Last week, the 10-year US treasury bond rate dropped to 1.75%. While it has risen since to about 2%, there can be no denying a basic fact. Government bond rates have dropped in almost all of the developed market currencies: the Euro, the British Pound, the Swiss Franc and the Yen. Since government bond rates are used as risk free rates to estimate discount rates in valuation or hurdle rates in corporate finance, there has been a great deal of hand wringing and angst among valuation practitioners on the consequences. In fact, if you allow for the increase in sovereign risk across the globe, you could argue that the "true" risk free rates are even lower than the already low government bond rates. In my previous post on the sovereign rating downgrade for the US, I noted that the default spread would have to be netted out against the government bond rate to get to the risk free rate. If, for instance, you accepted the S&P rating of AA+ for the US and estimated a default spread of 0.20% for that rating, the US dollar risk free rate right now would be about 1.80% (2% minus 0.20%).

So what effect do lower risk free rates have on value? The answer, if you follow conventional valuation practice, seems obvious. Lower risk free rates, holding all else constant, result in lower discount rates, and lower discount rates, all else held the same, will result in higher value. In fact, this seems to be the implicit message in the Fed's Operation Twist 2: that lower risk free rates are good for the economy and markets. It is also this facile conclusion that makes some practitioners uncomfortable with using today's rates in valuations; the angst gets deeper when the practitioner in question wants a "low" value for an asset (for tax assessments or to tilt the scales in a legal tussle). It is not surprising then that these practitioners flirt with an alternative: why not use "normalized" risk free rates instead of today's "abnormally" low risk free rates? The normalized risk free rates are generally computed by looking at the past: thus, the average 10-year treasury bond rate over the last 30 years, which is closer to 4%, is suggested as an option. Alluring though this option seems, not only is it the wrong solution to the perceived problem (of low risk free rates and out of control valuations), there may be no problem to solve in the first place. And here is why..

1.

2.

Risk free rate = Expected inflation + Expected real growth

Viewed through these lens, it is quite clear that a very low risk free rate is not generally compatible with a vibrant high growth economy. In fact, the biggest factor driving down ten-year bond rates this year from 3.29% to 2% has been the increasing pessimism about global economic health, pushing down both expected real growth and expected inflation. That is the basis for my argument that the Fed has become a side player in this game and that its push for lower risk free rates is actually at odds with its desire that the US return to healthy economic growth.

3.

Note that while the overall expected return on stocks (backed out from level of the S&P 500 index and expected cash flows from stocks) has been in a fairly tight range (8%-9%), the proportions coming from the risk free rate and equity risk premium have changed. And there are consequences for value as well. To see why assume that you are valuing a mature, average risk company (growing at the same rate as the economy) with $ 100 million in cash flows to equity currently in a market where the risk free rate is 4% and the equity risk premium is also 4% (thus creating a cost of equity of 8%). Since the risk free rate is the proxy for nominal growth in the economy, this company's value is:

Value of company = 100 (1.04) / (.08-.04) = $2,600 million

Now consider valuing the same company when the risk free rate is 2% and the equity risk premium is 6%. Since the nominal growth rate expectation is down to 2%, the value of the company is:

Value of company = 100 (1.02)/ (.08 - .02) = $1,700 million

The effect on value will be greater for higher risk companies, where the risk premium is magnified, and lower for lower risk companies, but it will be significant across the board. Note that the first scenario resembles the market numbers in 2007 whereas the second is close to where we are today. The shift in risk free rates/ risk premiums may explain why stocks look cheap today, relative to historic metrics.

So, what do we do about low risk free rates? As I see it, you can choose one of four routes, ranging from dysfunctional to dynamic:

Value of company = 100 (1.04)/ (.06-.04) = $5,200 million

You will find everything you look at to be dramatically under valued, but the model is internally inconsistent. In effect, though, you are combining a crisis risk free rate with a good times risk premium/growth rate to estimate too high a value.

Value of company = 100 (1.02) / (.10 - .02) = $1,275 million

Here, the inconsistency is that you have combined a good times risk free rate with a crisis risk premium/growth rate to estimate too low a value.

I would steer away from the internally inconsistent valuations, either dysfunctional (giving you too high a number) or depressed (giving you too low a number) because your inputs are at war with each other. As for denial and dynamic valuations, I prefer dynamic valuations because I am not sanguine that reversion back to historic norms will happen soon. I can see why long term, value investors may be attracted to denial valuations but they better have a road map to their alternate pre-crisis universe, or the valuations will not come to fruition. But the bottom line about risk free rates is worth repeating. Lower risk free rates do not always translate into higher values for risky assets and it is not necessarily a "problem" that needs to be solved.

