How much is a promise in the future worth?

Let’s start with a series of thought experiments.

Thought experiment #1

How much would you pay for the right to receive £1 in 1 seconds' time?

Let me assert for the purposes of this thought experiment that there is no credit risk, and that the obligation is guaranteed to be honoured.

You may want some compensation for your time, and say you’ll pay £0.50, so that you obtain a £0.50 return, but now imagine the same single opportunity is offered to a room of 1,000 people. It’s likely that someone will offer £0.51, and someone else £0.52 and so on, all the way up to £1.00 exactly.

The risk-free right to receive £1 in 1 second is worth almost exactly £1.

Thought experiment #2

How much would you pay for the right to receive £1 in 10 years' time?

Even if we again assume that there’s no credit risk, there are other issues that start to come to the fore over this time horizon. What will £1 be worth in 10 years’ time? What will inflation have done to the purchasing power of the currency in the interim? It should be obvious that paying £1 now for the right to receive £1 back in 10 years is unwise, and the amount you’d be willing to pay depends on your expectation of inflation/currency devaluation in the interim.

You may predict that inflation will run at 2% per year over the next 10 years, and so you calculate that £0.82 in today’s money will have the same purchasing power as £1.00 in 10 years' time, and would be willing to pay £0.82 for the right on that basis. Someone else offered the right may predict that inflation may run at only 1% per year, and so they may be willing to pay £0.91 for the right.

Either way, it should be obvious that a contract for £1 ten years from now is worth less than having £1 in your hand right now (there are some exceptions to this that I won’t go into here).

This idea that money in the future is worth less than money now is captured in the concept of a discount rate, that is, how much the future monetary value should be discounted by in order to work out its current value.

Discount rates are typically expressed on an annual basis (that is, how much to discount the amount by per year passed).

Thought Experiment #3

Now let’s imagine that someone is offering the right to receive £1 today, and on this same calendar day every year for the rest of time. How much would you pay now in exchange for this right? (Again, assuming there is no credit risk.)

It’s worth being explicit that this is the right to receive an infinite amount of money (a fixed amount each and every year for an infinite amount of time), however it’s also worth being explicit that the purchasing power of the £1 received each year decays dramatically (exponentially, in fact), to the point where the purchasing power in the 400th year is almost negligible.

We can use the same logic as above to apply a discount rate to each £1 we receive in the future, and the calculation looks as follows (let’s use a 2% discount rate, or 0.02)

Today we receive £1 of value

At the end of year 1 we receive £1 * (1 - 0.02) of value in today’s money

At the end of year 2 we receive £1 * (1 - 0.02)^2 of value in today’s money

And so on.

The sum value received over an infinite time horizon is:

£1 + £1 * (1-0.02) + £1 * (1-0.02)^2 + …

Which equates to:

£1 + £0.98 + £0.96 + …

This expression is what mathematician’s refer to as a geometric series, and, with discount rates between 0% and 100%, actually has a finite answer equal to[2]:

$S_\infty = \dfrac{a}{1 - r}$

Where a = the recurring annual payment, and r = (1 - ‘discount rate’). Putting our numbers in, a = £1, and r = (1-0.02), yields:

$S_\infty = \dfrac{a}{1-r} = \dfrac{\pounds 1}{1 - (1 - 0.02)} = \dfrac{\pounds 1}{0.02} = \pounds 50$

It should be noted that the value here is highly sensitive to the discount rate. At a discount rate of 2% the right is worth £50. At a discount rate of 1% the right is worth £100, and at a discount rate of 0%, the right is worth an infinite amount of money.

You may wonder after all of this, who on earth engages in these kind of risk-free loans. The reality is almost everyone reading this post has likely lent the US treasury money on this basis, either by directly holding a treasury bond, or via their pension funds, regardless of their domicile. Lending money to the US treasury is considered to be the closest you can get to a risk-free bond.

What is a share in a company?

A share in a company is the right to receive a portion of the profits of that company for the life of the share (amongst other things). The amount of profits accrued to each single share is equal to the total profits of the company divided by the total number of shares outstanding in the company.

Companies do not have a fixed lifetime, some last for centuries, some last for only a few months (my previous attempt at a startup is unfortunately included in this latter count), but it should nevertheless be apparent that a share in a successful, sustainable, profit making company bares a lot of similarity to the instrument in our third thought experiment above. If a company accrues £1 of profit per share per year, then the value of that share can be calculated on a similar basis to the formula above.

