I suspect that many readers of this blog also follow the blog of Aswath Damodaran, and if they don’t, I would recommend doing so. Damodaran is a Professor of Finance at the Stern School of Business and he often posts how he would approach valuing high-profile stocks such as Tesla or Uber. It’s very unlikely I would ever be interested in investing one of those companies, but his valuation methodology is pretty solid and his posts are educational. I think having a good grasp on the fundamentals of valuing a company is very important, even when most of the time you revert to simple back of the napkin valuations using a P/E multiple (which I do).

However, I disagree strongly with his approach in his latest post about how to incorporate truncation risk in the valuation of Aramco. To quickly recap Damodaran point of view, a DCF valuation is only suitable for going concerns because you assume cash flows are continuing long-term and potentially indefinitely. Because of that you can’t properly model survival risk where a company might cease to exist after a certain point in time. In the case of Amerco it could be a possible regime change at some point in time to fundamentally change the value of the business going forward. But it’s also very relevant in startups that have a high probability of not making it past the first few years. His suggested solution is to value the company as a going concern in one scenario, value it with a regime change in a second scenario, put some probabilities on the two scenarios and call it a day.

I think there are two problems with his post:

- This approach doesn’t work well because it doesn’t fit reality
- A DCF is perfectly capable of incorporating truncation risk

The reason that this methodology is flawed is because it doesn’t incorporate the timing of the potential event. It matters a whole lot if the regime change is in the next year, or if it happens in 30 years time. So what does a 20% probability mean? Is that a 20% probability of regime change happening this year? the next 5 years? the next 50 years? And because that question isn’t answered, you also can’t really do a equity valuation in the regime change scenario. What he is doing I think is just taking the going concern equity valuation, and reducing it by 50% in the regime change scenario to account for higher taxes and royalties. That implicitly assumes that the regime change will happen on day one, which is of course highly unlikely and unrealistic.

A better model would be to assume that there is a small probability of a regime change every single year. That sounds complicated, but luckily, that’s exactly what we can model in a DCF by increasing the discount rate. If you would for example assume a 1% probability that there is a regime change in a given year you can simply increase your discount rate by 1%. It’s that easy! A one percent chance for 10 years in a row would imply a probability of regime change of 9.56% during that period and 39.5% during a 50 year period.

Note that by increasing the discount rate by one percent in case of a regime change we are assuming that the residual value is zero. If you think there is a 1% probability with 50% residual value we should increase the discount rate by just 0.5%. And a second small note, if you want to do things mathematically correct, you can’t simply add probabilities. In the example below you can see how that becomes especially relevant when probabilities become large.

If you want you can also easily change the model to customize probabilities of failure for each time period. Perhaps you have the view that there is in the next five years a higher chance of regime change, followed by a period of lower probabilities. But it’s probably more useful when thinking about a startup. The probability of failure is presumably very high in the first few years while lower in subsequent year.

Let’s imagine a project that will costs 10 million to start and 10 million more at the end of the first two years, and will produce 100 million at the end of year 3 and 4. In the first two years there is a 25% probability of failure. We could (and should!) model that as following:

Note that the discount rate is calculated by taking: `(1 + "equity risk") / (1 - "truncation risk") - 1`

and the NPV factor is the factor of the previous year multiplied by `1 / (1 + "discount rate")`

of the current year. That way you carry a risk of total failure that you introduced in the early years forward for the calculations of the NPV of later years^{1}.

The great thing about this model is that it matches actual reality. If you have a failure in the first year you don’t have to put more cash in the project, that’s why the net present value of the cash outlays at the end of year one and two are severely discounted. And of course, the same is true about the payoff in year 3 and 4 because there is a high probability that the project will never get to that point. If you would want to model this using Damodaran’s approach you would be in big trouble, because one single failure scenario can’t easily account for the fact that failure can occur at different times, and that the impact on the net present value depends on the timing.

