In any version of a risk and return model for discount rates, where you start with a riskfree rate as a base and build up to costs of equity, debt and capital, it seems blindingly obvious that as interest rates go lower, discount rates will follow and that value will increase. It is this logic that has led to the hand wringing about how central banks have both created pricing bubbles and made discounted cash flow valuations implode by “lowering’ rates. In a recent article, Sanford Bernstein proclaimed DCF all but dead in a world with near-zero risk free rates, because as they see it, the resulting low discount rates were pushing up the value of future cash flows, and since these cash flows are inherently more difficult to estimate, DCFs were less reliable. I have no problem with Bernstein's equity research analysts abandoning DCF and switching to pricing stocks instead, but I believe that they need to do it for the right reasons, not the ones outlined in that thought piece.

**Risk free Rates in a Static World**

A few months ago, I posted on the hubris of central bankers who (a) believe that they control the level of interest rates and (b) that by changing the level of rates, they can affect stock/bond prices as well as real investments at companies. It is this misguided view of the world that, in my view, has given us years of ever-lower central banking rates, without the promised for results (of more capital investment and higher real growth). It is instructive that almost a decade into quantitative easing, the global economy still seems to be struggling to find its footing.

Unfortunately, this delusion that you can change the risk free rate and leave all else in the process unaffected is not restricted to central bankers and seems to have spread like a virus among valuation analysts, leading to many following the Bernstein script and abandoning DCF. The mathematics are impeccable. If you leave risk premiums (equity risk premiums and default spreads) unchanged, hold on to old growth rates and lower just the risk free rate, you will see value increase as the risk free rate decreases and perhaps approach infinity at really low or negative risk free rates.

To see why, let's assume that you had valued a company in 2007, when the risk free rate was close to 4% and the equity risk premium was also 4% and that you had assumed that this company's cash flow to equity, $100 million in the most recent year, would grow at 10% a year for the following five years and 4% thereafter. The value that you would obtain in a DCF would be $3.378 billion. Now assume that you have been revaluing the company every year in the years since, keeping the rest of your parameters fixed and changing just the risk free rate. As the risk free rate has dropped to levels not seen in recent history, your valuations will have zoomed:

Download spreadsheet |

Your value of this company increase from $3.4 billion to $9.1 billion , as the risk free rate dropped to 1.5%, and lowering the risk free rate further will only increase value. In fact, at a 0% risk free rate (which is where the Euro and the Swiss Franc are at in November 2016), your valuation would approach infinity. As an added feature, as your risk free rate decreases, a greater proportion of your value comes from the terminal value, accounting for almost 94% of your value at a 1.5% risk free rate compared to 84% of value at a 4% risk free rate. That is the crux of the Bernstein argument against DCF, with the twist that estimating future cash flows is always difficult and that lower risk free rates have tilted valuation towards cash flows even further into the future.

**Risk free Rates in a Dynamic World**

Let's get real. When risk free rates change substantially, it is not because central banks will them to be lower or higher, but because of shifts in the fundamentals, and those shifts will affect your other inputs into valuation. In this section, I aim to start by showing how changing risk free rates affect growth rates and risk premiums and then argue that the value effect of a change in the risk free rate can be complicated (as market watchers have found out over the decades).

*Risk free Rates and Growth (Real and Nominal)*

If you have read my prior posts on interest rates and central banks, one of my favorite tools for understanding interest rates is the Fisher equation, which breaks down a riskless rate into two components: an expected inflation rate and an expected real interest rate. Using a proxy of real GDP growth for the real interest rate, I derive an "intrinsic" risk free rate as the sum of the inflation rate and real GDP growth. I may be stretching but it works surprisingly well at explaining why interest rates move over time, as evidenced in the graph below, where I compare the T.Bond rate to the sum of inflation and GDP growth each year from 1954 to 2015.

So, what's the point of this graph? In addition to emphasizing the fact that central banks can affect rates only at the margin, it brings home the reality that low interest rates are indicative of a market that expects both inflation and real growth to remain low. It is entirely possible that the market is wrong but if you are doing valuation, you cannot selectively override the market on one variable (growth in the static example) while holding on to it on the other (risk free rate).

