free? And what do we do when we cannot find such an asset?**

Through this post, I discuss “risk-free rate” used when estimating discount rates. You may heard a lot about this topic if you’re [or ever] engaged in corporate finance valuation.

### Requirements for an Asset to Be Risk Free

**An asset is risk free if we know the expected returns on it with certainty [i.e., the actual return is always equal to the expected return]**.

Under what conditions will the actual returns on an investment be equal to the expected returns?

**There are two basic conditions that have to be met**:

**[1]. The first is that there can be no default risk**. Essentially, this rules out any security issued by a private firm, since even the largest and safest firms have some measure of default risk. The only securities that have a chance of being risk free are government securities, not because governments are better run than corporations, but because they control the printing of currency. At least in nominal terms, they should be able to fulfill their promises.

**[2]. There is a second condition that riskless securities need to fulfill that is often forgotten**. For an investment to have an actual return equal to its expected return, there can be no reinvestment risk. To illustrate this point, assume that you are trying to estimate the expected return over a five-year period and that you want a risk-free rate. A six-month Treasury bill rate, while default free, will not be risk free, because there is the reinvestment risk of not knowing what the treasury bill rate will be in six months. Even a five-year treasury bond is not risk free, since the coupons on the bond will be reinvested at rates that cannot be predicted today. The risk-free rate for a five-year time horizon has to be the expected return on a default-free (government) five-year zero coupon bond.

**This clearly has painful implications for anyone doing corporate finance or valuation, where expected returns often have to be estimated for periods ranging from one to ten years**. A purist’s view of “** risk-free rates**” would then require different risk-free rates for each period and different expected returns. Here again, you may run into a problem with emerging markets, since governments often borrow only short term.

### Risk-Free Rates When There Is No Default-Free Entity

**The assumption that you can use a government bond rate as the risk-free rate is predicated on the assumption that governments do not default, at least on local borrowing. There are many emerging market economies in which this assumption might not be viewed as reasonable**.

Governments in these markets are perceived as capable of defaulting even on local borrowing. When this is coupled with the fact that many governments do not borrow long term locally, there are scenarios in which obtaining a local risk-free rate, especially for the long term, becomes difficult. In these cases, there are compromises that give us reasonable estimates of the risk-free rate:

[1]. Look at the largest and safest firms in that market and use the rate that they pay on their long-term borrowings in the local currency as a base. Given that these firms, in spite of their size and stability, still have default risk, you would use a rate that is marginally lower than the corporate borrowing rate.

[2]. If there are long-term dollar-denominated forward contracts on the currency, you can use interest rate parity and the treasury bond rate (or riskless rate in any other base currency) to arrive at an estimate of the local borrowing rate.

For instance, if the current spot rate is 38.10 Thai baht per U.S. dollar, the 10-year forward rate is 61.36 baht per dollar and the current 10-year U.S. treasury bond rate is 5%, the 10-year Thai risk-free rate (in nominal baht) can be estimated as follows.

Solving for the Thai interest rate yields a 10-year risk free rate of 10.12%. The biggest limitation of this approach, however, is that forward rates are difficult to obtain for periods beyond a year for many of the emerging markets, where we would be most interested in using them.

You could adjust the local currency government borrowing rate by the estimated default spread on the bond to arrive at a riskless local currency rate. The default spread on the government bond can be estimated using the local currency ratings that are available for many countries. For instance, assume that the Indian government bond rate is 12% and that the rating assigned to the Indian government is A. If the default spread for A-rated bonds is 2%, the riskless Indian rupee rate would be 10%.

Riskless Rupee rate = Indian Government Bond rate – Default Spread

= 12% – 2% = 10%

### Cash Flows and Risk-Free Rates: Consistency Principle

**The risk-free rate used to come up with expected returns should be measured consistently with how the cash flows are measured**. Thus, if cash fows are estimated in nominal U.S. dollar terms, the risk-free rate will be the U.S. Treasury bond rate. This also implies that it is not where a project or firm is domiciled that determines the choice of a risk-free rate, but the currency in which the cash flows on the project or firm are estimated. Thus, Ambev, a Brazilian company, can be valued using cash flows estimated in Brazilian real, discounted back at an expected return estimated using a Brazilian risk-free rate or it can be valued in U.S. dollars, with both the cash flows and the risk-free rate being the U.S. Treasury bond rate.

Given that the same firm can be valued in different currencies, will the final results always be consistent?

**If we assume purchasing power parity, then differences in interest rates reflect differences in expected inflation rates. Both the cash flows and the discount rate are affected by expected inflation**; thus, a low discount rate arising from a low risk-free rate will be exactly offset by a decline in expected nominal growth rates for cash flows and the value will remain unchanged.

**If the difference in interest rates across two currencies does not adequately reflect the difference in expected inflation in these currencies, the values obtained using the different currencies can be different**. In particular, projects and assets will be valued more highly when the currency used is the one with low interest rates relative to inflation. The risk, however, is that the interest rates will have to rise at some point to correct for this divergence, at which point the values will also converge.

### Real versus Nominal Risk-free Rates

**Under conditions of high and unstable inflation, valuation is often done in real terms**. Effectively, this means that cash flows are estimated using real growth rates and without allowing for the growth that comes from price inflation. To be consistent, the discount rates used in these cases have to be real discount rates. To get a real expected rate of return, we need to start with a real risk-free rate. While government bills and bonds offer returns that are risk free in nominal terms, they are not risk free in real terms, since expected inflation can be volatile.

The standard approach of subtracting an expected inflation rate from the nominal interest rate to arrive at a real risk-free rate provides at best an estimate of the real risk-free rate.

Until recently, there were few traded default-free securities that could be used to estimate real risk-free rates, but the introduction of inflation-indexed treasuries has filled this void. An inflation-indexed treasury security does not offer a guaranteed nominal return to buyers, but instead provides a guaranteed real return. Thus, an inflation-indexed treasury that offers a 3% real return will yield approximately 7% in nominal terms if inflation is 4% and only 5% in nominal terms if inflation is only 2%.

The only problem is that real valuations are seldom called for or done in the United States, which has stable and low expected inflation. The markets where we would most need to do real valuations, unfortunately, are markets without inflation-indexed default-free securities. **The real risk free rates in these markets can be estimated by using one of two arguments**:

**The first argument is that as long as capital can flow freely to those economies with the highest real returns, there can be no differences in real risk free rates across markets**. Using this argument, the real risk free rate for the United States, estimated from the inflation-indexed treasury, can be used as the real risk-free rate in any market.**The second argument applies if there are frictions and constraints in capital flowing across markets**. In that case, the expected real return on an economy, in the long term, should be equal to the expected real growth rate, again in the long term, of that economy, for equilibrium. Thus, the real risk-free rate for a mature economy like Germany should be much lower than the real risk free rate for an economy with greater growth potential, such as Hungary.

I had a question on your comments on risk-free rates. Let’s say a Swedish company has most of its assets in Sweden, uses both SEK and EUR-debt

but is listed at the Amsterdam stock exchange. Which risk-free rate would be the most appropriate to use to calculate the cost of equity, the Swedish or Dutch risk-free rate (whose yield is comparable to the yield in other countries in the EU-zone)? Many thanks

Gidday Yvo,

Since it is governed by Sweden and most of its assets are in sweden, swedish would be more appropriate to be taken as a free rate base on the calculation. However, you would need consider the currencies factor as well. Even it uses both SEK and EURO-debt. If SEK is larger, then swedish’s yield defeintely the best choice. Otherwise, you would need a more up-front analysis on currency factor.

Goodluck!