# Discounting equity

Hello everyone,

A quick question regarding using the proper discount rates for equity. I’m assuming no debt (or FCFE/RI) to simplify matters.

So we all know that we should be discounting cash flows at their matching opportunity costs of capital. In this case we use the CAPM model. But the inputs used here will be variable and more or less, unique, at least for the explicit forecast period. So with that said, the inputs of the Rfr, and the market premiums associated with that matching time period should differ in all periods. Generally, we are implying an upward sloping yield curve in this case.

So my question becomes, how do you measure the equity risk premium for different spot rates? Do you use the geometric mean of 1 year returns of market over Rfr for t=1, then excess of two year holding period for t=2…etc, or do you use the arithmetic mean for the same sample selection?

This obviously gets more complicated when you include debt into the picture and discount cashflows to all claimholders, because then you’d have to adjust the cost of capital weights dynamically towards the target weight, and increment the beta (and thus Ke) accordingly to adjust for different levels of leverage.

Before you mention it, I’m not interested in using the dirty method of target weights and betas, nor the 20-year YTM from the start. This is for both practical application in my work, and a better theoretical understanding of proper valuation practise.

you use 6.0%. Let me know if any part of your original question remains unclear.

I’ll be building an ERP model at work this week after reviewing Damodaran’s papers (in fact, just sent him an email). You’re 6% ERP for the US market is way off, it rarely reaches that high, not to mention the current ZIRP’s effect will leave it even lower than normal. I’m in the MENA region so that doesn’t apply either way.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2450452

I generally use 5 for the US, though 6 may be sensible, given zirpiness.

I noticed that the survey had Egypt pegged at 13%

I figured out the most likely way I’d use the ERP for my country. I would most likely calculate the ERP for the US as well to derive it.

There is no such thing as a fixed integer for a discount rate that’s constantly changing. The implied ERP for the US today is around 5.2%, although that is really higher than historicals (mostly due to implicit growth assumptions and inflation). The historical geometric mean for the US’s ERP over the T-bond rate (although that would be accurate only for the CFs at t=10) is 4.2%, which is 1.2% higher than the global average of the developed markets,

Now the way I’d calcualte the ERP for my emerging market using a mature market (US most likely) is the following. I’ve spent all day today at work brainstorming and writing down notes for how I would execute the model.

So Ke = Rfr + B (Rm - Rfr). To keep consistent, the Rfr we use, beta, and the ERP has to match each single cash flow to equity (assume no debt for simplicity).

For the first input, we use the Rfr spot rates matching CF at time t. For the Beta, we will assume that no change in operating or financial leverage, or a structural change in the business mix. I will use the local index for now, although I’d have to derive the betas from a global index at some point down the line (I’ll leave this aside for later).

Now building an ERP for an EM, we can use a proxy from the US’s implied ERP, and build on top of it a country risk premium inherit for my country. There are three ways to do this.

1. The equity method: I’d derive the relative standard deviation of equity between a US index, and EG index, by dividing the latter’s standard deviation by the former’s. It would be more accurate if I convert the returns for the EG index to dollars using the spot rates of FX at their respective day. Pretty easy with excel and Reuters. The RSD would then be multiplied by the US’s ERP to derive our own. The ERP for the US to use would be the implied ERP today at 5.2%, and slowly reverting to normal levels at 4.2% over the explicit forecast period. This might not take into account current country risk factors, so most likely I’ll give it more weight towards a terminal phase.

2. The bond method: Deriving the government bond default spread of tenors up to 10 years over the US’s YTM. The defualt spread for each period would be multipled by the RSD of the equity returns of the EGP stock market over the bond market, simillar to above. The ratio is around 1.5-1.6 for EMs, so I’d likely use that. Since there aren’t many soverign T-bonds issued in foreign currency for my country, I’ll most likely extrapolate a default spread based on credit ratings of SP, Moody’s and Fitch using peer’s BDS. This should be the most accurate for nearer cash flow discounts, but should get less weight as the country’s rating improves and the spread narrows. Again, I’d use the BDS of short term and long term periods (ratings give both) to match the cash flows of say t= 1-5, then 5+.

3. The CDS method: This one is a little more tricky. Again, I’d use the RSD to capture country risk premiums (of equity) not reflected in the government’s credit default spread. So the SD of equity gets divided by the SD of each CDS’s tenor. Which would give me 10 RSDs for each period. Then multiply the respective RSD by their CDS rate. I’ll probably use current CDS rates for the first few years, but use historical CDS rates for later years in case a mean reversion occurs. The problem is the CDS might not reflect the country risk premium properly, because even the US has a CDS (0.46%?), I’d either net out the EG-US spreads, or just use the gross rates for EG. I haven’t thought about which would be more proper yet, this is an obstacle for me at the moment.

Any constructive feedback would be nice.

Most analyst don’t get too granular on this issue. The ERP should be thought of a the premium you wish to make over the risk free rate, which in equity valuation, the RF rate is the 10 year US T-note. For most people, the ERP is usually between 4-8%. I think 6% is a good figure. This is a fair return to make over the risk free rate, especially in today’s high-risk, low rate environment. As the global economy improves and as rates rise, the argument can certainly be made to use a lower ERP like 4% or something.