*Dynamic Implication: As the risk free rate changes, your estimates of nominal growth will have to be stepped down, not because you have changed your beliefs about a specific company, but because you should be lowering the base growth rate for the economy (global or domestic).*

*Risk free Rates and ERP*

The second variable that goes into play when risk free rates change is the equity risk premium. Again, you have to let go of the notion that equity risk premiums are static numbers that come out of historical data but are reflections of market worries about the future and investor risk aversion. Not surprisingly, the same forces that cause interest rates to move also affect the market's perception of risk and will cause equity risk premiums to shift. This can be seen when you look at implied equity risk premiums, where you back out what the market is demanding as an expected return on stocks from cash flows and subtract the risk free rate. In the graph below, I outline this effect since 2008.

The most striking finding, at least for me, is how little the expected return on stocks has changed since 2008, staying around 8%, while risk free rates have more than halved. The net effect is that the equity risk premium, close to 4% prior to 2008, has now moved to 6% and above.

*Dynamic Implication: As the risk free rate changes, the equity risk premiums you use will also have to change to reflect the market's updated expectations. A crisis that causes rates to plummet will also make risk premiums rise. If you stick with historical risk premiums, while using current risk free rates, you will misvalue companies.*

*Risk free Rates and Default Spreads*

The same forces that cause equity risk premiums to rise as risk free rates drop also come into play in the bond market in the form of default spreads on bonds. In the graph below, I estimate the default spread on a Baa rated bond by comparing the Baa bond rate to the T.Bond rate each year from 1960 to 2015.

As with the equity risk premium, default spreads have widened since 2008, from 2.02% in 2007 to 3.23% in 2015.

*Dynamic Implication: As the risk free rate changes, the default spread used to estimate the cost of debt should also change, thus ensuring that the cost of debt will not move in lock step with the risk free rate.*

*Risk free Rates and Debt Ratios*

To complete the story, the final ingredient that you need for the cost of capital estimation is a debt to capital ratio in market value terms. If as risk free rates change, both the equity risk premium and default spread also change, it should come as no surprise that the relative benefits of using one (debt) over the other (equity) will also shift. To chronicle these change, I looked at the aggregate debt to capital ratios, in market and book value terms, for all US stocks, each year from 2000 to 2015.

If you divide the time period into pre-2008 higher risk free rate and post-2008 lower risk free rate sub periods, it seems quite clear that US companies are borrowing more money than they used to. The facile explanation is that this is exactly what you would expect with lower interest rates but remember that those lower rates feed into both the cost of equity and debt. This effect is a more subtle one and reflects the relative risk premiums for equity and debt, perhaps suggesting that the price of equity risk has risen more than debt risk.

*Dynamic Implication: As the risk free rate changes, the debt ratios for companies will also change as they reevaluate the trade off of using debt as opposed to equity. That change, in conjunction with tax and default risk assessments, will lead to a change in the cost of capital.*

*Risk free Rates and Value: The Full Picture*

Now that we have a fuller picture of how risk free rates are interconnected to risk premiums and growth rates, let me revisit the example that I initiated in the static world of valuing equity in a company with a base year cash flow to equity of $100 million. Rather than let the growth rates and the risk premiums stay unchanged, here is what I assumed:

- The nominal growth rate in the economy will be equal to the risk free rate, reflecting how closely the T.Bond rate has tracked the nominal GDP growth rate.
- The company will grow at a rate 6% higher than the nominal growth rate of the economy for the next five years. Thus, with a 4% riskfree rate, the growth rate is 10%, matching the original assumption, but at a 2% riskfree rate, the nominal growth in cash flows will be 8%. In perpetuity, the company will now grow at the riskfree rate = nominal growth rate of the economy,
- The equity risk premium is the trickiest component, but if the market's behavior over the last decade is any indication, the expected return on stocks will stay at 8%, with the equity risk premium adjusting to the new risk free rate. Thus, if the riskfree rate drops to 2%, the equity risk premium will be 6%.

The effect on value of changing the growth rate is captured in the picture below:

Download spreadsheet |

Note that the neither the value nor the percentage of the value from terminal value change much as the risk free rate drops; in fact, they both decline marginally. Furthermore, I can now explore the effect on value of having a zero or negative riskfree rate and it is benign.

I can only give you my personal perspective on how lower interest rates have affected my valuations. With lower rates, contrary to the Bernstein thesis, I find myself less worried about terminal values and the assumptions that I might have made incorrectly. When my nominal growth rate in perpetuity is capped at 2%, 1% or even 0%, I can do far less damage with my assumptions about what a firm can do in perpetuity, than I did in 2007. If anything, low risk free rates makes my intrinsic valuations less volatile, not more so. It is true that these are dangerous times for auto-pilot DCFs where a combination of inertia, trust in historical data (on risk premiums and growth rates) and failure to check for internal consistency can lead to explosively bad DCFs. If Bernstein's point is that a good pricing (based upon multiples and comparable firms) is better than an auto-pilot DCF, I am in agreement!

