# «What is the riskfree rate? A Search for the Basic Building Block Aswath Damodaran Stern School of Business, New York University ...»

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What is the riskfree rate? A Search for the Basic Building Block

Aswath Damodaran

Stern School of Business, New York University

adamodar@stern.nyu.edu

www.damodaran.com

December 2008

What is the riskfree rate? A Search for the Basic Building Block

In corporate finance and valuation, we start off with the presumption that the riskfree rate

is given and easy to obtain and focus the bulk of our attention on estimating the risk

parameters of individuals firms and risk premiums. But is the riskfree rate that simple to obtain? Both academics and practitioners have long used government security rates as riskfree rates, though there have been differences on whether to use short term or longterm rates. In this paper, we not only provide a framework for deciding whether to use short or long term rates in analysis but also a roadmap for what to do when there is no government bond rate available or when there is default risk in the government bond. We look at common errors that creep into valuations as a consequence of getting the riskfree rate wrong and suggest a way in which we can preserve consistency in both valuation and capital budgeting.

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 how do we estimate a riskfree rate? We will consider these questions in this paper. In the process, we have to grapple with why riskfree rates may be different in different currencies and how to adapt our estimates to reflect these differences. We will also look at cases where estimating a riskfree rate becomes difficult to do and the mechanisms that we can use to meet the challenges. We will also look at questionable practices, when it comes to riskfree rates, and the consequences for valuations.

What is a risk free asset?

To understand what makes an asset risk free, let us go back to how risk is measured in investments. Investors who buy assets have returns that they expect to make over the time horizon that they will hold the asset. The actual returns that they make over this holding period may by very different from the expected returns, and this is where the risk comes in. Risk in finance is viewed in terms of the variance in actual returns around the expected return. For an investment to be risk free in this environment, then, the actual returns should always be equal to the expected return.

To illustrate, consider an investor with a 1-year time horizon buying a 1-year Treasury bill (or any other default-free one-year bond) with a 5% expected return. At the end of the 1-year holding period, the actual return that this investor would have on this investment will always be 5%, which is equal to the expected return. The return distribution for this investment is shown in Figure 1.

This investment is risk free because there is no variance around the expected return.

There is a second way in which we can think of a riskfree investment and it is in the context of how the investment behaves, relative to other investments. A riskfree investment should have returns that are uncorrelated with risky investments in a market.

Note that if we accept the first definition of a riskfree asset as an investment with a guaranteed return, this property always follows. An investment that delivers the same return, no matter what the scenario, should be uncorrelated with risky investments with returns that vary across scenarios.

Why do riskfree rates matter?

The riskfree rate is the building block for estimating both the cost of equity and capital. The cost of equity is computed by adding a risk premium to the riskfree rate, with the magnitude of the premium being determined by the risk in an investment and the overall equity risk premium (for investing in the average risk investment). The cost of debt is estimated by adding a default spread to the riskfree rate, with the magnitude of the spread depending upon the credit risk in the company. Thus, using a higher riskfree rate, holding all else constant, will increase discount rates and reduce present value in a discounted cash flow valuation.

If we categorize companies, based upon assets in place and growth assets, growth companies should be affected much more adversely when the riskfree rate increases than mature companies, holding all else constant.

Changes in the riskfree rate also have consequences for other valuation inputs.

The risk premiums that we use for both equity (equity risk premium) and debt (default spreads) may change as riskfree rates change. In particular, a significant increase in the riskfree rate will generally result in higher risk premiums, thus increasing the effect on discount rates. Investors, who settle for a 4% risk premium, when the riskfree rate is 3%, may demand a much larger risk premium, if riskfree rates rise to 10%. Finally, the factors that cause the shift in riskfree rates – expected inflation and real economic growth – can also affect the expected cash flows for a firm.

Estimating a Riskfree Rate In this section, we will look at how best to estimate a riskfree rate in markets where a default free entity exists. We will also look at how riskfree rates in nominal terms can be different for real riskfree rates, and why riskfree rates can vary across currencies.

Requirements for an investment to be riskfree If we define a riskfree investment as one where we know the expected return with certainty, under what conditions will the actual return on an investment always be equal to the expected return? In our view, there are two basic conditions that have to be met.

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. Even this assumption, straightforward though it might seem, does not always hold up, especially when governments refuse to honor claims made by previous regimes and when they borrow in currencies other than their own.

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 5-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.

In summary, an investment can be riskfree only if it is issued by an entity with no default risk, and the specific instrument used to derive the riskfree rate will vary depending upon the period over which you want the return to be guaranteed.

The Purist Solution If we accept both requirements – no default risk and no reinvestment risk –as prerequisites for an investment to be riskfree, the risk free rates will be vary with time horizon. Thus, we would use a one-year default free bond to derive the riskfree rate for a

5 2.50% 98.00 2.55% 2.9543% 6 2.75% 99.00 2.78% 2.9510% 7 3.00% 98.00 3.06% 3.3789% 8 3.25% 97.00 3.35% 3.7884% 9 3.50% 99.00 3.54% 3.7174% 10 3.75% 98.00 3.83% 4.1522% If we accept the proposition that the riskfree rate should be matched up to the time period of the cash flow, we would use the rates in this table as the riskfree rates by period – 1.5% for year 1, 2.27% for year 2 and so on.

From a pragmatic standpoint, refining riskfree rates to make them year-specific may not be worth the effort in mature markets for two reasons. The first is that with any well reasonably well behaved yield curve1, the effect on present value of using yearspecific risk free rates is likely to be small, since the rates do not deviate significantly across time. The second is that the rest of the parameters that we use in analysis now have to be defined relative to these riskfree rates; the equity risk premium that we use for the cost of equity in year 1 has to be defined relative to a one-year riskfree rate rather than the more conventional computation, which uses ten-year rates. This will usually result in higher equity risk premiums for the short-term risk free rates, which may nullify the eventual impact on the cost of equity. For instance, assume that the one-year rate is 2% and that the ten-year rate is 4% and that the equity risk premium, relative to the ten-year rate, is 4.5% but is 6% against a one-year rate. The cost of equity for an average risk investment will then be 8% for the one-year cash flow (2%+6%) and 8.5% for the 10year cashflow (4%+4.5%).

When would it make sense to use year-specific riskfree rates? If the yield curve is downward sloping (short term rates are much higher than long term rates) or excessively upward sloping, with long term rates exceeding short term rates by more than 4%, there is a payoff to being year-specific. In market crises, for instance, it is not uncommon to see big differences (in either direction) between short term and long-term rates. If we decide to use year-specific rates, we should also estimate year-specific equity risk premiums and default spreads to be consistent.

1 We use historical norms to define “well behaved”. In the United States, for instance, yield curves over the last century have been upward sloping, with long term (10-year) treasury rates about 2% higher than short term (3-month) treasury bill rates.

A Practical Compromise If we decide not to estimate year-specific riskfree rates, we have to come up with one riskfree rate to use on all of the cash flows. But what rate should we use? One answer exists and it has its roots in an interest-rate risk management strategy that is widely used by banks called duration matching. Put simply, banks that faced interest rate risk in their assets (generally loans made to corporate and individual borrowers) face two choices.

The first is to try to match up the cash flows on each asset with a liability with equivalent cash flows, which would fully neutralize interest rate risk but would also be difficult to put into practice. The other is to match up the average duration of the assets to the average duration of the liabilities, resulting in less complete risk hedging, but with far less cost.