Finance-Theory · Unit 8 · Video 6 · Interactive Practice

Correlation Moved the Money: Ratings, Winners, and the Government's Pricing Problem

IKey Formulas

Formula	Name	What it needs
$P_S = 1000\,(0.99 - 0.09\rho) = 990 - 90\rho$ $P_J = 1000\,(0.81 + 0.09\rho) = 810 + 90\rho$	Tranche prices in the two-bond example	Each bond pays $1,000 or nothing and defaults 10% of the time; $\rho$ is the correlation of the two defaults, so both default with probability $0.01 + 0.09\rho$ . Risk-free rate zero, so price = expected payoff
$P_S + P_J = 2 \times 0.9 \times 1000 = 1800$	The pool keeps its value	Each bond is worth $0.9 \times 1000 = 900$ whatever the correlation, so a change in $\rho$ moves value between the tranches and destroys none
$\text{result} = \text{sale price} - \text{price paid}$	Bailout or investment	Paying more than the estimated value hands the seller the difference; the final verdict waits for the price at which the paper is eventually sold
$\dfrac{31.1}{443.5} \approx 0.07$	TARP's net cost per dollar disbursed	The GAO's lifetime net cost ($31.1 billion, as of September 30, 2023) against the $443.5 billion disbursed, out of an authority cut from $700 billion to $475 billion

Key Insight: Inside the two-bond model, correlation rising from 0 to 1 moved $90 from each senior tranche to its junior tranche. In the real crisis the junior tranches of subprime deals were wiped out, and the largest gains went to investors who had bought protection against these securities, such as John Paulson, whose 2007 pay was reported at some $3–4 billion.

IIWould a Conservative Correlation Have Saved AAA?

A correlation higher than the one used in pricing takes value from senior holders and hands it to junior holders.

Assumed ρ (pricing) 0.30 Realized ρ 0.90

💡 The video gives no single realized figure, only that realized correlations went much, much higher than 25–30% once the housing market turned, particularly for subprime mortgages; the starting value of 0.90 is illustrative.

IIIA Decline Four Times the Worst

In the two decades before 2007 the national house-price index never fell more than 2.2% in a year; how far had it fallen by September 2008?

Your guess, 12 months to Sep 2008 (%) −3.0

IVBailout or Investment: Two Prices

Whether a purchase is a bailout or an investment depends on two prices: the price paid and the price sold.

Price paid 100 Eventual sale price 60

💡 An 80% equity stake adds the company's upside if the paper recovers, which can make the deal outright profitable; the video describes this as roughly the deal struck with AIG and close to the terms Warren Buffett negotiated with Goldman Sachs.

VQuiz Questions

Problem 1 · Concentration That Lasted

Given: As of December 31, 2023, ten firms were registered with the SEC as nationally recognized statistical rating organizations, and Fitch, Moody's and S&P together accounted for 94% of all outstanding credit ratings. In 2007 the same three accounted for 98.8%. Find the share of outstanding ratings held by the other seven registered firms together at the end of 2023, and how many times larger that is than the share held by all firms other than the three in 2007.

Other seven firms' share, end of 2023?

How many times the 2007 share?

✅ Correct! The rest of the industry held

100\% - 94\% = 6\%

in 2023 against

100\% - 98.8\% = 1.2\%

in 2007, and

6 / 1.2 = 5

. A fivefold rise, yet three firms still held 94% of all outstanding ratings: the concentration has lasted.

❌ Check the share. The other firms hold whatever the three do not: 100% minus the three firms' share in 2023.

❌ That is the change, not the share.

98.8 - 94 = 4.8

is how many percentage points the three firms' share fell between 2007 and 2023. The other seven hold

100\% - 94\% = 6\%

❌ Firms are not ratings. Seven of the ten registered firms is 70% of the firms, but the question asks for their share of outstanding ratings:

100\% - 94\% = 6\%

❌ Wrong year.

100\% - 98.8\% = 1.2\%

is the other firms' share in 2007. At the end of 2023 it was

100\% - 94\% = 6\%

❌ Check the multiple. Divide the other firms' 2023 share by their 2007 share.

❌ A difference, not a multiple.

6 - 1.2 = 4.8

is how many percentage points the other firms' share rose. "How many times" asks for a ratio:

6 / 1.2 = 5

❌ Upside down.

1.2 / 6 = 0.2

divides the 2007 share by the 2023 share. The 2023 share is the larger one:

6 / 1.2 = 5

❌ Wrong firms.

98.8 / 94 \approx 1.05

compares the three firms' own shares. The question asks about everyone else:

6 / 1.2 = 5

Show solution

The other firms hold whatever share the three do not:

2023:\ 100\% - 94\% = 6\% \qquad 2007:\ 100\% - 98.8\% = 1.2\%

Their share grew by a factor of

\frac{6}{1.2} = 5.

Even after a fivefold rise, seven registered competitors together held only 6% of outstanding ratings, while Fitch, Moody's and S&P held 94%. As the video puts it, starting a new rating agency remains extremely difficult, because regulators impose standards that are close to impossible for a startup to meet.