So what effect do lower risk free rates have on value? The answer, if you follow conventional valuation practice, seems obvious. Lower risk free rates, holding all else constant, result in lower discount rates, and lower discount rates, all else held the same, will result in higher value. In fact, this seems to be the implicit message in the Fed's Operation Twist 2: that lower risk free rates are good for the economy and markets. It is also this facile conclusion that makes some practitioners uncomfortable with using today's rates in valuations; the angst gets deeper when the practitioner in question wants a "low" value for an asset (for tax assessments or to tilt the scales in a legal tussle). It is not surprising then that these practitioners flirt with an alternative: why not use "normalized" risk free rates instead of today's "abnormally" low risk free rates? The normalized risk free rates are generally computed by looking at the past: thus, the average 10-year treasury bond rate over the last 30 years, which is closer to 4%, is suggested as an option. Alluring though this option seems, not only is it the wrong solution to the perceived problem (of low risk free rates and out of control valuations), there may be no problem to solve in the first place. And here is why..

1.

__The risk free rate is not just a number in a discount rate computation but an opportunity cost.__One way to think about the risk free rate is that it is the rate you will earn if you choose not to take the risky investments that are out there (stocks, corporate bonds, real estate, a business venture). So, let's carry this to its logical extreme. Let's assume that you do replace today's risk free rate (2% or lower) with your normalized rate (4%) and that the resulting high discount rate gives you a low value for your risky asset. Let's then assume that you choose not to invest in that risky asset. Where do you plan to invest that money instead? In your normalized bond earning 4%? Since it exists only on your spreadsheet, I am afraid that you will have to settle for that "abnormally" low 2% interest rate.2.

__The risk free rate is a reflection of what people expect in the overall economy for the foreseeable future.__Harking back to an equation that I have used before, note that the risk free rate is the sum of two market expectations: an expectation of inflation for the future and an expectation of real growth.Risk free rate = Expected inflation + Expected real growth

Viewed through these lens, it is quite clear that a very low risk free rate is not generally compatible with a vibrant high growth economy. In fact, the biggest factor driving down ten-year bond rates this year from 3.29% to 2% has been the increasing pessimism about global economic health, pushing down both expected real growth and expected inflation. That is the basis for my argument that the Fed has become a side player in this game and that its push for lower risk free rates is actually at odds with its desire that the US return to healthy economic growth.

3.

__The risk free asset is also where investors flee when the fear factor rises, the much vaunted "flight to safety" during crises.__But this flight does not just affect the risk free rate.... It affects risk premiums for all risky asset classes: equity risk premiums rise, default spreads on corporate bonds widen and cap rates on real estate become higher. If you define the expected return from stocks as the sum of the risk free rate and the equity risk premium, the last decade has seen changes in that composition:Note that while the overall expected return on stocks (backed out from level of the S&P 500 index and expected cash flows from stocks) has been in a fairly tight range (8%-9%), the proportions coming from the risk free rate and equity risk premium have changed. And there are consequences for value as well. To see why assume that you are valuing a mature, average risk company (growing at the same rate as the economy) with $ 100 million in cash flows to equity currently in a market where the risk free rate is 4% and the equity risk premium is also 4% (thus creating a cost of equity of 8%). Since the risk free rate is the proxy for nominal growth in the economy, this company's value is:

Value of company = 100 (1.04) / (.08-.04) = $2,600 million

Now consider valuing the same company when the risk free rate is 2% and the equity risk premium is 6%. Since the nominal growth rate expectation is down to 2%, the value of the company is:

Value of company = 100 (1.02)/ (.08 - .02) = $1,700 million

The effect on value will be greater for higher risk companies, where the risk premium is magnified, and lower for lower risk companies, but it will be significant across the board. Note that the first scenario resembles the market numbers in 2007 whereas the second is close to where we are today. The shift in risk free rates/ risk premiums may explain why stocks look cheap today, relative to historic metrics.

So, what do we do about low risk free rates? As I see it, you can choose one of four routes, ranging from dysfunctional to dynamic:

__1. The dysfunctional valuation:__You leave risk free rates at today's low levels, while your risk premiums and growth rates come from happier, more stable times. Implicitly, this is exactly what you will do, if you use equity risk premiums from historical data (Ibbotson, for instance) and earnings growth rates that reflect the "good old days". Using the example above, you would value the average risk, mature company, using a 2% risk free rate, a 4% nominal growth rate and a 4% equity risk premium:Value of company = 100 (1.04)/ (.06-.04) = $5,200 million

You will find everything you look at to be dramatically under valued, but the model is internally inconsistent. In effect, though, you are combining a crisis risk free rate with a good times risk premium/growth rate to estimate too high a value.

