**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:

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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*