Mar 15, 2010

Why can't the discount rate be lower than the growth rate in terminal value?

What is the theoretical reason for it.

Thanks.

### Ways to Calculate Terminal Value

Terminal value is an important part in determining company valuation. Before digging in to the theoretical explanation to the above question, here’s a quick review of the calculation. Depending on various factors, you may want to use an exit multiple or perpetual growth method, such as the Gordon Growth Model for determining terminal value in a DCF model.

For more information on how to find your growth rate and discount rate, check out these posts:

• explains:

Growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it.

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Um, the discount rate is higher than the growth rate buddy, not the other way around.

The model does not work because it would give you a negative number (impossible). You would need to use EBITDA and an exit multiple to find terminal value.

"Greed, in all of its forms; greed for life, for money, for love, for knowledge has marked the upward surge of mankind. And greed, you mark my words, will not only save Teldar Paper, but that other malfunctioning corporation called the USA."

Uh, it has to be, otherwise you get negative terminal value.

n - terminal year, r - discount rate, g - growth rate

Perpetuity PV = (FCF @ n+1) / (r - g)

If r = g, PV = 0 (or really it's probably infinite, but the formula gives you a negative value).

Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

nick_123:
Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

BobbyLight:
nick_123:
Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

Hmmm, I'm still not seeing it. Why wouldn't a company's growth rate be able to surpass the required return of investors?

After all, growth and risk are driven by different variables.

nick and Bobby, you were saying the same thing.

giants92:
nick and Bobby, you were saying the same thing.

No, the original post has been corrected.

If a company were to grow faster than the expected rate of return in perpetuity, in effect growing faster than the market itself, then the company would be on pace to eventually become larger than the entire market. Impossible.

Nick, growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. This is kind of like asking, "why don't trees just keep growing past the clouds?" or "why can't a stock be worth less than zero?" You're just playing with numbers here and have forgotten the underlying reality. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! Seriously, unless you think this is a likely scenario, it just simply cannot be that the growth rate exceeds the risk. Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

With an answer like that, you should be working at a hedge fund.

Absolute truths don't exist... celebrated opinions do.

jhoratio:

Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! ... Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

Maximum effort.

A company with 300% annual growth rate, like Intuit had at the start, for example, shouldn’t have a discount rate of 301%. It don’t work for startups.

Is there a dcf based valuation method for startups or intrapreneurial projects within an established firm?

+1 SB. Thank you posting a good part of the answer.

....this whole thread is really really sad. It shows that a lot of people know modeling well but don't understand the underlying...here go some additional details:

1. Economic reasons for this include the attraction of competition. If you're growing at a tremendous rate, it's not that investors just want to invest in your company. It's that competitors will jump into your industry as well.

2. There is a limited amount of market share out there. If you're growing at 20% per year, your model will soon have more customers than exist for the product on the entire planet.

3. Reasons one and two lead to a predictable life cycle of a company. We have seen this over and over again with companies that initially have super strong growth rates and as they become bigger and bigger set into a fairly predictable steady state. Consumer goods are a great way to think about this. At one point, Coca-Cola was growing like crazy. As it became a mature and larger company, it's simply not going to sell 20% cokes next year. The opportunity set and markets have been for the large part exhausted. Or think of even something like a Microsoft. It grew like crazy and now it's a blue chip. Low terminal growth rate is the fate of us all if we do well enough.

4. Lastly, as someone else said later on here, when you've completely exhausted your obvious opportunities and taken all the share you can, guess how you grow? Some relationship to GDP which will almost necessarily be below your discount rate.

Nothing of the above is wrong but there is a mathematical reason that it must follow R>G (other than it’s not realistic, which is above answer)  see below basic formulatic definitions that are relevant

R = annual FCF / Invested Capital (market value).
G = reinvestment rate * R

reinvestment rate*invested capital + invested capital = 1-yr forward invested capital

so what this tells us is that invested capital grows at the reinvestment rate and FCF grows at G. When G(t+1)> R, the forward return (FCF / Forward Invested Capital) would be greater than R, which cannot be true since terminal R represents the future annual return of every year in perpetuity. Practically, I’d think of it as if G>R than more capital pours in, which right-sizes R to below G.

