Finance-Theory · Unit 4 · Video 1 · Interactive Practice

Leverage: How a 10% Fall Becomes a 200% Loss

IKey Formulas

Formula	Name	What it says
$L = \dfrac{\text{assets}}{\text{equity}} = \dfrac{A}{E}$	Leverage	Everything the price move acts on, divided by the money you actually put in
$r_{\text{equity}} = L \cdot r_{\text{asset}}$	The return rule	The asset's return multiplied onto your stake, in both directions
$\text{wipe-out fall} = \dfrac{1}{L}$	Wipe-out threshold	The fall that takes the whole equity, since $L \cdot \tfrac{1}{L} = 1$
$E_1 = A_0\,(1 + r_{\text{asset}}) - D$	Equity after the move	The lender's claim $D$ is fixed, so the entire change in $A$ lands on $E$

Key Insight: Leverage is not the debt-to-equity ratio: $L = 1 + D/E$ , because the price move acts on the whole house and not only on the borrowed part. And leverage is not risk by itself — it multiplies whatever the asset does, so a $20{:}1$ position is safe exactly as long as the price never falls by $1/L$ .

IIThe Down Payment Sets the Ratio

The down payment fixes the ratio, and with it how far the price can fall before your equity is gone.

IIIA Ten Percent Move, Two Positions

The lender's claim is fixed, so the whole price move lands on the money you put in.

IVWhy Twenty to One Was Not Frightening

Leverage only hurts if the price moves far enough, and the years before 2007 never came close.

💡 Financial engineering and the design of derivative securities widened the market on the way up, and the same instruments amplified the decline. None of the arithmetic above changed in 2008 — only the size of the moves it was applied to.

VQuiz Questions

Problem 1 · Leverage from a Down Payment

Given: an apartment priced at $600,000, bought with 25% down — find the leverage ratio.

✅ Correct! The price move acts on the whole apartment, so leverage counts all $600,000 of assets against the $150,000 you put in:

L = 4

❌ That is the debt-to-equity ratio.

450{,}000 / 150{,}000 = 3

leaves your own $150,000 out of the numerator. Leverage counts everything the price acts on:

L = 1 + D/E = 1 + 3 = 4

❌ You divided the assets by the loan.

600{,}000/450{,}000 = 1.33

compares the house with the bank's claim. Leverage divides the assets by your money.

❌ That is the down-payment fraction. Leverage is its reciprocal:

L = 1/0.25 = 4

❌ Not quite. Work out the equity first, then divide the full price by it.

Show solution

The down payment is the equity; the bank lends the rest:

E = 0.25 \times 600{,}000 = 150{,}000, \qquad D = 600{,}000 - 150{,}000 = 450{,}000

Leverage is total assets over equity:

L = \frac{A}{E} = \frac{600{,}000}{150{,}000} = 4 \quad (4{:}1)

Two shortcuts worth keeping. Since the equity is a fraction $f$ of the price, $L = 1/f$ , and here $1/0.25 = 4$ . And since $A = E + D$ ,

L = \frac{E + D}{E} = 1 + \frac{D}{E} = 1 + 3 = 4

so the debt-to-equity ratio is always exactly one less than the leverage ratio.

Problem 2 · The Return on Your Own Money

Given: an apartment priced at $400,000, bought with 10% down, after which prices fall 8% — find your return $r_{\text{equity}}$ .

✅ Correct!

L = 10

, so

r_{\text{equity}} = 10 \times (-8\%) = -80\%

— and directly,

-32{,}000 / 40{,}000 = -0.80

❌ That is the apartment's return, not yours. Your $40,000 carries the whole $400,000, so the asset's move arrives multiplied by

L = 10

❌ You divided the loss by the loan.

-32{,}000/360{,}000

prices the loss against the bank's money, and the bank takes none of it: a lender holds a fixed claim. Divide by your own $40,000.

❌ Not the whole stake — not yet. The equity is gone only at a fall of

1/L = 10\%

. An 8% fall destroys $32,000 of a $40,000 stake and leaves $8,000 standing.

❌ Not quite. Put the whole loss over the money you put in, or multiply the asset's return by the leverage.

Show solution

Step 1 — the position.

E = 0.10 \times 400{,}000 = 40{,}000, \qquad D = 360{,}000, \qquad L = \frac{400{,}000}{40{,}000} = 10

Step 2 — the loss. The fall applies to the whole apartment:

0.08 \times 400{,}000 = 32{,}000

and the bank loses none of it — it wants its $360,000 back with interest whatever the apartment is now worth. The entire $32,000 lands on you.