**Playing Devil's Advocate**

If you are skeptical about my arguments, I don't blame you! In fact, I will preempt you and bring up some counter arguments that you can make against my thesis.

__Mean Reversion__: The essence of mean reversion is that when something looks unusually low or high, it will be revert back to historic norms. Using this argument on risk free rates, there are some who use "normalized" risk free rates (with the extent of normalization varying across users) in valuation. There are two problems with this argument. The first, and I referenced it in a different context in my post on CAPE, is that assuming things will revert back to the way they used to be can be dangerous, if there has been a structural shift in the process. The second, and perhaps even stronger, argument is that you cannot selectively mean revert some numbers and not mean revert others. Thus, if you decide to replace today's risk free rate with a normalized risk free rate of 4%, reflecting 2007 levels, you have to also adjust your growth rates and risk premiums to reflect 2007 levels. In effect, you will be valuing your company in 2016, as if your were back in 2007. Good luck with that!__Central Bank as Master Manipulators__: The conventional wisdom is that the Fed (and central banks) are all-powerful and that the low rates of today have little to do with fundamentals and more to do with central banking policy. If you believe that and you also believe that markets are being led by the nose, you do have the basis for a "bubble" argument, where "artificially" low interest rates are leading all financial assets into bubble territory. The problem, though, is that if this were the case, the cost of equity should be tracking down, in step with the risk free rate, and as the figure on equity risk premiums (in the section above) notes, that does not seem to be the case.

That is not to say that I am sanguine about low interest rates. The low growth and low inflation that these numbers signal are having their effect on companies. Real investment has declined, cash flows to investors (in dividends and buybacks) have increased and cash balances have surged. The increase in debt at companies will not only increase default risk but make these companies more sensitive to macro economic shifts, with more distress and default coming in the next downturn. Finally, to the extent that central banks send signals about the future, the desperation that is being signaled by their policies does not evoke much confidence in them.

**Conclusion**

The risk free rate is an input into a discounted cash flow valuation but it is not an input that can be changed in isolation. When risk free rates change, they reflect shifts in fundamentals that should also show up in risk premiums and growth rates, making any resulting change in value difficult to forecast. As the hysteria mounts ahead of the next FOMC meeting, my suggestion is that you step back and take a big-picture perspective. This too shall pass!

**YouTube Video**

**Attachments**

- Risk free rates, Inflation and GDP Growth
- Risk free rates and ERP
- Risk free rates and the Baa Default Spread
- Risk free rates and Debt Ratios over time
- Static and Dynamic Valuation Spreadsheet

**DCF Myth Posts**

Introductory Post: DCF Valuations: Academic Exercise, Sales Pitch or Investor Tool

- If you have a D(discount rate) and a CF (cash flow), you have a DCF.
- A DCF is an exercise in modeling & number crunching.
- You cannot do a DCF when there is too much uncertainty.
- It's all about D in the DCF (Myths 4.1, 4.2, 4.3, 4.4 & 4.5)
- The Terminal Value: Elephant in the Room! (Myths 5.1, 5.2, 5.3, 5.4 & 5.5)
- A DCF requires too many assumptions and can be manipulated to yield any value you want.
- A DCF cannot value brand name or other intangibles.
*A*DCF yields a conservative estimate of value.- If your DCF value changes significantly over time, there is something wrong with your valuation.
- A DCF is an academic exercise.

## 8 comments:

Hi Ashwath, Great to read your argument. You do a great job of straddling the academic world and the marketplace and that makes you very interesting.

You seem to suggest that DCF model that uses “adjusted” assumptions will justify the current valuations. What is the use of a valuation tool in which all parameters can be changed to justify prevailing prices? Doesn’t that defeat the purpose? The utility of a reliable tool is that it provides users with an anchor in reference to which one measures the moving object. If the anchor keeps moving will you ever get valuations that are incorrect? Given the adjusted inputs the market would be fairly valued at all times.

Of the two parameters under debate, viz., interest rates and growth rates, the former has a giant market maker (central banker) who steers the rate by providing two way quotes for unlimited quantity, at all times. Therefore we take that as a given.

The source of growth rate in the economy is a complex derivative of multiple forces such as demographics, government policies, innovation, competition, technology, politics and many others. Investors attempt to arrive at a growth rate (for an economy or a company) by factoring in these myriad factors. Clearly a complex but not entirely scientific process. The variance in the rate thus arrived is the reason there is a buyer and a seller.