# Concluding remarks

Let me know in the comments if you found this post useful, otherwise I will be back to posting the occasional merger with a contingent value right ;). And of course, perhaps you want to point out that I’m completely wrong. It’s totally possible that I don’t know things better than someone who is a professor of Finance.

# Disclosure

No intention to ever initiate a position in Aramco

1. A previous version of this post was using the wrong formula to calculate the discount factor and the combined discount rate. Thanks to a reader in the comments this has now been corrected.

ZebI enjoyed reading this post. It was a nice change of pace from your investing ideas, which I also enjoy. Thanks for taking to time to write this blog!

Alpha VulturePost authorThanks, nice to hear! π

German readerGreat post and please more of these!

You make some valid points especially about timing, which Aswath Damodaran does not consider.

I struggle, however, to understand your modified DCF. If you assume 25% risk of failure in a specific year why do you just multiply it onto the discount rate and don’t consider it further in the subsequent years?

” If you have a failure in the first year you donβt have to put more cash in the project, thatβs why the net present value of the cash outlays at the end of year one and two are severely discounted.”

Isn’t that exactly what you fail to implement with your approach?

Wouldn’t you instead have to make a decision tree branching for every year where failure could occur, leave the discount rate as is and add all the branches probability weighted up in the end?

Alpha VulturePost authorBy using previous years discount factor to calculate next years discount factor you do consider it in all subsequent years, so if you put a total risk of failure of 25% in year 1 you basically reduce the net present value of all cash flows that are modelled after year one with 25%.

In the example I give the discount factor for year 3 is for example 48%, which is calculated by (1 – 29%) * (1 – 29%) * (1 – 5%). So because the two probabilities of total failure of 25% are incorporated in this discount factor the net present value of this cash flow is 1-(75%^2)=43.75% lower than it would have been if there would have only been the equity risk discount factor of 5%/year.

Hope this explanation makes sense, otherwise try recreating my example in a spreadsheet.

Israeli readerI indeed follow both you and Damodaran, and found both your posts instructive.

Alpha VulturePost authorGood to hear! π

PeterisThanks! Great post. Happy to read more π

Alpha VulturePost authorThanks π

DannyHey Alpha,

Really enjoyed this. I must admit that I’ve done this exact type of analysis where I create a few different cash flows and discount rates, apply a probability to each scenario and then add up the weighted expected present values, and there’s the risk adjusted present value. I now see why this method is flawed, since it doesn’t really represent reality as it ignores timing.

I agree that Damodaran’s method is unrealistic since it doesn’t account for the timing of a regime change, but I also don’t understand your alternative because it doesn’t account for the impact on cash flows. In other words, I don’t understand how simply increasing the discount rate on the status quo scenario accounts for the entirety of a regime change scenario.

Perhaps a better way is to normalize the cash flows by averaging the weighted probability of a regime change scenario with the status quo scenario, and then using a normal market discount rate?

Also, could you explain the math behind your 9.56% chance of regime change in 10 years, if 1% chance of regime change in any given year?

Best,

Danny

Alpha VulturePost authorHi Danny,

Let me start with answering the easiest question first, if you have a 1% probability of something happening in one year the cumulative probability is 1 – 0.99^10 = 0.0956. You can’t simply add probabilities, because otherwise you would have a 100% probability after 100 years (and more than 100% after that which obviously doesn’t make sense).

With regards to adding a probability to the discount rate, it incorporates the regime change scenario by effectively adding a branch to the set of outcomes with a zero value. If you increase the discount rate with 1% in year 1 (and leave it unchanged for all future years) you basically put a branch with a 1% probability in your valuation with a zero outcome at that point. If your view is that value halves in a regime change scenario you should put a 0.5% probability (keeping half of the value there). If you then want to have this possibility in any given your you simply increase the discount rate for every year (and that is a lot easier to do in a DCF than selectively changing the discount rate for a few specific years such as I did in the example in my post).

(Note that while I talk about adding probabilities, it really is multiplying as in the 9.5% after 10 years example!)

Hope this was helpful π

DannyThis was helpful, thanks!

Could it also be asked like this?

