## “Risk free” rates and discount rates for DCF models

In the discussion to the Piquadro valuation, I quickly mentioned that the concept of “risk free” rates is a difficult concept at the moment.

Let’s have a quick look at the “academical” world:

**CAPM**

If we look at the CAPM (no matter if one beliefs this or not) we can see that the risk free rate of return plays an important role there. First, it is the basis return on needs to achieve with any investment, secondly it also influences the equity risk premium.

**Risk free rate of return**

The definition of the risk free rate itself is quite “fishy”. Investopedia for example states:

Investopedia explains ‘Risk-Free Rate Of Return’

In theory, the risk-free rate is the minimum return an investor expects for any investment because he or she will not accept additional risk unless the potential rate of return is greater than the risk-free rate.In practice, however, the risk-free rate does not exist because even the safest investments carry a very small amount of risk. Thus, the interest rate on a three-month U.S. Treasury bill is often used as the risk-free rate.

This is of course not really applicable for any serious long term investor. Damodaran has a nice paper about “risk free rates” here.

His major points are as follows:

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.

So this is important: **No default risk !!!** So it is wrong for instance to use current yields of Italian Govies for valueing Italian stocks, as considerable default risk is embedded in current spreads. The “country” risk could/should be embedded into the equity risk premium, not into the risk free rate. A hypothetical Italian company with 100% of its business in Germany for example, should only get a very small country risk charge if any.

A second point is the following:

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.

In theory, one should discount annual cash flows with the respective annual risk free rates. With a flat yield curve, this is not so important but for steep yield curves the differences can be significant. However in practice Damodaran recommends using the **duration of the cash flows **of the analysed investment as proxy for the risk free rate. As the best proxy if we don’t want to do this, he recommends the 10 year rate.

For the EUR, he recommends specifically the following:

Since none of these governments technically control the Euro money supply, there is some default risk in all of them. However, the market clearly sees more default risk in the Greek and Portuguese government bonds than it does in the German and French issues. To get a riskfree rate in Euros, we use the lowest of the 10-year government Euro bond rates as the riskfree rate; in October 2008, the

German 10-year Euro bond rateof 3.81% would then have been the riskfree rate.

With regards to **currencies** he says this:

Summarizing, the risk free rate used to come up with expected returns should be measured consistently with the cash flows are measured. Thus, if cash flows are estimated in nominal US dollar terms, the risk free rate will be the US Treasury bond rate. This will remain the case, whether the company being analyzed is a Brazilian, Indian or Russian company. While this may seem illogical, given the higher risk in these countries, the riskfree rate is not the vehicle for conveying concerns about this risk. This also implies

that it is not where a project or firm is domiciledthat determines the choice of a risk free rate,but the currency in which the cash flowson the project or firm are estimated.

The most common mistake with currencies is usually to use current exchange rates for future cashflows which then results in a preference for projects in countires wiht high nomnal rates.

About **Inflation**, he is not really clear in my opinion. He argues basically, inflation does not matter because we get the same result if we use yields of inlfation linked bonds combined with inflation adjusted growth rates.

Especially the current situation, where we see **negative real yields in many markets**, one could argue about his appoach. A negative real yield means for an investor, that the “risk free” nominal asset would have a guaranteed loss in real purchasing power over the investement horizon.

Consider for instance the UK: 10 year gilts run at 2.158% yield, this would be the proxy for the risk free rate. Current inflation runs at 5%, UK 10 year implied inflation from inflation linked bonds is around 3%.

So if I would use the 10 year gilt as proxy as the risk free rate, I woul dalready accept a loss of -1% p.a. in real terms p.a. or almost -3% p.a. based on current inflation rates.

I think this topic might justify even a doctorate thesis, but in my opinion, one could go the following pragamatic way:

Proxy for risk free rate: ** Higher of 10 year risk free Govie Yield in currency or inflation ).**

So in the case of the risk free rate for an Italian company I would compare:

a) 10 year risk free EUR rate = 10 year bunds = 1.89%

b) Inflation: Currently =3.4%

I would the use the **higher of the two rates, 3.4 %**. This would be a pragmatic way to avoid unnecessary country risk premium and still make sure, the risk free rate does not imply a guaranteed loss in real terms.

In various teachings, both Graham & Buffett advocate cyclical adjustments. Why not use a cyclically adjusted 10y government bond rate?

This a good post on risk free rate that you may find useful

Rethinking the Risk-Free Rate: Offering Alternatives https://t.co/CmHSx1MJ4j via Enterprising

I wonder if this blog is still live?

If yes I have a question.

In a tender which has seen responses from vendors in different currencies with different stage payments over a period of time how would the best evaluated price be determined fairly? what should be the discount rate for each currency to arrive at a common currency NPV?

are there any World bank or FIDIC Norms on such a situation?

hi,

i am not sure if I understand your question correctly.

With different currencies, you first have to adjust for currency forward prices. Depending on the currencies we are talking about, you can use traded forward prices to translate into onebase currency.

mmi

Hello,

first of all thanks alot for sharing your thoughts. Yours is the only blog i truly enjoy to read regularly. As well I do appreciate the vivid discussions around the topics.

Concerning the choice for an adequate risk free rate in valuation I did get your point. I totally agree on your point, that the application of current low riskfree rates in valuation drives numbers into unrealistic heights.