Step 3 — the return.

r_{\text{equity}} = \frac{-32{,}000}{40{,}000} = -0.80 = -80\%

which is the rule read directly: $10 \times (-8\%) = -80\%$ .

Check the equity that remains: the apartment is worth $0.92 \times 400{,}000 = 368{,}000$ against a debt of $360{,}000$ , so

E_1 = 368{,}000 - 360{,}000 = 8{,}000 = 40{,}000 \times (1 - 0.80)\ \checkmark

Problem 3 · Five Percent Down

Given: the video's $500,000 house bought with 5% down — a $25,000 stake against a $475,000 loan — after which prices fall 12%.

What fall in the price takes the entire equity?

What is your equity after the 12% fall?

✅ Correct! At

L = 20

the wipe-out fall is

1/20 = 5\%

, and a 12% fall leaves a house worth $440,000 against a $475,000 debt: equity of −$35,000.

❌ Check the threshold. The equity is gone when

L \cdot r_{\text{asset}} = -1

, that is at a fall of

1/L

. Here

L = 500{,}000/25{,}000 = 20

❌ Equity does not stop at zero. The loan is a fixed claim: you still owe $475,000 on a house worth $440,000, so the equity is negative.

❌ That is the change, not the level.

r_{\text{equity}} = 20 \times (-12\%) = -240\%

destroys

2.4 \times 25{,}000 = 60{,}000

, and the equity ends at

25{,}000 - 60{,}000 = -35{,}000

❌ The bank does not share the fall. A lender takes no downside risk, so the whole $60,000 swing lands on your $25,000 stake.

❌ Not quite. Value the house after the fall, then subtract the debt, which has not moved.

Show solution

The position.

E_0 = 0.05 \times 500{,}000 = 25{,}000, \qquad D = 475{,}000, \qquad L = \frac{500{,}000}{25{,}000} = 20

The wipe-out fall. The equity is exactly used up when the leveraged return is $-100\%$ :

L \cdot r_{\text{asset}} = -1 \iff r_{\text{asset}} = -\frac{1}{L} = -\frac{1}{20} = -5\%

After a 12% fall. The house is worth

A_1 = 0.88 \times 500{,}000 = 440{,}000

while the debt is still $475,000, so

E_1 = 440{,}000 - 475{,}000 = -35{,}000

Read as a return:

r_{\text{equity}} = 20 \times (-12\%) = -240\%, \qquad 25{,}000 \times (1 - 2.40) = -35{,}000\ \checkmark

Past the wipe-out fall the loss keeps going: you have lost the stake and owe $35,000 beyond it.

Problem 4 · Sixteen to One at a Firm

Given: a firm reporting a net leverage ratio of $16{:}1$ , whose asset base then falls 7% — find the loss as a percentage of the firm's capital.

✅ Correct!

16 \times (-7\%) = -112\%

: the capital is gone and 12% of it is owed on top, because 7% is already past the

1/16 = 6.25\%

fall that takes all of it.

❌ That is the assets' decline, not the capital's. The ratio says each dollar of capital supports $16 of assets, so the asset return arrives multiplied by 16.

❌ You divided by the leverage.

7\%/16

is the loss per dollar of assets spread over the capital. The rule multiplies:

r_{\text{equity}} = L \cdot r_{\text{asset}}

❌ That is the wipe-out fall, $1/16$ . It is the size of the asset move that costs

100\%

of the capital — not the loss from a 7% move, which is larger.

❌ Not quite. Multiply the asset's return by the leverage ratio.

Show solution

Apply the same rule that priced the house to the firm:

r_{\text{equity}} = L \cdot r_{\text{asset}} = 16 \times (-0.07) = -1.12 = -112\%

Check it against the threshold. The fall that takes the whole capital is

\frac{1}{L} = \frac{1}{16} = 6.25\%

and $7\% > 6.25\%$ , so a return past $-100\%$ is exactly what we should expect: the capital is exhausted and the remaining $0.75\%$ of the asset base, worth $12\%$ of the capital, has nowhere to land but the firm's creditors.

The same ratio run across a few sizes of move:

-5\% \Rightarrow -80\%, \qquad -7\% \Rightarrow -112\%, \qquad -10\% \Rightarrow -160\%

No capital base survives losses of that proportion for long — which is why one line on a page of financial highlights carried the whole story.

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