__2. The depressed valuation__: You could replace the risk free rate today with a higher, normalized risk free rate, while using the higher risk premiums and growth rates that characterize crisis marks. Thus, in the valuation example, you would be using a 4% risk free rate in conjunction with a 2% nominal growth rate and a 6% equity risk premium, leading unsurprisingly to a low value:Value of company = 100 (1.02) / (.10 - .02) = $1,275 million

Here, the inconsistency is that you have combined a good times risk free rate with a crisis risk premium/growth rate to estimate too low a value.

__3. The denial valuation__:You could be a normalizer, replacing current numbers with normal numbers, not just on the risk free rate but on the other inputs (equity risk premiums, cash flows, growth rates) as well. This faith in mean reversion leaves the intrinsic value of the hypothetical company stuck at $2,600 million, as risk free rates and risk premiums change, and views the crisis as "nightmare" that will soon be forgotten. Unlike the first two choices, this one is internally consistent and may, in fact, be the valuation that is used by a classic contrarian investor, who believes that markets over react and adjust back to norms over time.__4. The dynamic valuation__: You could use today's combination of a low risk free rate, high risk premium and low nominal growth to estimate a value of $1,700 million for the company. The valuation is internally consistent but the downside is that it will be volatile and change as the macro environment changes, creating discomfort for those who believe that intrinsic value is a stable number that stays unchanged over time.I would steer away from the internally inconsistent valuations, either dysfunctional (giving you too high a number) or depressed (giving you too low a number) because your inputs are at war with each other. As for denial and dynamic valuations, I prefer dynamic valuations because I am not sanguine that reversion back to historic norms will happen soon. I can see why long term, value investors may be attracted to denial valuations but they better have a road map to their alternate pre-crisis universe, or the valuations will not come to fruition. But the bottom line about risk free rates is worth repeating. Lower risk free rates do not always translate into higher values for risky assets and it is not necessarily a "problem" that needs to be solved.

## 30 comments:

With the macro environment so unstable, I find valuation to be even more subjective than normal. Correct me if I'm wrong -- as I probably am -- but let's say you take the Treasuries to be a good indicator of the risk-free rate. Let's also say your time horizon is long term. And then operation twist happens.

Your risk-free rate would be pushed down, correct? So your PV rises. But your growth expectations -- grounded in the teachings of Bernanke -- rise. So your PV falls? Finding those numbers seems to leave a lot to user preference/error. And then there's inflation...

my take away from this post is that inherently valuations probably don't drive stock prices as valuation is a subjective issue, a matter of assumption and choosing values. someone can justify a valuation of 1275mn and someone 5200mn. In good times market would choose to give credence to the 5200 valuation levels, and in bad times to 1275. This is why the crazy moves, when underlying fundamentals don't change that dramatically. I mean could a change in growth rates from 3% to sub 2% mean a change in fundamental valuations by 20% or more? I thought equities discounted long term cash-flows, but it seems they don't look beyond the next 3 yrs.

Hi,

I am fairly new to this but I can't see that there is any real option other than valuing a company relative to the current risk free rate. You look at the current risk free rate and see how a company measures up against it because the risk free rate is the best that you can do (risk-lessly) at this point in time. If the current, say "dollar" based risk free rate really is too high or low then the market will adjust it quite quickly.

As far as I can make out valuation seems to me to be a valuation of the future for the fixed point in time - now. An intrinsic valuation seems to be a valuation of a risky asset that is relative to a risk-less asset that takes into account the risk. This seems to be a really sound way of doing things as if suddenly say you could buy a bond, available in unlimited supply, that would risk-lessly yield 10% every year for 10 years then you would be nuts to buy an asset that yielded less. So the value of your risky asset would have to change. Somebody will see that $10 on the ground ;-)

The other way that I have found that I think of things is that the risk free rate is a foundation. Assets are valued relative to this foundation. If there is an earthquake and the foundations start moving (becoming volatile) the all the stories above it will move as well - their valuations will change. If the foundations change very rapidly than the intrinsic valuations (which are relative to the foundations) will change rapidly as well - no matter how stable the asset's cash flows. The real value of the business will change rapidly there is no option.