Most analyst bypass the CAPM model all together and use a build-up model based on multiple risk premiums or use a discount rate of the margin of safety or return they hope to make on the trade. By using a higher discount rate your DCF model will be more conservative and robust. Using too low of a discount rate could lead you believe that many securities are undervalued when in-fact they’re not.

I know that analysts don’t get too granular, that’s beyond the point and doesn’t validate improper valuation. The ERP is not fixed and changes from week to week. Not only that, but it is also different at any given point in time for all the future periods because of the term structure. Think of ERP as a matrix in this case.

The Rfr is not the 10 year T-note except when you discount the cash flow you are expecting to recieve ten years from now. All the other cash flows have to use the Rfr’s matching their duration.

You shouldn’t use conservative discount rates, or any other inputs in your model, nor optimistic ones either. Your model should be as objective as possible, without double counting, underestimating, or overestimating risk.

I’m just telling you what most professional analysts and PMs do that I have come across and talked to. I think you are over thinking this whole issue. Unlike Fixed Income, equity is a game of horse shoes. You’re never going to be spot on. The best you can hope for is to be “close enough.” That is why most people use some arbitrary number that makes sense like use 6% or something. I agree that the risk free rate should reflect your expected holding period or time horizon, but when you are on the sell side, you don’t know what your audience’s time horizon is and if you’re in the buy side, it is highly probably that you could hold the stock for 10 years, that is why the 10 year note can be argued to be the appropriate RF rate. Remember, that equities are usually considered long-term investments, espcially in value investing circles, which I think you might be apart of since you are concerned with valuation. All that aside, in today’s low interest rate environment, using CAPM or some other build up model, you will often get some discount rate that is too low and spits out a valuations that highly undervalues the holding.

Trying to keep with the CAPM theme, you then get into a whole other mess of what is the appropriate period over which to calculate beta. Is the trailing 1 year monthly returns? is it trailing 3 year daily returns? Is it trailing 10 year monthly returns or is it trailing 6 month daily returns versus the benchmark? Each one of those will give you a different beta figure that will significantly change your valuation. That is also why many analyst scrap CAPM all together or use a combination of various valuation models to come to their final conclusion. You’re effort to uncover the perfect ERP is a lost cause. This number should be forward looking based on the risk in the market. This is often derived from years of experience and market knowledge. Otherwise, equity prices wouldn’t change as they do, would they. DCF valuation is nothing more than garbage in garbage out. But if you don’t believe me, go ahead and invest all your money into one stock that you believe to be undervalued by a 50% margin of safety and lets see how you do.

It’s funny, you’re asking for advise and I am just guessing that you have zero professional experience and you just passed level I and are telling people they are wrong. 6-9% is often the ERP that I have seen PMs and Analyst’s use, if they even use the CAPM model in the first place. But ok, you try using a lower ERP and the entire equity universe is going to look cheap. Buy Away!

I think Mr.Smart just needs to spend the hours nailing down the ERP to five decimal places, and compare the investment results and decisions with the results he’d get if he used a round number like “5%” or “6%” and then come to a conclusion on his own about how much extra money those extra decimal points got him, and whether all that effort was time well spent, versus using it to discover the next value opportunity.

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But Chad, 5-6% is way too high. It would be unreasonable to expect a premium of 5-6% for investing in equities. Completely unreasonable, according to Mr. Smart, so it must be true. =)

There’s the ERP that is “implied” by some calculation of expected growth minus treasuries, and then there is the ERP which is “justified” by the level of risk in the market. Ultimately, as an investor, your risk tolerance is what tells you how much ERP you need in order to choose to invest in equities. The implied ERP tells you if the market is likely to give that to you in the short term, and the long-term historical ERP tells you if you are likely to get that rate if you wait long enough.

Yes, you want to be objective, but the “right” ERP is as much determined by your risk preferences as it is by anything that the market measures. Every investor has their own ERP. The market ERP is not objectively “real” or meaningful, except that that is where the marginal buyers and marginal sellers are presently in balance.

What MrSmart is saying (perhaps without realizing it) is not that 5% or 6% is way too high. He’s just saying that “current prices would be rational if the ERP were much lower than that,” which is basically the same thing as saying that the entire market is overvalued.

Over very long (50+ year) histories, a broad basket of equities has delivered about 5% more annually (4.9% when I last did the calcualtion). Given that ZIRP unwinding is likely to be risky, perhaps an extra % or two is justified, which means that you would require prices to drop a bit before buying in. Admittedly unwinding could hit bonds too, I suppose (though not if you hold them to maturity, then it’s just buyer’s remorse for not waiting to get a better rate). Even so, the ERP is relative to treasury bonds/notes so it just means that if bonds drop due to QE ending, stocks will drop too (though there’s probably some duration effects on stocks, so I wouldn’t assume it’s 1-to-1).