It's a convergent geometric series (infinite series that converges to a finite sum) as long as growth r. However, it becomes a divergent geometric series (diverges to infinity) when g > r. Hence, the model blows up when g > r.

This isn't the underlying theoretical reason why 'g' can't be greater than 'r'... this is just an application of the theory that produces a mathematical error if the theory (i.e. r>g) has not been accounted for.

Bro you're from Australia, nobody is trying to hear your take on anything besides what it's like to be retarded.

Perpetuity Growth rate higher than required rate of return = a bubble that will never burst.. like puff in the microwave...

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If you have any perpetual yearly cash flow that grows at a rate greater than the discount rate, your NPV will be infinite. Think about it this way - every future year's present value will be greater than the previous - because your cash flow is growing faster than you can discount it - and thus you'll will not obtain a finite net present value.

Question:

i have a project planned to start on 1 jan 2010, is able to produce revenue cash flow starting from 1million on 1 jan 2011 and this is expected to grow at 13% per year until 1 jan 2020.

The initial cost of the project is 20million. Beta: 1.1, risk free rate: 1.7%, risk premium is 9.5%.

So the cost of capital should be 0.017 + 1.1(0.095) = 0.1215

but it is lower than the growth rate of 13%.

is my calculation wrong?

No since your growth rate is not perpetual.

This is a purely mathematical question, it can be answered with zero economic reasoning (although you can go further with it to give the economic interpretation like the guys went here). The sum of perpetual "cash flows" (or whatever) growing at a constant rate and being discounted through time only converges to a value (the "V = F/(k-g)" formula) if the growth rate is lower than the discount rate. If it is higher, this value (the sum of the cash flows, not the formula, which is conditional to this specific case: k > g) is infinite (as all the terms of the sum are higher than one, and the sum is infinite). It can be negative if the numerator is negative (the "cash flows"), and if the growth rate is higher than the discount rate, then it is infinitely negative.

"Never believe in anything until it has been officially denied"

Any model for k higher than g?

So in a 2 stage model, can the growth rate in the initial short-term high growth period exceed the firm's WACC, as long as the perpetual growth rate is WACC?

absolutely, imo you can model any scenario you like and can defend. however these assumptions (as always) need to be reasonable, meaning that the stages should not have abrupt changes in the pace of growth. A way to prevent this is adding a mechanical appendix to the initial forecast (say 6 years initial forecast + 4 years appendix), where growth rates are slowly converging to the perpetuity growth rate assumed.

i believe this method often increases the validity of a model

This response is from a purely conceptual economic point of view (ignoring all of the other valid practical arguments like trees don't grow to the sky and mathematical arguments around divergent series).

If you think about a discount rate as a required rate of return, this becomes an easier question to understand.

Roughly speaking, a security's return / discount rate = 1. yield plus 2. operations growth (EBITDA, FCF, whatever) plus 3. changes in multiple.

Re #3, in perpetuity, you don't get any of the effects of changes in multiple. So we just have yield + operations growth.

Re #1, if we're talking about a strictly positive FCF business, the yield has to be positive.

Therefore, your growth rate must be less than your overall rate of return since your rate of return should take into account both growth and positive yield.

The standard perpetuity with growth formula no longer is valid. It does not give you a negative value and anyone who says so is mathematically mistaken. The formula is derived from a geometric series, with geometric part less than 1. If r g the perpetuity of the cash flows is positive infinity.

If you try to explain theoretically why growth rate can never be greater than the discount rate, you have to keep the assumption in mind that while calculating terminal value, we have assumed the growth to be a stable growth rate and that the firm you are valuing is a going concern.

Now we know that the stable growth rate of the model can never be greater than nominal rate of GDP growth. If the growth rate of a firm is higher than the growth rate of the economy in the long term, then the firm will be bigger in size than the economy which is not feasible

Also, we know that in the long term, real growth rate of economy becomes equal to the real risk free rate. So the nominal growth rate of the economy will be equal to the nominal risk free rate

We know that the discount rate used is calculated by adding a premium to the nominal risk free rate and that the premium will always be positive

From the above two statements, we can argue that since the stable growth of the model is less than the nominal growth rate of GDP, it will also be lesser than the discount rate used in the WACC model

you are one smart monkey - be my tutor ;)

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