Finance-Theory · Unit 8 · Video 4 · Interactive Practice

Tranching: How Two Risky Bonds Become a Triple-A Claim and Toxic Waste

IKey Formulas

FormulaNameWhat it needs
P=(1−p) F+p⋅0,y=FP−1P = (1-p)\,F + p \cdot 0, \qquad y = \dfrac{F}{P} - 1Price and promised yield of one risky bondFace value FF and default probability pp. The risk-free rate is set to zero, so a price is just an expected payoff: $900 and 11.1% for FF = $1,000 at p=10%p = 10\%
Pr⁡(2F)=(1−p)2,Pr⁡(F)=2p(1−p),Pr⁡(0)=p2\Pr(2F) = (1-p)^2, \quad \Pr(F) = 2p(1-p), \quad \Pr(0) = p^2The pool's three outcomesTwo bonds that default independently: 81%, 18% and 1% in the video
Pr⁡(senior defaults)=p2,Pr⁡(junior defaults)=1−(1−p)2\Pr(\text{senior defaults}) = p^2, \quad \Pr(\text{junior defaults}) = 1 - (1-p)^2Tranche default probabilitiesSenior is paid first from the pool, junior only from what is left: 1% and 19%
PS=(1−p2) F,PJ=(1−p)2 F,PS+PJ=2(1−p) FP_S = (1-p^2)\,F, \quad P_J = (1-p)^2\,F, \quad P_S + P_J = 2(1-p)\,FTranche pricesThe same expected-payoff rule: $990 + $810 = $1,800 = 2 × $900

Key Insight: Nothing is hidden and nothing is added. Priority alone moves default risk from the senior claim to the junior claim, and $90 of price moves with it; the total stays at $1,800. Every number rests on one assumption: the two bonds default independently.

IIWho Gets Paid in Each Outcome

The senior claim takes the pool's first $1,000; the junior claim gets only what is left.

IIIDefault Risk, Sorted by Priority

At every bond default probability pp, the senior claim defaults less often than one bond and the junior claim more often.

💡 Every curve assumes the two bonds default independently, like the video's two separate coins; the next video asks what changes when they do not. As in the video, the claim's default probability is read as a whole-life figure and set against published 10- and 20-year cumulative rates, so the yardsticks are a rough guide to a rating, not a rating.

IVValue Reallocated, Not Created

Priced at expected payoff, the two tranches always sum to the two bonds' value; priority only moves price between them.

VQuiz Questions

Problem 1 · The Senior Claim at 20%

Given: Two bonds, each with face value $1,000, each default with probability 20% and pay nothing on default, independently of each other. They are pooled and split into a senior and a junior claim of $1,000 face value each. With the risk-free rate at zero, find the senior claim's default probability and price.

✅ Correct! The senior claim misses its payment only when both bonds fail: 0.2×0.2=0.040.2 \times 0.2 = 0.04. It is paid in the other 96% of cases, so it sells for $960.
❌ Not quite. Those are the numbers for one original bond. The senior claim is paid first, so one default alone still leaves it the pool's $1,000; it fails only when both bonds default.
❌ Not quite. Those are the junior claim's numbers: it defaults when either bond fails, 1−0.82=0.361 - 0.8^2 = 0.36. The senior claim is paid first and fails only when both bonds do.
❌ Not quite. The default probability is right, but the price multiplies the face value by the probability of being paid, 96%, not by the probability of default. Default contributes 4% × $0.
Show solution

The senior claim takes the first $1,000 the pool receives, so it is short only when the pool receives nothing, which happens when both bonds default. With independent defaults:

Pr⁡(senior defaults)=p2=0.2×0.2=0.04\Pr(\text{senior defaults}) = p^2 = 0.2 \times 0.2 = 0.04

Price is the expected payoff (risk-free rate zero), in dollars:

PS=0.96×1000+0.04×0=960P_S = 0.96 \times 1000 + 0.04 \times 0 = 960

The senior claim defaults 4% of the time and sells for $960, against 20% and $800 for an original bond.

Problem 2 · The Junior Claim at 6%

Given: The same structure, but each bond now defaults with probability 6%, independently. Find the junior claim's default probability and its price.

Junior default probability?

Junior price?

✅ Correct! The junior claim is paid only when neither bond defaults, 0.942=0.88360.94^2 = 0.8836 of the time, so it defaults 11.64% of the time and sells for $883.60.
❌ Not quite. 2×0.06×0.94=0.11282 \times 0.06 \times 0.94 = 0.1128 counts only the cases where exactly one bond fails. The junior claim also gets nothing when both fail: add 0.062=0.00360.06^2 = 0.0036.
❌ Not quite. Adding 6% + 6% counts the both-default case twice. The chance that at least one bond fails is 1−0.942=0.11641 - 0.94^2 = 0.1164.
❌ Not quite. 0.062=0.00360.06^2 = 0.0036 is the chance both bonds fail, which is when the senior claim defaults. The junior claim fails as soon as either bond does.
❌ Not quite. The junior claim is paid only when both bonds pay.
❌ Not quite. $880 uses a 12% default probability (6% + 6%), which double-counts the case where both bonds fail. The junior claim is paid with probability 0.942=0.88360.94^2 = 0.8836.
❌ Not quite. $887.20 uses an 11.28% default probability, which leaves out the 0.36% chance that both bonds fail. The junior claim is paid with probability 0.942=0.88360.94^2 = 0.8836.
❌ Not quite. $940 is the price of one original bond at 6%. The junior claim is riskier than that bond, so it must sell for less.
❌ Not quite. Price = probability of being paid × $1,000.
Show solution