To force this complex growth rate down to a number that matches the diktat of central banker is to raise the central banker to the status of God that knows everything there is to know.

Can you be more specific on how you calculate ERP? It may be worth a separate post to go over which numbers you use, the logic, etc.

Dear Aswath Damodaran,

Great, that you won't let go of DCF because of low rates.

How come, you estimate, that the real risk free rate should be about the same size as the growth in GDP? (I cannot find the blog where you argue this view)

The Fisher Equation is to my knowledge about inflation and the real versus the nominal rate - not about GDP or growth.

Also, if you take out inflation from your illustration and compare real rates to real GDP growth they doesn't seem to follow each other very well.

Finally, the logic also escapes me. Of course you should demand and expect something (in real terms) for lending out your money - even to a very solid country - but why should it be around the GDP growth rate?

Should we really be able to participate in and get the growth of the world without taking any risk?

Dear Prof. Damodaran,

Thank you for your insightful post! The only question I have so far is on which inflation measure you use in your calculations. From the information in your attached file, the inflation rates do not seem to correspond to the DoL average annual inflation rates.

Kind regards,

Tim

Quick question professor. Do you think there are special implications for high / low beta companies when rates are near zero? It looks like Cost of Equity is relatively steady ~8%. If a higher percentage of Ke comes from ERP should this drag valuations of higher beta companies, at least theoretically based on CAPM (Ke = Rf + B x ERP). Or is this more theory and practically less likely to actually happen?

Thanks for the post.

How come you too keep the cost of equity constant (rather than making it dynamic)? Sure you use a dynamic ERP as rf rate changes, but it may not change so as to keep Ke constant.

Another way to look at this is: When treasury yields 2%, bonds yield a little more, why not settle for say 5% on equities? As an investor I should (so too should other investors) be happy to get say, 5% including dividends from equity compared to the alternative investments available at this point. So shouldn't it be fair to use any rate over treasury and bond yields? Why keep at 8%, just because the implied ERP gives that number? Note that the implied ERP is calculated based on cash flow estimates which can be incorrect too.

Dear Professor Damodaran,

Thank you for your post.

It makes perfect sense to adjust ERP in lower growth enviroment.

But then, if we try to value fair value of the S&P 500 with a 1,5% risk free rate (4 months ago), even with an ERP of 6,25% and with estimated dividends nominal growth of 4,5% (2,25% real growth + 2,25% inflation) I get a fair PE of 28! Using current 10 year T-bond, 2,35%, I get a fair PE of 24! These are much higher than real current numbers.

The point is I think markets are assuming higher ERP but also higher stable LT risk free rates (due to central banks intervention).

And this is why I think that again markets can perform well in an increasing rates environment because it is already incorporating that in discount rates (I thing 3,5%-4% for risk free rates).

What do you think?

Best regards,

Hugo Roque

Dear Prof. Damodaran,

I must begin by highlighting the importance, impact and usefulness of your role on the 'world of finance' through your books, articles or available data. At least, it has been helping me a lot on my also not so profound thoughts about valuation and macroeconomic issues.

Relatively to this post, I do believe/support that relatively low interest rates can be the basis for economic growth as long as financial institutions are able to reasonably evaluate their clients' credit risk and governments/central banks maintain a watchful eye on inflation. Working on the assumption that both premises are guaranteed, low interest rates would release cash outflows for financing activities to investment activities. Additionally, it would force private investors to seek for opportunities within their organizations/companies or look for entrepreneurial endeavours. From my point of view, high interest rates only serve the interests of financial institutions in detriment of real economic growth.

To close this argument, I believe that if, in the presence of low interest rates, companies are able to find investment opportunities higher than the cost of capital than value is created both through ROIC and lower cost of capital. However, in the absence of value-creating investments, companies also would be punished by the same reasons (decrease on cash inflows and higher credit risk - spread). It would be interesting to see if your relation between T.Bond rates and GDP growth rate still holds for several individual countries from different regions.

At last, with no right to ask something of this kind, I would be overwhelmed if the Professor would have any time or interest in valuing a Portuguese (yes, i'm portuguese) bank named 'NovoBanco' (former Banco Espírito Santo). This historical bank was bailout by the Portuguese government and received €4.9bn. After declining an offer of ~€2bn last year, we are at this moment on a bidding process and rumours talk on a acquisition price less than half of what was offered last year. I confess I would be delighted to hear your thoughts on this company since our country we lack financial discussions regarding some truly impactful issues.

Always great reading as much as possible and learning at the same time from Professor.

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