How much would you pay for $1 in 10 years, assuming no inflation, if there’s a 1% probability that dollar goes to 0 in any given year? I think the answer is .9053 cents or 90.53% probability that dollar will be there at the end of 10 years.

(For some reason, this is a slightly higher probability than your calculation which implies 90.44% chance.)

Yes, so I also don’t really understand why Damodaran says: “The problem, though, is that this higher discount rate still goes into a DCF where expected cash flows continue in perpetuity, creating an internal contradiction, where you increase the discount rate for truncation risk but you do nothing to the cash flows.”

He is basically saying that increasing the discount rate does not equate to increasing the probability of truncation risk, because there are still cash flows, but you are clearly showing that it does take into account future cash flows.

So, in Aramco’s case, a 1% chance of regime change in any given year, and only 50% negative impact to cash flow, the real discount rate added should be 1%-(1%*.5)=.5%. If there is a 100% negative impact to cash flow if the regime change occurs, the real discount rate added should be 1%-(1%*0)=1%.

Also, if the impact is expected to be 50% positive to cash flow, then we can have 1%-(1%*1.5)= -.5%, thus decreasing the discount rate and increasing the present value.

I guess the rule of thumb for adding in scenarios to the discount is as follows: adjust the probability of each scenario by the impact to cash flow it might have, then add this risk to the discount rate!

Alpha VulturePost authorHi Danny,

Not quite sure how you get to 90.53% because 0.99^10=0.9044 according to my calculator. But yes, your idea is right (although instead of assuming zero inflation the real assumption should be a risk-free rate of zero, you can have a risk free rate of zero while having positive inflation, see the Euro).

And yes, that’s one of the reasons I disagreed with Damodaran’s post: a DCF is extremely suited to incorporate truncation risks if it’s a recurring risk in every year (which is very applicable to the regime change risk in Saudi Arabia I think).

But yeah, think you got it π

renge54Wonderful, thanks Alfa Vulture, Danny & all. Working off AV’s probability adjusted discount rate formula (with 5.00% “equity risk”, 1% “truncation risk”), the discount rate adjustment is probably 47.5 bps assuming 50% value, and 95 bps zero value (wonkish; good grasp of the fundamentals, BOTE calculations and a dose of common sense are never out of fashion).

DannyI have a BA II Plus calculator and solved for the PV.

Inputs:

N=10

Discount rate=1%

Future Value=$1

Payments=0

Alpha VulturePost authorAh okay, I see what is causing the difference. I start with a present value of $100 and then after 10 years of 1% risk you get a future value of $90.44.

You are doing it the other way around. If you have a future value of $100, but a present value of $90.53 you didn’t lose money, but you made 1%/year. What you put in the calculator is a growth rate, not a discount rate. You could do it by having a -1% growth rate, but then it’s still not very intuitive because your starting point won’t be a nice round 100, so you can’t easily see what percentage of money you lost because of the risk.

DavidThe CVR now trades under the BMY/R ticker on the NYSE. I can’t get access to a Schwab or Etrade account. Does anyone know where to buy this as a European?

Alpha VulturePost authorI can buy it at Interactive Brokers, and I assume there are other options if that doesn’t work for you.

Fred McArthurI have an issue trying to figure out what we are doing, adding probabilities to a discount rate. The discount rate is not a probability is a measurement of risk about variations in the estimated cash flow as continuous risk and truncation is a discrete risk.

What I understood from your model is that you are adding a premium to the discount rate.

That premium is not telling any probability unless you attach some distribution to that premium, and in that case, the probability will be about the premium value.

Alpha VulturePost authorHi Fred,

The discount rate itself is not a probability (you can’t say a 6% discount rate is a 6% probability of something), but it does incorporate the probabilities of all the various risks you are exposed to. In theory you could start with the risk free rate, assign probabilities to the countless risks you are exposed to every single year as an equity investor for example, and reconstruct the equity risk premium (taking into account the magnitude of the various risks as well of course).

