Finance-Theory · Unit 2 · Video 2 · Interactive Practice

Who Pays for a Government Guarantee? GSEs, Shareholders, and Counterparties

IKey Formulas

Relationship	Name	What it settles
$\text{remainder} = 1 - d$	What a fall of $d$ leaves	An 83% session leaves $0.17 of each dollar of the previous close; a 90% to 95% fall from the peak leaves $0.10 to $0.05 of each peak dollar
$1 - d_{\text{total}} = (1 - d_{\text{before}})(1 - d_{\text{session}})$	Nested windows chain	The session is the last segment of the two-year window, so its 0.17 is already a factor of the peak figure; multiplying it in again counts the session twice
$\text{equity loss} = \min(L, E)$ ; $\ \text{shortfall} = \max(0,\ L - E)$	The order losses land (loss $L$ , equity cushion $E$ )	Equity absorbs the first losses; only the shortfall reaches the paper, or, once the paper is guaranteed, the taxpayer

Key Insight: A guarantee does not make a cost disappear; it decides who bears it. The equity absorbed the loss first and went to nearly nothing, the guaranteed paper was made whole, and the residual cost went to taxpayers, spread across generations in a way nobody has yet allocated.

IITwo Baselines, One Collapse

The one-session fall is the last segment of the two-year fall, so its remainder is already inside the peak figure.

Previous close, per peak dollar $0.50

IIIWhere Each Group Stood

Where each group stood in the capital structure, not what it deserved, decided who absorbed the loss.

Loss on the mortgages $25 Treasury guarantee

💡 The waterfall settles who bears a loss, not whether the rescue was worth it: weighing the borrower still in the house against the pension fund that lost its capital and the child who has not yet paid takes a valuation framework, which the rest of this lecture builds.

IVQuiz Questions

Problem 1 · Measured From the Peak

Given: over the preceding couple of years the common stock fell on the order of 90% to 95% from its peak. What was left of each peak dollar?

✅ Correct!

1 - 0.90 = 0.10

and

1 - 0.95 = 0.05

: a peak dollar had become somewhere between ten cents and five cents.

❌ That is the fall, not what is left. A 90% decline takes 90 cents of each peak dollar and leaves

1 - 0.90 = 0.10

❌ Different baseline. $0.17 is what the single session left of each dollar of the previous close,

1 - 0.83

. Measured from the peak, the remainder is

1 - d

with

d

between 0.90 and 0.95.

❌ That multiplies the session in a second time.

0.05 \times 0.17

and

0.10 \times 0.17

apply the session's 0.17 on top of a peak figure that already runs through that session. The remainder is simply

1 - d

Show solution

A fall of $d$ leaves $1 - d$ of every dollar at the start of the window. Measured from the peak:

1 - 0.90 = 0.10, \qquad 1 - 0.95 = 0.05

so each peak dollar had become somewhere between $0.10 and $0.05, the range in the lecture.

The $0.17 belongs to a different starting point: $1 - 0.83 = 0.17$ is what was left of each dollar of the previous day's close. Both describe the same stock on the same day, and either way shareholders' equity had gone to nearly nothing.

Problem 2 · Two Figures, One Close

Given: an analyst reads both figures in the coverage after today's close and reports: “The stock is down 90% from its peak and then another 83% today, so a peak dollar is now worth $0.10 × 0.17 = $0.017, a 98.3% total fall.” What is wrong with the report?

✅ Correct! Both figures end at today's close. The 0.17 is the last link of the chain that produces the 0.10, so multiplying them together applies today's session twice.

❌ Remainders chain only across windows that follow one another. These two windows end at the same close, with the session sitting inside the two-year window, so its 0.17 is already a factor of the 0.10.

❌ Percentages of different baselines cannot be added. The 83% is a share of yesterday's close and the 90% a share of the peak; and no holder of equity loses more than everything.

❌ A product of two falls measures nothing. It also lands below the 90% quoted from the peak, a figure that already covers today's session.

Show solution

Write the three prices per peak dollar: $V_{\text{peak}} = 1$ , $V_{\text{prev}}$ at yesterday's close, and $V_{\text{close}}$ today. The two quoted figures are

\frac{V_{\text{close}}}{V_{\text{peak}}} = 0.10, \qquad \frac{V_{\text{close}}}{V_{\text{prev}}} = 0.17.