So, you do a valuation, how long is it good for? Well it is good for now. If things change it will be out of date. If there is a huge spill in the Gulf of Mexico then the valuation of your company will change it it was responsible. If the risk free rate changes then the value of your asset changes.

Alan,

I think you have it just right. A valuation is an assessment of the future as of right now.... and you have to use the current risk free rate.

Jason,

You are right about the macro environment instability translating into valuation instability (why is it subjective? It is what it is...) As for your reasoning, out works only if you believe that Bernanke has immense persuasive powers left... I don't think he does.

A good quote from Atlas Shrugged..."A is A." I agree that there is no problem to solve here. The risk free rate is forward looking and incorporates future expectations of growth and inflation. It can't be too low or too high, it just is. Sometimes A is A.

Good points.

Here is an informative/scary article from a mises scholar, - it is an eye opener for me. It explains the effects of Fed's rate manupulation.

http://mises.org/daily/5223/media.aspx?action=author&ID=1619

Look..LOL... I can assure you that most practicioners don't use these frameworks in their investment decisions.

Those that do, do so only on the margin; it's never the decisive factor. Real investment decisions are ultimately made for other reasons.

I'd say the practicioners (i.e. the market) in a low rate environment functions more like the following:

Let's look for the greater fool, and play chicken until the village idiot buys; then sell sell sell, and run for the hills.

Unknown,

By "practitioners", you must mean investors, analysts and portfolio managers and I agree with you. Most of them don't do and are not interested in valuation. They want to stay ahead of the pack and most of the time, they are the pack.

However, I am referring to a much wider set of practitioners. About 90% of valuations are done by appraisers valuing private businesses for sale, accountants assessing fair value and others whose objectives don't include making money on the valuation. Those practitioners still have to make choices on risk free rates, risk premiums and growth rates..

as the new view of lower future nominal growth (let's assume lower real growth and not lower inflation, thus WACC does not change) becomes priced in, the discount rate increases (WACC - g) lowering present value, as you mention...but in your example, you do not change FCFE...if future g is lower, why shouldn't future FCFE increase? if it does not increase, you must believe future ROIC on old invested capital decreases as g rates decrease...in extreme cases, I believe they probably do (e.g. 1930's).

That is actually a great point about ROIC. I am implicitly assuming that the ROIC will decrease if nominal growth opportunities decrease but I should have been explicit.

An interesting question would then become: what would happen if the ROIC stayed unchanged? Here are the consequences. For firms with ROIC = Cost of capital, there would be no change in value when the risk free rate declined (and risk premiums go up). For firms with ROIC> Cost of capital, the value will go down but not by as much as in the example in the post. For firms with ROIC < cost of capital, it will actually be good if there is less growth and less reinvestment.

curious.. if the real risk free rate is that of a 10 year zero coupon rather than a coupon bond your problem is solved..

The "return to a 10 year" coupon bond is largely a result of the reinvestment rate of the coupon... and as you well know only if the reinestment rate remains at the coupon rate is YTM an accurate measure...

Ten Year Zero Coupon bonds have rallied close to 30% in "price" in the last months..

thats the real valuation exercise... not two hypothetical and imposible to predict future cash flows... i.e. the rate of reinestment and the dividend rate.

Sir,

I was not able to understand why

Risk free rate = Expected inflation + Expected real growth

does the above equation applicable for US only or for some other country like India?

Stan,

What problem are you solving? And a zero-coupon is a nominal rate, not a real rate...

Ankit,

The equivalence holds in all markets but it is an expected growth rate in the long term (and so will not be directly comparable to current growth in growing, emerging markets).

Thanks for sharing your views. Would you apply the same kind of "reverse" rationals for PIGS countries which have historically high risk free rates ?

Thanks, Sylvie

The PIGS countries all operate in Euros. The Euro risk free rate is at historic lows (not highs). The rates for these countries are high because of sovereign default risk being high and not because of the risk free rate. In other words, the risk free rate in Euros for a Greek company is still 1.5%... it is the rest of the equation (the equity risk premium) that is sky high.

Sir,

Should we not be using different WACC for each year ? For example, if we think that the economic outlook to restore to normal in 2 years, we can use the the normal risk-free rate, Risk premiums and growth projections from 2013 onwards while going for the current low risk-free rates etc for the next two years. I think that should take care of the above differences and further reduce the variance under each scenario that you have caluclated.

As long as you change the risk premiums every year as well...