Truncation risk is not different really than any other type of risk. In the case of Amerco perhaps the assumption is that expected cashflows will be 50% lower going forward. But that’s fundamentally not different to the fact that a random company can have some fundamental shift in business and is expected to have 1% less cash flow going forward, 10% less cash flow going forward or 25% less cash flow going forward.

Note that I’m not just multiplying the probability of truncation risk with the cash flow, it’s the size of the impact multiplied by the probability. In the simple example above the impact is a loss of value by 100%, so a 100% impact times a 25% probability is a 25% increase in discount rate. If the impact would be a loss of value of 10% a 25% probability of that event would be to increase the discount rate by 2.5%.

FredAlright, correct me if I’m wrong.

Your approach to incorporate truncation risk consists of assigning a probability of regime change per year (1%) multiply (1-residual). So if the residual value is 0 and the probability of regime change is 1%, truncation risk is 1% (premium for our discount rate), and the probability of suffering regime change in 50Y is 1-(1-0.01)^50 equal 39%.

Let me go to another question. Should we add truncation risk to the cost of equity as a risk over the dividends/FCFE or should we add it to the cost of capital as a risk over the firm cash flow?

Alpha VulturePost authorHi Fred,

Yes, to the first question.

And the second question, both approaches should work and should be equivalent. If you for example think that there is a 1% of truncation risk with a 50% loss of equity value for a 0.5% increase in discount rate you probably should model a 1% truncation risk with a lower than 50% loss of firm value (depending on the amount of debt).

GianlucaAlpha,

Damodaran’s method is simple but coherent. Yes, you could do a Monte Carlo simulation with state-dependent cash flows and discount rates etc. etc. But will the added complexity be offset by a more than proportional increase in accuracy? I highly doubt it. It’s better to be approximately right than precisely wrong.

I also think you got a bit carried away with your “method”: could you show us the math of why you should adjust the discount rates as you do? As per your above explanation it doesn’t make any sense.

Also, your solved example has at least two flaws:

a) If cash flows and probabilities are known you should discount at the risk-free rate, not an “equity” one, whatever that means.

b) the NPV of the project is 72.62 c.a. as can be found by drawing the tree and using backward induction. Your method, as presented in the post, is pure phantasy.

Maybe you could write another post to shed more light on your ideas.

One final thought: I read many people online taking exception to Damodaran’s valuations and presenting “the” genious model they’ve found: not that he’s perfect, but maybe taking into consideration that he’s being teaching and researching valuation for decades, we should all be a bit humbler, or else we might end up like those amateur mathematicians who think they solved the Goldbach conjecture in a weekend and get angry when professionals tell them their “proof” doesn’t make any sense.

All this said of course with great admiration for your blog and your posts: I’m great fan!

Alpha VulturePost authorHi Gianluca,

I recreated a tree of the example project, because I realize I could very well be wrong: working with probabilities is always tricky and often counter-intuitive.

Luckily it seems to me that my math adds up, the NPV I get from summing all the values in the tree is exactly the same as the NPV calculated in the DCF example above. If you want to see the math behind it, you can check out my shared Google sheet here: https://docs.google.com/spreadsheets/d/1V1D3QCXndVrvGbytvCQA-nZsH76_3t1kRO4Ax35mhgQ/edit?usp=sharing

Regards your other comments, yes the example could have worked just as well with a risk free rate instead of an equity risk rate. But I was imagining a project with normal business risks, not something with 100% certain payoffs (we were talking about valuing companies after all).

And Damodaran wrote on Twitter that I had a legitimate point, so I don’t feel like I’m ending up like those amateur mathematicians π

And great to hear that you like the blog! π

GianlucaThank you for sharing the spreadsheet :-), Alpha.

In your tree computations, though, it looks like you messed up the discount factors: you multiply by (1-0.05)^t instead of dividing by (1+0.05)^t.

If you correct that you get the 72.62 c.a.

Alpha VulturePost authorShit you are totally right!