Personally I also ignore the inflation in estimating the WACC, because higher inflation should theoretically also drive the cashflows.

Has anybody considered the money market yield curve (Zinsstrukturkurve) choosing a risk free rate? The question aims to my concerns, if choosing ONE risk free rate for a defined time pattern might lead to an error in discounted values. Using a couple of time frames with specific, more differentiated factors should lead to more precise results. In respect to your concern about choosing too low risk free rates in general, you could adjust easily for that in doing so without getting absurdly low vaues from your dcf’s (for example because you are using 5% for risk free rate, while the market is implying only 1,5%).

Kind Regards

Leon

Hi Leon,

you are completely right. Theoretically you should use the ZEro yield curve and discount each cashflow with its matching discount rate.

Howevr in practice, this is quite a lot of work, so Damodaran recommends calculating the duration of the projected cashflows and use the zero rate with the same duration.

As long duration bonds are not very liquid and market prices are questionable, at the end of the day most practitioners use the 10 year rate.

mmi

P.S.: Thanks for the compliments !!!

Jan,

that is the old discussion, some people recomend using 3m t-bills as a proxy.

But as I said before, i have the nagging feeling thath in case of negative real yields this might be a mistake. It seems howver that I can’t articulate this concern correctly.

mmi

a) I do not set individual performance targets for each security but rather an overall target return for my account. I always want to buy securities that have a much higher expected return than my target return (–> margin of safety). The idea of MoS is that it offers you a buffer in case something goes wrong. If you make an infinite number of investments which are all acquired at a high enough margin of safety (and your calculations are not complete bogus) then you end up with your target return…

…which is then “risk-free”.

At least it is a more useful “risk-free” concept than using govies as a proxy, at least to me.

Anyhow, even if you do not use my calculation above, why not use the money-market return as a risk free proxy. This seem to be more practical to me, as value investors tend to weight between cash/risky investments rather than 10y bond portfolio / risky investments.

Jan

Jan,

thanks for the comment.

I am not sure if I understand a) correctly.I might have a target return for an investment, like the Depfa T2 bond.

But I can never be sure thatmy assumptions are really correct. So how can asume that ths is risk free ?

Sideremark: If I would find a German Government guaranteed investment with 5.5 % yield, I could create a “synthetic” 11% investment by leveraging ,the investment. So I am notsure an absolute target return for an investment always makes sense.

With regard to b)

You arecorrect.In the case of negative real yields, one might actually neeed to deduct the negative yield form the growth rate.

mmi

Hi mmi,

great post on a vital topic. Agree with your line of thought.

Two comments:

a) Every investor has a target return, let’s say 11%. Is it too far fetched to argue that 11% would constitute the ‘base rate’ for all discounting operations? This is a unorthodox standpoint but as a value imvestor you might look for investments that bring you 11% without meaningful risk of terminally impairing your capital. If you happen to have a slightly more risky investment case, you add a risk premium to your 11% base rate to account for additional risks. As a value investor (who always has plenty of MoS), you might in fact deem 11% (or whatever you set as benchmark) risk free.

b) You raised the point of inflation. If we consider gordon growth as a simple tool for valuation, the IV of a stock is e/(wacc-g). earnings 10, wacc 10% = IV of 100. if you think about it, what you calculate is a ‘real wacc’ since if you considered inflation you would automatically have to adjust g to at least inflation. In the above example, paying 100 for a 10 royalty (that increases with inflation) yields real 10% to me.

Many very high qualiy businesses offer these real 10% yields currently which truly makes you wonder who buys LT bonds at the moment.

Jan

Hey, great blog!

Concerning inflation you write:

“Especially the current situation, where we see negative real yields in many markets, one could argue about his appoach. A negative real yield means for an investor, that the „risk free“ nominal asset would have a guaranteed loss in real purchasing power over the investement horizon.”

I would agree with Damodaran on this issue and ignore inflation. If you have negative real yields, that’s just “global macro” situation that you have, nothing you can do about it. If you now want to invest your money “risk-free”, you will have to accept negative rates. It is therefore the correct comparison to any other investment.

Think about it in terms of opportunity costs. By chosing not to invest in the rist free asset, but say in a stock, how much do you give up? a negative number, you get something already. so following your approach, you would demand a higher risk premium for holding the stock in times of negative real rates than in other times. Does that make sense?

Another minor thing, it is of course not “guaranteed” that you will lose purchasing power, it is only in the expected value that you do. That is because you don’t know what inflation is going to be. Maybe that also makes it a bit easier to accept my point.

(In that sense, the “risk-free” rate isn’t even risk-free, in real terms it is still only an expected return. This can be interesting when investing in different kind of “projects”, if your cash-flows are almost “real” you could argue for using the real interest rate as “risk-free rate.” Maybe this could be relevant for these kind of businesses people usually assume are an “inflation hedge”, say consumer staples.)

hi valueseeker,

my concern is that under the classic approach, discount rates currently might be too optimistic (too low) at the moment esp.in countries wih negative real yields, as interest rates are artifically low.

Especially assuming zero growth in nominal terms assumes already growth in real terms. So I think the trick might be to use a negative growth rate alreaady for the “steady state”.

mmi.