The unofficial, and actual, conclusion.

So, let’s get it over with.

Again, take an introductory economics course and you’ll be sure to hopefully happen upon the concept of the money multiplier.

The idea behind this one is as such:

banks can lend a portion of their deposits. The rest must be kept in reserve. Making the simplifying assumptions: that the reserve ratio remains constant; that all available funds are lended; and, that all money ultimately flows back to banks; that the process of new money creation can be modelled with a geometric series.

Imagine that regulators require banks to hold 10% of demand deposits in reserve. They can lend out 90%.

If $100 is deposited with a bank, they can lend out $90.

Now for the assumptions that all available money is lended, and that the money ultimately flows back to the bank:

The $90 is then deposited with the bank.

And the assumptions that the reserve ratio remains constant, and that all money available to lend will be so:

Required to keep 10% in reserve, the bank lends $81.

Which is circulated, and ultimately returned to the bank.

$72.90 is then offered to a keen borrower …

and the process continues.

This can be modelled with a geometric series.

First term: the amount initially deposited with a bank

Common ratio: the amount a bank can lend-out: (1 - Reserve Ratio)

Such that:

|(1 - Reserve Ratio)| < 1

The series has a sum to infinity:

For total circulation:

This gives us total circulation.

For ‘new money created’ / added to circulation the initial ‘already-existing’ money in circulation must be subtracted:

For the example with initial deposit $100 and reserve ratio 0.1:

This canon harbours the following basic principle: injections into aggregate spending produce an impact upon consumption which is ultimately larger than the initial input.

That is to say: money injected will be spent; its recipients will then spend it; as will the subsequent recipients; and so on, and so on, and so on.

Imagine that 90% of income is spent, and $100 of new income is introduced to the economy:

Person A spends $90 buying from Person B. Person B then spends $81 buying from Person C. Person C will spend $72.90 buying from Person D; who will in turn spend $65.61 buying from Person E.

Don’t worry about Person E, there is nothing wrong with him - he’ll go ahead and buy from F, but I presume that you get the point by now, and would rather avoid chasing this trail all the way through the alphabet and beyond.

Point being: total spending produced by the $100 injection amounts to more than $100.

Visually:

Which is just a geometric series:

With - a first term, being the injection (here 100)
And - a common ratio, being the portion spent on consumption / re-circulated (here 9/10)

The portion being circulated is designated the marginal propensity to consume (MPC): the portion of an increase in income which is spent on consumption.

The residual portion, that amount not spent on consumption, 1-MPC, is designated the marginal propensity to withdraw [sometimes save] (MPW).

Such that we make the simplifying assumption of a constant and homogenous MPC, we can take a sum to infinity:

Feeding through our inputs:

The spending multiplier is the amount by which we multiply the initial injection to get at the ultimate increase in consumption precipitant.

Note also that the common ratio, MPC, is assumed to be between 0 and 1 for the whole economy.

Individual households may borrow to spend above increases in income, however, it is assumed that the economy as a whole will ‘live within its means’.

Wrapping up with the example:

Thankless - sometimes

Perfectly accurate - impossible

Will it rain? How hot will things get? Are there sunny days ahead ?

Try harder to improve precision - risk falling further from the truth

And I’m not even talking about the weather … it’s cash flows again

Valuation, of companies, or those prospective projects thereof, often comes in the form of a DCF

That’s not: Dissonance, Conjecture, Fancy though all three may be involved,

but Discounted Cash Flows.

And, believe it or not, the process involves … discounting cash flows

So, let’s say that you’ve forecasted the first five years of a company’s cash flows:

Halcyon days over? Teething troubles done? Stable maturity set to persist?

All depends upon the company in question - and your inputs of course

But if we are expecting operations to continue, we had better find some way to address all those days rolling-off into the far future

Now, the concept of a growing perpetuity could come in useful.

Predicting the distant future with pinpoint accuracy is tricky, and sometimes it might just be appropriate to assume that in the long-run a company will grow at some rate, comparable to that of the economy as a whole.

Close-enough, slower, or thereabouts, because few companies can truly outpace the economy indefinitely.

But also, because: where r_d < g , the formula for a growing perpetuity breaks.

Taking the expected long-run rate of growth for the economy (upon which we expect the company to converge) as a placer for g, and an estimated long-run cost-of-capital as a stand-in for the discount rate r_d, we get something that looks a bit like this:

$700 is Cash flow C; the expected cash flow one year out into the future (after our forecast 5) - this is what we’ll grow-out. The long-run discount rate is 15%, and the expected continual growth-rate 10%.

Since the formula gives us the present value during the period before the sequence begins (in previous examples time 0, where the first cash flow arrived at time 1; here at time 5 since the first long-run forecast cash flow arrives at time 6), we then have to discount again (here from time 5 to time 0).

In this case, that would involve the following:

Where the short-run discount rate for years 1-5 was set to 20%..