And I have a related error in the calculating of the combined discount rate of the “equity risk” plus “truncation risk” as well. That should be calculated as (1+”equity risk”)/(1-“truncation risk”)-1.

I have now updated the shared spread sheet, and now both the DCF model and the tree give the same 72.62. The idea that you can incorporate truncation risk in the discount rate works, but there was a big mistake in my calculations!

I will now update the post to fix it for future readers π

SpecterYou can simulate the model with Monte-Carlo simulation without adding complexity, but it’s irrelevant for this problem. We are not trying to estimate a density kernel for the NPV but a rational way to add a new premium to the discount rate bearing the probability of regime change.

Regarding the “known” probabilities. He is calculating cash flows under uncertainty; therefore, he is using ERP and a truncation premium. The ‘known’ probability is just an assumption about the regime change event.

Nobody is making crazy math or creating new models, we are using the same model dealing with the problem with a different approach, and we respect pretty much professor Damodaran.

renge54Mathematical amateur hour, I am afraid. But kudos for trying.

Alpha VulturePost authorThanks for your comment!

renge54In your hypothetical example, the truncation risk is for zero value (start-up failure) during year 1 or 2, for which you’ve observed a way of adjusting the discount rate in the DCF valuation. In the Aramco case, that would be nationalization without compensation. Realistically, “regime change” would entail a range of outcomes (with varying degree of positive residual value). There is a reason why we still have DCF, Monte Carlo simulation and decision trees around, and more development work is forthcoming. Statements to the effect that “if you think there is a 1% probability with 50% residual value we should increase the discount rate by just 0.5%”, I would moderate out of the original post.

Alpha VulturePost authorProof me wrong then about that statement ;). Conceptually there is absolutely no difference between a 1% chance of 50% loss or a 0.5% chance 100% loss. Both have the same expected value.

If you want to make more complex assumptions or models, I’m sure that at some point you want to use Monte Carlo simulations or other tools.

Pingback: Best of What Weβve Read (11/24/2019) - Compounded Knowledge Press

MMI007Interesting perspective I heard: when investing in Aramco, do remind it’s a mafia business, hyper corrupt, totalirarian, and pseudo-talibanic. You are lending them money. Will you ? Really ?

pogonificHey AV,

An enjoyable post. I guess I’ll make two comments:

1. Both you and Damodaran start with the same DCF model structure (which has a set of assumptions) and then both of you add one more assumption in order to model the Aramco special case. Given that for every single change in the discount rate (in your world view) there exists a single probability of regime change (in Damodaran’s world view) which makes the end valuation the same, it’s really a semantical issue as to how exactly you’re defining the additional assumption parameter you’re both adding. And therefore in my mind the difference between approaches is, if you take a step back from the detail, minimal.

2. I’d argue it’s exactly as hard for you to pick the right change in discount rate to put into your assumption list as it is for Damodaran to pick the right probability to put into his assumption list – because both of you are as clueless as each other as to the true probability of regime change. And so therefore, both approaches are somewhat (and equally so) deceitful – they fundamentally produce a single number at the very end of the process which has been produced by taking something you have no clue about (true probability of regime change) and assigning some finger in the air number to it.

But I guess in my world view this is really an issue with the whole DCF approach (I hate it BTW and never use it), as at its core that’s what DCF does – it gives you the impression you know something by hiding what you don’t know behind a complex enough structure to make it look like you might know. If you read any of Mauboussin’s books, to me the whole thing’s a joke – I can’t believe so many people take that seriously.

Thanks for posting – always enjoy the blog!

Alpha VulturePost authorHey pogonific,

I would argue that there is a really big difference, even though you could pick in both valuations the numbers in such a way that you get the same result. The big difference is that in one model you put in a number that has actual explicit meaning while in the other options you just add some meaningless correction factor.

1. Increasing the discount rate. You can’t easily say the discount rate of X% higher means Y in the real world.

2. The single probability of regime change. That sounds like it’s a real world probability, but that’s my whole issue with his methodology. His 20% number doesn’t mean anything. It’s not a 20% probability of anything, because it’s undefined. It’s just another way of adding a correction factor.

