Track 8 · Arithmetic · lesson 8

Build the whole model yourself

15 min

Every model in this track has moved one thing while holding the rest still. That is how you learn what a lever does. It is not how a life works, because in a life all of the levers move at once and several of them are attached to each other.

This is the last model in the track and the first one where nothing is held still. Seven controls, both sides of the balance sheet, twenty years.

Net worth is one subtraction, and it has exactly two moving parts.

Assets grow at a rate and receive whatever is saved. Debts grow at a different rate and are reduced by whatever is repaid. Net worth is the first minus the second, every year.

That is the entire model. Everything difficult about it is that the two rates are usually different, and the difference decides which side is winning.

The number that decides everything

Before touching anything, look at the two rate controls. One is what assets earn. One is what debt charges. If the second is larger than the first, then every unit that goes to the asset side instead of the debt side is losing the gap, with certainty, every year.

That is not an argument for clearing every debt. It is the reason the comparison has to be made explicitly, because nothing on a statement makes it for you.

Net worth projection

Hypothetical model, not a forecast

Projects assets growing and debts being repaid side by side, and the difference between them.

It assumes

Assets grow at one constant rate and debt accrues at another, both unchanging for twenty years.
Debt interest is added first, then the year's repayment comes off.
Saving and debt repayment are separate flows funded from outside the model.
A debt that reaches zero stays there and takes no further payment.

It ignores

Income, tax and the fact that both flows compete for the same money
Illiquid assets that cannot be sold at the value shown
Any new borrowing during the period
Inflation — the whole projection is nominal

Every control moves. Find a set of numbers where the debt never clears and twenty years of saving still leaves net worth lower than it is today.

Assets today20,000

Debts today15,000

Added to assets each year6,000

Return on assets500%

Interest on debt800%

Debt repayment each year4,000

Years20

Net worth today: 5,000
Net worth at the end: 251,462
Assets at the end: 251,462
Debts at the end: 0
Year the debt clears: 5
Debt clears inside the period: 1

Net worth
Assets
Debts

Assets climbing while debts fall to zero, with net worth as the widening gap between the two.
Year	Net worth	Assets	Debts
0	5k	20k	15k
1	14.8k	27k	12.2k
2	25.2k	34.4k	9.2k
3	36.2k	42.1k	5.9k
4	47.8k	50.2k	2.4k
5	58.7k	58.7k	0
6	67.6k	67.6k	0
7	77k	77k	0
8	86.8k	86.8k	0
9	97.2k	97.2k	0
10	108k	108k	0
11	119k	119k	0
12	131k	131k	0
13	144k	144k	0
14	157k	157k	0
15	171k	171k	0
16	186k	186k	0
17	201k	201k	0
18	217k	217k	0
19	234k	234k	0
20	251k	251k	0

Not there yet — keep moving the controls.

Once the interest charged exceeds the repayment, the debt line stops being a line coming down and becomes a curve going up — the same curve as the asset side, pointed the other way. Every mechanism in this track works in both directions, and this is the chart where you can see both at once.

Read the omissions list on that model as carefully as the assumptions.

The largest one is that saving and debt repayment are funded from outside. In reality they come from the same pay packet, so every extra unit sent to one is a unit not sent to the other. The model lets you raise both to their maximum simultaneously, which no household can do.

A model that cannot represent the constraint you are actually under will always produce a more cheerful answer than the situation deserves.

Check

What does treating saving and debt repayment as separate, independent flows hide?

Nothing important — they really are separate transactionsThat they compete for the same money, so the real choice is how to split one amountThat debt repayment is more urgent than savingThat income exists at all

The decision the model cannot make

What would you do

There is 12,000 a year of surplus. There is also 40,000 of debt at a fixed 11%. The asset side is expected to return something in the region of 6% a year, which is an expectation rather than a promise, and over five years the range of plausible outcomes around it is wide.

Optimising for: Be certain of the result in five years, rather than have the best expected result.
You cannot know: What the asset side actually returns over these particular five years
Whether the 11% is genuinely fixed for the whole period
Whether the money would be needed back before the five years are up
Whether repaying the debt removes access to it as a reserve

The whole surplus goes to the debt until it is goneThe whole surplus goes to the asset sideSplit it evenly

Building the same thing on paperOptional

Everything in this model fits on one page, and it is worth writing out once by hand rather than trusting a slider.

Two columns. Assets at the top of one, debts at the top of the other. For each year: multiply the asset column by one plus its rate and add the saving; multiply the debt column by one plus its rate and subtract the repayment; write the difference in a third column.

Twenty rows. The value of doing it by hand once is that you find out how few inputs there are, and therefore how much of any projection is somebody's assumption rather than anybody's data.

Worth remembering

Net worth is one subtraction: assets, which grow and receive saving, minus debts, which grow and receive repayment.
When the rate charged on debt exceeds the rate earned on assets, sending money to the asset side loses the gap with certainty.
A debt whose interest charge exceeds its repayment grows on exactly the same curve as a compounding asset, pointed the other way.
Treating saving and debt repayment as independent flows hides that they compete for one surplus.
Two people can agree on every number and disagree completely, because they are optimising for different criteria.