All combined it may look something like this:

The terminal value might well account for the bulk of company value - so, be wary, because intentional meddling small input tweaks could alter a forecast entirely.

I’m talking drought to blizzard at the click of a key.

Where economic optimism is heightened only slightly:

Increasing the estimated long-run growth-rate of the economy from 10%-11% increases firm value by almost 21%.

Where she’ll stop, nobody knows

Imagine the following stream of cash flows:

Where you are to receive “$C” in one year. This amount scaled-up by some growth rate, g, two years from today. At the end of the third year, the second year amount grown-out by that same rate of growth.

etc - etc - etc - forever

Holding the time-value of money closely in mind, the present value of this cash flow stream can be enumerated accordingly:

Where C is the first cash flow, and is the r_d discount rate

Notice something?

Such that |the common ratio| < 1

Each successive term gets smaller

That is, so long as g < r_d

We can take the sum to infinity

Padding out the formula for the sum to infinity of a geometric series:

We get:

The present value of a growing perpetuity with first payment C and growth-rate g:

Well - now we know

That’s correct - ceaseless cash flows … but not quite the money printing machine it first seems

Imagine the following stream of cash flows placed on a timeline like so:

Where you are to receive “$C” at the end of every year … forever

With the time-value of money in mind, we can express the present value of this stream of cash flows as such:

Where C is the cash flow, and is the r_d discount rate

Notice something?

Such that |the common ratio| < 1

Each successive term gets smaller

And, so, we can take the sum to infinity

Populating the formula for the sum to infinity of a geometric series:

We get:

The present value of a perpetuity with cash payment C:

Unlike that stream of cash flows, this one won’t go on forever: it ends about …

…

now

Retirement:

You love cheese.

Indeed, you love cheese so dearly that you are prepared determined to spend $1000 on the moulding, yellow, stink-bricks; each and every year.

You are about to retire, and, quite naturally, plan to set aside a Gouda [not at all pronounced like this] stack of money to fund your murine predilection for the next decade.

Just one problem: inflation - and isn’t she a monster; like a plump ginger cat with its sharpened claws right down your rat-hole, she is cruel. At 10%/annum, she’s more than vexing; and if you don’t prepare, you’re going to get hurt.

Fortunately you' still have access to that bank account which pays 20% interest at the end of each year.

Now then - just how much to save …

You’ll need a stream of cash flows which looks something like this:

Such that you have $1000 to spend on cheese this year; $1100 to spend next year since prices will have risen 10%; $1210 to spend in the year thereafter, prices having risen by 10% again: etc, etc, etc.

Since your account will return 20% interest at the end of each period, to have $1100 available for a cheesy withdrawl in one year, you’d need to deposit $916.67 today. To take out the $1210 in two years, you’d be required to store $840.28 today.

The amount you’ll need today can be expressed as:

Such that $1000 is required for this year; $1210 for next year; and $2357.95 for the tenth year.

Since the first $1000 is needed for this year, the amount you’ll need to deposit to make cheese purchases during each of the subsequent 9 can be expressed as:

Equivalent to:

You could crunch the cheddar numbers one-by-one …

Or wedge them into the formula for a growing annuity:

Like so:

Adding-on the $1000 for this year’s expenditure, you’d need …

well …

a lot of money for cheese!

visual appendix: in case you are interested in the mechanics

Bond Pricing:

Imagine you are browsing ye old debt markets and see a bond (surprise surprise).

The bond pays a coupon of $100 at the end of each year for the next fifty years. It has a face value of $1000.

This bond doesn’t play by the rules of the real world and so is completely riskless.

You don’t much like risk - which is great for you.

You do however, have access to a bank account which pays annual interest of 20% on deposited funds; all of which is paid at the end of the year.

-

Aside: please tell me where I can get myself one of those accounts!

-

Taking 20% as your risk-free discount rate, you can establish the present value of the bond.

Visually:

Manually, this works out to something like:

Could crank it out one-by-one, but … not much fun …

However, such that the coupons take the form of a simple, fifty-year annuity, we can take a shortcut:

Discounting the coupons as an annuity, and the face value as a lump-sum single cash-flow:

stress button

This subscribe button is best-avoided if you are seeking peace - I am not a bond, I do not offer regular coupons; only the possibility of irregularly-delivered, irritating, in-mail.

Prepare for a growing sense of déjà vu

Imagine the following stream of cash flows placed on a timeline:

Where the amount you receive at each interval will grow by some constant growth rate.

Denoted at each stage by multiplying the previous cash receipt by (1+g).

Where g is the growth rate.

You are to receive “$C” one year from now. This amount multiplied by (1+g) two years from now. In three years, you’ll receive the amount paid in two years multiplied by (1+g).

Discounting each payment to present-day, we can express the present value of this cash flow stream as:

Notice something?

Walloping these into the formula for the sum to n terms of a geometric series:

We get:

The present value of a growing annuity stream of n periods, with first cash payment C, and growth-rate g:

Even if you are now completely fed-up with me (I certainly am), I’m sure that we are both agreed upon the present value of a break.