3. Finally, my method, were you assume an explicit probability of regime change every year and its impact on the value of the company.

And yes, I don’t really have a good idea about what the actual probabilities are. But I’m not investing in this company, and someone who would presumably does. But at least in my model, you can input a probability. Even if you would know that there is a probability of regime change of X% for the next Y years, you wouldn’t be able to figure out with what percentage to increase the discount rate or what the so called “single probability of regime change” would be.

PogonificWell, I think we 80% agree because I think we’re both sceptical as to how much we really know π And I also would never in a million years buy anything Damodaran values – for me it’s all about this year’s cashflows as a proxy for next – the rest I’m honest enough to myself to admit that I don’t know.

I completely agree with your point 2. Your point 1 I find difficult – let’s say you open a newspaper tomorrow and read the headline “King Whatever’s son (i.e. the next Saudi leader) publishes book espousing democracy”. How much would you change your discount rate by? 0.5%? Maybe 1%? How would you even compute that? My point is that both you and Damodaran have the exact same problem – how do you translate from newspaper headline space into something that changes the additional assumption you’ve added to your DCF? My contention is neither of you can, and therefore your approach hasn’t really made things more practicable – and hence why they are not that different. That doesn’t make it bad, flawed or whatever – just means it’s another theoretical approach with issues.

But I guess for me the elephant in the room (as I think discount rate or regime change probability really is a minor point) is the DCF approach. DCF takes a whole bunch of inputs. Each of those is really a probability distribution. And then it spits out a single number, when really it ought to spit out a probability distribution. But investment analysts don’t understand maths so they just hang their hat on the single number the DCF spits out. And here’s the thing, I strongly think the analysts know so little about their input assumptions, the input probability distributions are as near as you can get to a uniform distribution as possible. And this means that inevitably that makes the output distribution uniform as well. And therefore the value that’s spat out is meaningless – it’s like me telling you that I expect an unweighted die to throw a 3. Sure I expect that, but that doesn’t give me anyway to make money betting on the next throw if all the values are at the same price. And yet that’s exactly what people do when they take the DCF number out and ignore it’s really in a uniform distribution. And so debating the semantics of a minor change to a flawed approach for me has limitations.

Alpha VulturePost authorI think I’m a bit less skeptical about DCFs. Sure there are serious limitations, and if you input garbage you get garbage out. But theoretically the method is sound. And if you start with a theoretically sound model and combine that with some of your assumptions it’s presumably better than starting with a model that has fundamental flaws and the same assumptions. And there are many instances imaginable where a simple DCF can tell you more than just taking this years cash flow as a proxy for next year. Think a toll road with a concession that is about to end, a royalty stream that will expire or on the other side a high growth business.

With regards to your overal point of translating a headline into an adjustment of the model. Sure, that’s not easy. But if you have a specific probability as input you could read something and think, “okay, this sounds bad, maybe the chance of failure the next year is two times as high”. In the little model I have above you could for example change the 25% probability in year one to a 50% probability and you would be able to see what kind of impact that would have on the valuation.

With Damodaran’s model that kind of reasoning is also possible, except not that detailed. The 20% is not a meaningful probability, but if you think that going forwards everything has a two times as high chance of failure you could double it. The big problem is that the 20% is just not a meaningful number. If you read something and think there is a 50% profitability of X happening you cannot do anything with it. You can only make relative changes.

And just an addition to the discount rate is even more problematic as a model. If you increased the discount rate with a certain risk factor you can’t double that risk factor if you think the probability of failure doubles. But you can translate it do an actual probability first, and then do something with it. In my example 5% equity risk with a 25% truncation risk results in a 40% discount rate. Double the truncation risk to 50% and the discount rate goes up all the way to 110%. So you can’t make (intuitive) adjustments to the discount rate but you can do that with the truncation risk probability itself. And that’s the main gain of the model. Putting explicit meaningful numbers in it that can be used to reason with.