Both end at the same price, $V_{\text{close}}$ . The session is the last segment of the two-year window, so the chain runs

\frac{V_{\text{close}}}{V_{\text{peak}}} = \frac{V_{\text{prev}}}{V_{\text{peak}}} \times \frac{V_{\text{close}}}{V_{\text{prev}}}

and the 0.17 is already one of the factors inside the 0.10. Multiplying again, $0.10 \times 0.17 = 0.017$ , describes a second 83% session that never happened.

A peak dollar is worth $0.10: nearly nothing, measured from either baseline.

Problem 3 · Back to Yesterday's Close

Given: suppose that at today's close the stock stands 93.2% below its peak, a figure inside the lecture's 90% to 95% range, and that today's session alone was the 83% fall from yesterday's close quoted from the floor. How far below the peak was the stock at yesterday's close?

✅ Correct! Yesterday's close was

0.068 / 0.17 = 0.40

of a peak dollar, 60% below the peak; the session then took 83% of that $0.40.

❌ That divides by the fall.

0.068 / 0.83 \approx 0.082

undoes a session that took only 17%. Going back through a session means dividing by what it left, 0.17.

❌ That subtracts percentages of two different baselines.

93.2 - 83 = 10.2

treats the 83% as a share of the peak; it is a share of yesterday's close.

❌ That runs the session forward a second time.

0.068 \times 0.17 = 0.01156

is a price after another 83% fall. Going back to yesterday's close means dividing by 0.17.

Show solution

Step 1: today's close per peak dollar. $1 - 0.932 = 0.068$ .

Step 2: undo the session. The session left 0.17 of yesterday's close, so

V_{\text{prev}} \times 0.17 = 0.068 \quad\Rightarrow\quad V_{\text{prev}} = \frac{0.068}{0.17} = 0.40.

Step 3: read it from the peak. Yesterday's close was $0.40 per peak dollar, so the stock stood $1 - 0.40 = 0.60$ , or 60%, below the peak.

Check the chain: $0.40 \times 0.17 = 0.068$ ✓. Before the session the stock had already lost 60% over the preceding stretch; the session then took 83% of what was left.

Problem 4 · Who Bears a Loss of $20

Given: an illustrative issuer holds $100 of mortgages, funded by $92 of paper and $8 of common equity. The mortgages lose $20 of value. A government guarantee stands behind the paper; no part of it is pointed at the shares.

What do the shareholders keep?

How much do taxpayers pay?

✅ Correct! The equity absorbs the first $8 and keeps nothing; the $12 shortfall is paid by taxpayers, so the paper is made whole at $92.

❌ The guarantee stands behind the paper, not the shares. Equity sits in the bottom tier and absorbs losses first.

❌ That shares the loss pro rata, 20% of every claim. Losses land in order: the equity takes the first $8 before the paper loses anything.

❌ Limited liability puts the floor at zero. Shareholders can lose their whole $8 but no more; the $12 beyond it has to come from someone else.

❌ That charges taxpayers for the whole loss. The equity cushion takes the first $8; only the shortfall

20 - 8 = 12

reaches the guarantee.

❌ Somebody still pays. The printing presses settle how the promise gets honored, not who bears the cost: the paper is owed $92 and the mortgages cover only $80.

❌ That is the paper's pro-rata share, 20% of $92. Losses do not land pro rata: the equity goes first, leaving a shortfall of $12.

Show solution

Losses land from the bottom of the capital structure, with $L = 20$ and $E = 8$ .

Step 1: equity absorbs first. $\text{equity loss} = \min(L, E) = \min(20, 8) = 8$ , the shareholders' whole stake. They keep $0, and limited liability means they owe nothing beyond it.

Step 2: the shortfall. $\text{shortfall} = \max(0,\ L - E) = 20 - 8 = 12$ . The mortgages are now worth $100 - 20 = 80$ , which covers only 80 of the 92 owed on the paper.

Step 3: the guarantee. The paper is made whole at $92, so the $12 gap is paid by taxpayers. Check: $80 + 12 = 92$ ✓.

Without the guarantee, the paper would have received $80 and its holders would have taken the $12 loss themselves. The guarantee did not remove the cost; it moved it from the counterparties to the taxpayers.

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