Yes I know what common practition is like. No, I’m not interested in using arbitary numbers. Yes estimating the cost of equity is close to being a shot in the dark, I agree. But leaving it at that and ‘hope’ that you are close enough is not justified, especially when you try to calculate with decent percision what your opportunity cost should be.

I didn’t say the risk free rate should reflect your time horizon, if you are using a discounted FCFF/FCFE, then you are taking on a control prespective, and you should use all points on the zero curve. At least to derive the IRR (which would not matter then). It doesn’t matter if you intend to hold them for 1 week or 20 years, the valuation is exactly the same in this case.

The beta explains a decent part of a stock’s activity, you should effectively use a bottom-up beta by breaking down the company to their relevant SBUs, calculate an weighted unlevered beta, then relever based on current capital structure while reverting to target if any. Adding premiums to the CAPM is justified, since CAPM only explains systematic risk. But for the global marginal investor with a well diversified portfolio and insignificant frictional or structural limitations, the CAPM is most likely sufficent if done properly.

I’ve never heard of 6-9%, not even in short term arithmetic historical values.

~5.5% is the accepted number with US asset managers, while in reality, the current implied is closer to 5%, and the normal ‘real’ levels over a long period is closer to 4%.

Normally, you’d be right. But you are forgetting that I’m using the US ERP as proxy to derive my country’s own ERP, so it pays to be intricate. One small error in each step compounds significantly, and there are a lot of inputs in this case.

The concept of the “term structure of the ERP” is a theoretical abstraction that is so disconnected from reality that even academics don’t bother discussing it; the concept of the “term structure of the ERP in Egypt” is downright laughable.

Why would the term structure of an equity risk premium be a theoretical abstraction? I wouldn’t even use the term structure of ERP in Egypt because I’m not relying on the Egyptian historical market index. I’m deriving them from a mature equity market, where it clearly exists with more data to extract.

Well, it sounds like we can’t offer you any help here. Good luck!

The model panned out really well. The output was strinkingly simillar to survey results, and the multiple Ke values of an international asset managment firm for my market (according to their ex-employee).

Now I’m facing a final problem, which is something I hope one of you can chime on it.

First of all, why is the risk free rate in the CAPM model, or virtually all investment benchmarks, reported on a before tax basis? In other words, if the governement decides to raise the tax rate to 90% for a treasury security, then the market yield is not really indicative of the ‘risk free’ interest rate.

By going through this step, I can adjust for differences in tax rates, and discount the tax effect where applicable. But I must understand the tax treatment of RfRs first, in theory, and practically for the US and here.

This is because different investors have different tax rates, including many institutions that are tax free. It’s up to the investor to adjust and compute the tax rate that applies to them. Sharpe does do some modifications to establish a methodology, I recall.

In any case, most investment results don’t change people’s or institutions’ marginal tax brackets, so the untaxed expected returns are simply proportional (to a first order approximation) to the taxed expected returns. You can compensate for this effect by simply changing your level of risk aversion in a model. Since risk aversion is subjective and not measureable anyway, it doesn’t do much damage to assume it accounts for after tax risk preferences, as long as you know what the proportionate difference is.

If you do asset-liability modeling, the tax effects are much more pronounced, but most ALM people are almost by definition fixed income types and therefore don’t need to calcultethe cost of equity

Makes sense. But I guess the CAPM is built around the global marginal investor where the portfolio is completely diversified, and transaction costs are not a factor.

As I understand it, the CAPM is built on an after-tax basis, since the earnings, or cash flows, you discount to equity are already post-tax. So it doesn’t pay to subtract a tax effect from the risk free rate in that model. Although that sounds good in theory, it doesn’t keep cleanly consistent as far as I can tell. Because even though the market return is net of tax, the risk free rate is subject to so, and comparing returns should be done on an effective basis. Unless the institutional investor has the luxury of investing in a risk free security without being subject to income tax on interest. Seems to mesh well with the fact that I’ve never seen the RfR, nor the Ke being discounted by (1-t) in any official or literate textbook. But the assumption has to hold that the CAPM is targeted at the marginal institutional investor who is tax exempt on T-securities. Something that I’m pretty sure does not happen, at least in my country.

Untaxed returns are observable, whereas the average investor marginal tax rate (assuming it’s not a fixed rate for all investors), so it makes sense to run the regression off of observable data and make corrections for your individual tax rate than to make some finger in the air guess about what the average tax rate is that applies to all investors in a market, dollar weighted, and whether they are using swaps or futures or ETFs or mutual funds or or are non-profits or pensions, or family accounts protected by tax treaties in the Bahamas, etc.

It is true that when tax regimes change, observable returns will change because of them, such as when dividends and capital gains are taxed at different rates, or investment returns get taxed differently than ordinary income. When these things change, it does mean that historical estimates of thugs like the ERP and Betas are going to be noisier than they would be otherwise, but they are noisy anyway, and anyone who did do this work must have put in so much work to do it (going through historical records and making estimates of the effects and adjustments), that they would be unlikely to publish it if they are not academics. And academics might have strong incentives to peddle the results to the private sector anyway.