Very textbook … but perhaps you can see where this one is going

Imagine the following stream of cash flows, placed on a timeline as such:

Where you are to receive “$C” at the end of every year, for the next five years.

Meaning a payment made in one year from today; the next in two years from today, and so on, and so forth.

Remembering the time-value of money, we can express the present value of this stream of cash flows as:

Where C is the cash flow, and r_d is the discount rate

For n terms, we’d have a stream of cash flows:

The present value of which could be expressed as:

Notice something?

Whacking these into the formula for the sum to n terms of a geometric series:

We get:

The present value of an annuity stream of n periods with cash payment C:

The picture gets bigger in the next one: on growing annuities …

Where 0 < |r| < 1, each successive term will get smaller. The amount we add on at each stage therefore gets smaller.

Visually, one might picture it like this:

Or this:

Where we never quite reach the line or second box, but can get infinitesimally close. Taking limits we are essentially there.

We can crank out a formula for the sum to infinity:

Arrived at like this: (sort of)

remembering the formula for the sum to n terms:

when n is infinity:

taking limits, r^∞ will tend toward 0 since:

where 0 < |r| < 1 , r will shrink each time it is multiplied by itself; here occurring an infinite number of times (shrinking continuously toward zero).

tying things up:

The maths here is not quite ‘proper’, but I hope it gets the idea across (conceptually at least)

Tested for the geometric series with first term 1 and common ratio ½ :

I don’t want this to drag on forever - not sure if your patience has a sum to infinity, but I fear it is getting smaller with each installment …

Taking the series with first term 2 and common ratio 2:

2 + 4 + 6 + 8 + 16 + …

We can generalise for the sum to n terms:

Here is how we get there:

The sum to n terms can be written as:

Multiplying through by r:

Taking the second from the first:

Factorising:

Dividing through by (1-r):

Taking the sum of the first five terms for the geometric series with first term 2 and common ratio 2:

Short episode, not much dialogue, no substance, rushed … 2/10.

Not like to win any awards, but why not stick around for episode 3? - it’ll be out soon enough - and hey, things might just get better (only so much further downhill this rock can roll).

Take the form:

Essentially, they function very similarly to those basic arithmetic sequences which were, during childhood lessons of math, drilled intensely, horrifically, and oh-so-unrelentingly, through the impervious breadth of our skulls.

Yes, dare I say it: 2, 4, 6, 8 is back with a vengeance. Except this time it’s his bigger, angrier, cousin: 2, 4, 8, 16.

That is to say, we still have a first term (a) chased by a trail of descendants, though now each successive term is generated, not with the addition or subtraction of some fixed amount, but via multiplication, by a common ratio (r).

For example:

2, 4, 8, 16 is a geometric sequence with first term 2 and common ratio 2.

We begin with 2, and multiply by 2 to generate each successive term.

An expression for the n^th term can be generalised:

where:

n represents position

a represents the first term

r represents the common ratio

Note: r can be fractional and, or, negative

Hardly an explosive cliffhanger, but the series continues in part two

Danger Button

Warning: do not press unless you want me to haunt your inbox

I’m certain that you’ve heard it.

Hasty refresher inbound.

Unless of course you are already unseasonably familiar; in which case it is probably time to skip ahead; only so many hours in a day.

… frighteningly few when fiendish finance theory steals a proverbial bite from your cash wad at each circular passage of the clock.

Let’s say that you have a $110 rental bill to pay in one year.

How much do you need today? $0? $110?

…

Wrong - right now, you have access to a riskless bank account paying 10% annual interest. Shove $100 in there, let it sit, and you are good to go.

How about this: you lend me $100 today, and I’ll pay you back $105 in a year.

That’s the round hundred in full, plus 5% interest … for being such a wonderful person.

Good deal?

…

No.

Beside putting your faith in an internet weirdo, you’d be forgoing $5.

Had only you stuffed the one-double-o in that magical account, and you’d be printing off $110 instead.

Not quite cruising off into the sunset money; but it might just cut a Happy Meal.

[with inflation as it stands, even here I can make no promises]

Point is, money has a time value.

It can be invested in competing projects.

That $105 you may have expected to receive is probably worth no more than $95.46 to you now. Were you to deposit $95.46 today, then sit on your hands for a year, a ripe $105 would be primed for withdrawal - and so you probably shouldn’t be prepared to part with any more than $95.46 today for the promised delivery of $105 upon the passage of twelve months.

When you think time value of money, think timeline.

The rate by which we move cash flows about is the discount rate. This represents the rate of return available on investments of equivalent risk.

Assuming the lending and deposit were both risk free, the 10% interest rate might suffice.

Money is compounded forward in time.

Money is discounted backward in time.

Note: the discount rate used by individuals and organisations is likely to reflect an expected, or required, rate of return on/for a project.

this rate may change over time - i.e. 10% for one year, 20% for two years, 30% for three.

And, that’s it, done, all out of time …