What is a share in a company worth?

There are two obvious answers to this question:

The discounted value of future profits that will accrue to this share (as we have just examined)

Whatever price you can sell it for

The catch with the first approach is that the future is unknown, and unknowable. Unlike in our third thought experiment where you’re guaranteed to receive £1 per year in perpetuity with no credit-risk, companies are not guaranteed to make a profit every year or to survive in perpetuity. The level of profit in each future year can be forecast and predicted but not guaranteed. Companies can at times face unforeseen existential risks due to changes in their operating environment (see airlines in a pandemic) or technological disruption (see Kodak).

The discounted value of future profits is therefore subjectively assessed by the investor at the time of investment, and is done on the basis of modelling the size of the market (the TAM), the expected market share, the expected margins etc. Shares in loss-making companies can still have value if the investors can reasonably suspect that the losses will subside, and ultimately be replaced by substantial profits. This is the argument that underpins the valuations of high-growth negative cashflow startups - at some point in the future, the heavy investment in growth will reduce, and substantial profits will begin to accrue to the initial investors. This argument is born out in reality by some of the companies referenced previously (Facebook, Google), but it’s also worth noting that investors get it wrong from time-to-time (WeWork).

There are some that would say that a share in a company is worth only what other people are willing to pay for it, but this is wrong. The wonderful thing about shares in profitable companies is that they are valuable even if nobody is willing to buy them at any price, as they entitle the owner to a flow of money regardless.

As stated above, changes in the underlying discount rate can result in substantial changes to the value of an asset. Many investors use the yield on 10-year US treasury bonds as an approximate measure of an appropriate risk-free discount rate. This yield has been in secular decline for four decades, from a high of 15.84% in Sept-1981, to a low of 0.52% in Aug-2020. If you use this as the discount rate and apply the same maths as above, a perpetual £1 coupon shifts from being worth £6.31 in 1981, to being worth £192 in 2020. If this trend ever reverts, there will be significant implications for asset prices.

There are also times at which markets become detached from the fundamentals, where buyers buy assets at any price in the belief that they’ll be able to sell them for more later (regardless of the actual expected future profits of the company). The dot-com bubble[3] is the canonical recent example of this happening. The initial surge in asset prices was justified by reasonable beliefs about the future profits of companies leveraging new internet technologies, but was soon overtaken by irrational exuberance bidding the prices up to unsustainable levels.

Lastly, even in normal market conditions, there are buyers to whom particular companies have special acquisition value, as they are in a related industry and the buyer would benefit from acquiring the team/expertise/product or eliminating a competitor. In this case, the buyer may be willing to pay significantly more than the discounted value of future profits due to the subsequent value that can be created by merging the companies.

How do a company’s shares grow in value?

As discussed in the previous post, companies generate profits by creating value and capturing some proportion of it. A company’s shares grow in value when the company’s future profits grow in value, which happens when the company either creates more value, or captures a higher proportion of it.

If the proportion of value captured is held constant, then either developing new products and/or growing the customer base of existing products are effective ways of growing the value of a company’s shares. Conversely, if the product features and customer base are held constant, then increasing the proportion of the value captured also grows the value of a company’s shares, but is typically much harder to do in practice due to competition (and unhappy customers who have a shrinking consumer surplus).

For this reason, successful startups generally have a laser-like focus on product development and customer acquisition. Doing either in isolation is rarely enough.

Summary

£1 tomorrow is worth less than £1 today (assuming discount/interest rates are positive and non-zero).

A fixed stream of future income has a finite value today (under the same assumptions).

A share in a company represents a stream of future income, the precise value of which is unknowable, and is subjectively assessed by investors.

Changing discount rates can have a substantial effect on the current value of future income.

Companies can increase the value of their shares by either developing new products which create and capture new value, or by extending the reach of existing products and acquiring new customers.

References

Graham, B., & Dodd, D. (2008).

Security Analysis: Sixth Edition, Foreword by Warren Buffett

. McGraw-Hill Education.

Geometric series. In

Wikipedia

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Geometric series

Dot-com bubble. In

Wikipedia

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Dot-com bubble