Problem 2 · TARP's Cost per Dollar

Given: TARP's authority was cut from $700 billion to $475 billion, $443.5 billion was ultimately disbursed, and the GAO put the lifetime net cost at $31.1 billion as of September 30, 2023. What was the net cost for every dollar disbursed?

✅ Correct!

31.1 / 443.5 \approx 0.070

: about seven cents of net cost for every dollar disbursed.

❌ Wrong denominator.

31.1 / 700 \approx 0.044

divides by the original authority. The question asks per dollar actually disbursed: 443.5 billion.

❌ Wrong denominator.

31.1 / 475 \approx 0.065

divides by the reduced authority, not by the 443.5 billion that was actually disbursed.

❌ Not the cost.

443.5 / 700 \approx 0.63

is the share of the original authority that was disbursed; the net cost of 31.1 billion does not enter it.

Show solution

Divide the lifetime net cost by the amount disbursed (both in billions of dollars):

\frac{31.1}{443.5} \approx 0.0701

That is about 7 cents of net cost per dollar disbursed. The authority figures (700 and 475 billion) are ceilings on what could be spent, not what was spent.

Problem 3 · A Margin Is Not a Result

Given: The government's best estimate of an asset's value is $60 per unit. It pays $50 per unit. Years later the paper sells for $45 per unit. Find the margin at purchase against the estimate, and the result per unit on the paper once it is sold.

Margin at purchase?

Result per unit at sale?

✅ Correct! The margin

60 - 50 = +10

held only if the estimate did; the paper sold for 45, so the result is

45 - 50 = -5

per unit. Whether the whole deal lost money also depends on anything else the government received, such as an equity stake.

❌ Check the margin. The margin compares the estimated value with the price paid: estimate minus price.

❌ Sign flipped.

50 - 60 = -10

subtracts the wrong way. Paying 50 for something estimated at 60 leaves a cushion of

60 - 50 = +10

❌ Wrong price.

60 - 45 = 15

uses the eventual sale price. At purchase the government compares its estimate with the 50 it actually pays.

❌ Wrong scenario.

60 - 100 = -40

is the video's other column, where the government pays 100. Here it pays 50.

❌ Check the result. Once the paper is sold, the result is the sale price minus the price paid.

❌ The estimate did not hold. The +10 margin assumed the paper was worth 60. It sold for 45, so the result is

45 - 50 = -5

❌ Wrong comparison.

45 - 60 = -15

measures the sale against the estimate, but the government paid 50, not 60:

45 - 50 = -5

❌ Sign flipped. Buying at 50 and selling at 45 loses money:

45 - 50 = -5

, not

+5

Show solution

At purchase, against the best estimate:

\text{estimate} - \text{price paid} = 60 - 50 = +10

a $10 margin per unit, if the estimate holds.

At sale, the estimate no longer matters:

\text{sale price} - \text{price paid} = 45 - 50 = -5

a $5 loss per unit on the paper. The margin at purchase was protection against an estimate that turned out too high by 15, and it covered only 10 of it. This is why the video says the verdict depends on two prices: the price paid and the price at which the paper is eventually sold.

Problem 4 · The Conservative Analyst

Given: In the two-bond example with default correlation $\rho$ , the senior tranche is worth $990 - 90\rho$ dollars and the junior tranche $810 + 90\rho$ dollars. An analyst prices the senior tranche at a deliberately conservative $\rho = 0.30$ , and the correlation actually realized is $\rho = 0.90$ . How much does each senior tranche lose against the price paid, and where does that value go?

✅ Correct! The conservative assumption cut the loss from 81 to 54, but did not remove it:

963 - 909 = 54

moves from the senior to the junior holders.

❌ Not quite. 81 is the loss for an analyst who assumed zero correlation:

990 - 909 = 81

. This analyst priced at

\rho = 0.30

and paid

990 - 27 = 963

❌ The pool keeps its value. The amount is right, but

P_S + P_J = 1800

at every

\rho

: the junior tranche rises from

810 + 27 = 837

810 + 81 = 891

, a gain of exactly 54.

❌ Not quite. 90 is the full move from

\rho = 0

\rho = 1

. Here the move is from 0.30 to 0.90:

90 \times (0.90 - 0.30) = 54

Show solution

Price paid at the assumed correlation:

P_S(0.30) = 990 - 90 \times 0.30 = 990 - 27 = 963

Value at the realized correlation:

P_S(0.90) = 990 - 90 \times 0.90 = 990 - 81 = 909

Loss per senior tranche: $963 - 909 = 54$ , which equals $90 \times (0.90 - 0.30)$ .

The junior tranche goes from $810 + 27 = 837$ to $810 + 81 = 891$ , a gain of 54. The pool stays at $909 + 891 = 1800$ : the conservative assumption shrank the transfer but, as the video says, would not have avoided the dislocation.

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