The junior claim gets the pool's second $1,000, which exists only when both bonds pay:

Pr⁡(junior paid)=(1−p)2=0.942=0.8836\Pr(\text{junior paid}) = (1-p)^2 = 0.94^2 = 0.8836 Pr⁡(junior defaults)=1−0.8836=0.1164=2×0.06×0.94+0.062\Pr(\text{junior defaults}) = 1 - 0.8836 = 0.1164 = 2 \times 0.06 \times 0.94 + 0.06^2

Its price is the expected payoff, in dollars:

PJ=0.8836×1000+0.1164×0=883.6P_J = 0.8836 \times 1000 + 0.1164 \times 0 = 883.6

So the junior claim defaults 11.64% of the time (almost twice the 6% of one bond) and sells for $883.60.

Problem 3 · Price and Promised Yield of the Junior Claim

Given: Two independent bonds, each with face value $1,000 and default probability 12%, are tranched into senior and junior claims of $1,000 face value each. The risk-free rate is zero. Find the junior claim's price and its promised yield (what it earns if paid in full).

Junior price?

Junior promised yield?

✅ Correct! The junior claim is paid with probability 0.882=0.77440.88^2 = 0.7744, so it costs $774.40 and promises 1000/774.4−1≈29.1%1000/774.4 - 1 \approx 29.1\%, far above the 13.6% of one original bond.
❌ Not quite. $880 is one original bond, paid 88% of the time. The junior claim is paid only when both bonds pay.
❌ Not quite. $985.60 is the senior claim, which fails only when both bonds do (0.122=0.01440.12^2 = 0.0144). The junior claim is the one that absorbs the first losses.
❌ Not quite. $760 prices the junior claim at a 24% default probability (12% + 12%), which counts the both-default case twice. It is paid with probability 0.882=0.77440.88^2 = 0.7744.
❌ Not quite. Price = probability of being paid × $1,000.
❌ Not quite. 13.6% is the promised yield of one original bond, 1000/880−11000/880 - 1. The junior claim costs less, so it promises more.
❌ Not quite. 22.56% is the junior default probability, 1−0.77441 - 0.7744, which is also the discount from face as a share of the face value. The promised yield divides the gain by the price: (1000−774.4)/774.4(1000 - 774.4)/774.4.
❌ Not quite. Doubling the original bond's 13.6% does not give the junior yield: yields do not add up like that. Compute 1000/774.4−11000/774.4 - 1 from the junior price.
❌ Not quite. Promised yield = face value / price − 1.
Show solution

Step 1: Probability the junior claim is paid. It needs both bonds to pay:

Pr⁡(junior paid)=0.88×0.88=0.7744\Pr(\text{junior paid}) = 0.88 \times 0.88 = 0.7744

Step 2: Price as the expected payoff, in dollars:

PJ=0.7744×1000=774.4P_J = 0.7744 \times 1000 = 774.4

Step 3: Promised yield, what it earns if paid in full:

yJ=1000774.4−1≈1.2913−1≈29.1%y_J = \frac{1000}{774.4} - 1 \approx 1.2913 - 1 \approx 29.1\%

For comparison, one original bond costs $880 and promises 1000/880−1≈13.6%1000/880 - 1 \approx 13.6\%; the senior claim costs $985.60 and promises about 1.5%. The more often a claim defaults, the higher the yield it has to promise.

Problem 4 · Did Tranching Create Value?

Given: Two independent bonds, each with face value $1,000 and default probability 25%, are tranched as in the video, and every claim is priced at its expected payoff. A dealer points out that the senior claim sells for $187.50 more than an original bond and says tranching created value. Which statement is right?

✅ Correct! Priority moves $187.50 of price from the junior claim to the senior claim, and the total is unchanged: value is reallocated, not created.
❌ Not quite. A promise is not a price. The junior claim is paid only with probability 0.752=0.56250.75^2 = 0.5625, so it sells for $562.50, which is $187.50 below an original bond.
❌ Not quite. Both move by the same amount. The senior claim falls from 25% to 0.252=6.25%0.25^2 = 6.25\% and the junior claim rises from 25% to 1−0.752=43.75%1 - 0.75^2 = 43.75\%: 18.75 points each way.
❌ Not quite. Demand from pension funds explains why the market grew, but here every claim is priced at its expected payoff, and the two tranches' expected payoffs still add up to the pool's: $1,500.
Show solution

With p=0.25p = 0.25, prices in dollars (risk-free rate zero):

Pbond=0.75×1000=750P_{\text{bond}} = 0.75 \times 1000 = 750 PS=(1−0.252)×1000=0.9375×1000=937.5P_S = (1 - 0.25^2) \times 1000 = 0.9375 \times 1000 = 937.5 PJ=0.752×1000=0.5625×1000=562.5P_J = 0.75^2 \times 1000 = 0.5625 \times 1000 = 562.5

The senior claim costs 937.5−750=187.5937.5 - 750 = 187.5 dollars more than a bond and the junior claim 750−562.5=187.5750 - 562.5 = 187.5 dollars less. Together:

937.5+562.5=1500=2×750937.5 + 562.5 = 1500 = 2 \times 750

In general the shift is p(1−p)Fp(1-p)F in each direction, so PS+PJ=2(1−p)FP_S + P_J = 2(1-p)F at every pp: tranching reallocates value between the claims; it does not create any.

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