Great topic, I've been thinking about this multiple times. Using the current 'risk free rate' to value riskier investments is problematic MOSTLY because the rate is rigged by the FEDs - so is that rate real (is it incorporating actual expectations of inflation and growth, if you know for a fact that rate is being pushed down). I know a lot of you will say that the markets are efficient and they would push the rate back up if the investors felt that the rate is lower than where it should be... I have my doubts about that and about market efficiency for short periods of time.

So now the issue is, a lot of investors feel the need to value assets (not only buy and sell on momentum, but on a valuation basis) - how should those investors approach that valuation - I believe it's about time horizon.

Inconsistent models don't make sense for sure, so short term investors would make sense to value assets by using the dynamic model (if fact they don't care if the rate is gamed, they only care what looks cheap based on that). Long term investors shouldn't go for the same logic, since the chances are that the rates will reverse to a certain degree once the multiple QEs end(God knows how many there will be.

But one thing I do hate, and that is hearing people on radio saying that stocks are cheap based on the FED model (they were also saying that months ago based on the same argument) - I never was a fan of that model

Hi

We calculate Equity Risk Premium for a longer term period and takes a average of it as our basis of calculating required returns. In the given case how should we calculate the Equity Risk Premium as the long term average seems to give us a low required return. Is it implied equity risk premium or it is based upon some option methos? Please explain.

Necesito el valor de la tasa libre de riesgo de los bonos del tesoro de eeuu, a 10 aÃ±os...por favor.. es urgente..

I need the value of risk-free rate of U.S. Treasury bonds, 10 years ... please .. urgent ..

I need the value of risk-free rate of U.S. Treasury bonds, 10 years ... please .. urgent ..

Necesito el valor de la tasa libre de riesgo de los bonos del tesoro de eeuu, a 10 aÃ±os...por favor.. es urgente..

Hi, In the text you refer to the 30-year historical US government 10-year bond yield to be 4%. I find the historical number for the same period to be around 7% (nominal). Have you corrected for any items (high inflation in the 1980s?)? As I am a "normalizer" I am trying to find the best forecast for the normalized risk-free rate. Forecasts for real growth and inflation for the US imply 4-5%. Historical 10-year US government bond yield around 7% (I thought). So I am uncertain which rate to apply - the range 4-7% is quite large. The WACC for this purpose will be used as discount rate for long-term investments in the minings and metal industry. Thanks in advance.

Dear Prof. Damodaran !

I hope you will find this in the best of your health and spirits.

I am afraid my question is not related to this particular post.

My question is related to FCFF.

The formula is:

EBIT(1-t)+ Depreciation - CAPEX + Decrease in Working Capital...

I want to ask:

The resulting FCFF will give the free cash flows for all supplers of capital and shareholders.

What about the OPENING CASH BALANCE (in case of retail companies, they may have a lot), what about CASH INTEREST ON DEPOSITS, and finally what about DIVIDENDS RECEIVED for a holdings company, which almost every year receives dividends.

Why don't we use them in FCFF calculation?

Thank you for your kind cooperation.

Bye

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Thanks a lot, this is very interesting post.

One questions that keeps bothering me is: Why should growth matter for valuation?

It might sound strange because we are all used to using formulas such as Value = CF / (k-growth), but shouldn't it matter whether growth is profitable or not? And shouldn't the standard assumption be that growth has a value added of zero, i.e. there is no free lunch?

If this is the case, I guess (?) it means the following:

1) Lower growth means higher free cash-flows (yeah, we don't have to invest anymore!). In this case 2% of the 2,600mn cash-flow would be freed for payouts and not needed for investments.

2) Using this higher value for cash-flows and lower value for growth, valuation remains unchanged (100.1.04+2%*2,600)/(0.08-0.02)=2,600

Current bank regulations allow banks to hold “solid” sovereign debt against much less capital than when lending to “The Risky” like small businesses and entrepreneurs.

That translates directly into an artificial lowering of the “risk-free-rate”, call it subsidy, and so in fact we have not the faintest idea what that rate would be, in the absence of the distortions or manipulations carried out by the regulators.

http://subprimeregulations.blogspot.se/

Dear Professor Damodaran

I am working on my master thesis, where I am doing a valuation of a Danish company.

I am struggling to find the right cost of equity! I have read this blog post and decided to go with the "dynamic valuation" method. As of today the effective rate of the Danish 10-year government bond is as low as 0,73%! Thus, my question is - what should the risk-premium be to "compensate" for this? As far as I know the historical market risk-premium in Denmark has been around 5%, but using this information results in (in my opinion) a value that is way too high.

Can you help me?

Best regards

Stephanie

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