Finance-Theory · Unit 8 · Video 2 · Interactive Practice

Bonds That Might Not Pay: Default Risk, Credit Ratings and Debt That Behaves Like Equity

IKey Formulas

FormulaNameWhat it needs
P(y′)≈P(y)[1−Dm∗(y′−y)+12Vm(y′−y)2]P(y') \approx P(y)\left[1 - D_m^*(y'-y) + \tfrac{1}{2}V_m(y'-y)^2\right]Recap: duration and convexityA default-free bond, whose one risk is a change in yield. Like every formula so far, it assumes each promised payment arrives
default:  paidt<promisedt\text{default:}\ \ \text{paid}_t \lt \text{promised}_tDefaultA promised payment missed, of either interest (a coupon) or principal (the face value). A government borrowing in its own currency can print money to pay; a corporation cannot
Aaa, Aa, A, Baa⏟investment grade ∣ Ba, B, Caa, Ca, C⏟junk\underbrace{\text{Aaa},\ \text{Aa},\ \text{A},\ \text{Baa}}_{\text{investment grade}} \ \Big|\ \underbrace{\text{Ba},\ \text{B},\ \text{Caa},\ \text{Ca},\ \text{C}}_{\text{junk}}Moody's rating scaleA rating indicates how likely the issuer is to default; below Baa is non-investment grade, or junk. S&P and Fitch use the same steps in capitals, AAA down to BBB, then BB down to C, plus D for an issuer already in default
bondholder’s payoff=min⁡(F, V)\text{bondholder's payoff} = \min(F,\ V)Risky debt in bankruptcy (a simple model)Our simplification of the video's bankruptcy story: a firm owing one bond, whose holder recovers the full value of the assets. FF is the promised payment, VV the value of the firm's assets at maturity. When V<FV \lt F the firm cannot pay, the bondholder takes over the assets, and the payoff is their value: random, like a stockholder's

Key Insight: Investment grade and junk are degrees of the same risk, not different kinds of bond. The further down the scale a bond falls, the more its payoff depends on what the firm's assets turn out to be worth, and the more it behaves like equity.

IIPromised Is Not Paid

A default is any promised payment that is missed, interest or principal, and of these two promises only the corporation's can break.

IIIOne Risk, Three Agencies, One Line

Moody's, S&P and Fitch grade the same default risk on matching scales, split into investment grade and junk.

IVRiskier Debt Pays Like Equity

A simple model with hypothetical numbers: a firm owes one bondholder $100, and its assets at maturity take one of five equally likely values; when they fall short, the bondholder's payoff becomes whatever the assets are worth.

💡 Most bond investors want exactly the flat part of that line: as of 2025 US fixed income outstanding was about $61.2 trillion against $68.9 trillion of US stock-market value, and worldwide $160.7 trillion against $157.8 trillion, a market that size because most of its participants use it as a safe haven.

VQuiz Questions

Problem 1 · Which Side of the Line

Given: Four bonds carry these ratings. Which one is investment grade?

✅ Correct! BBB is the lowest investment-grade step at S&P and Fitch, the same step as Moody's Baa.
❌ Not quite. Moody's investment grade ends at Baa. Ba, with one "a", is the step below the line: somewhat speculative, the first junk grade.
❌ Not quite. At Fitch the line sits below BBB. BB is one step lower, the first non-investment-grade step.
❌ Not quite. Three letters do not mean top quality here: Moody's top grade is Aaa, while Caa is highly speculative, three steps below the line.
Show solution

Investment grade runs down to Baa at Moody's and to BBB at S&P and Fitch; everything below is non-investment grade (junk).

  • Moody's Ba: one step below Baa, junk (somewhat speculative).
  • Fitch BB: one step below BBB, junk (somewhat speculative).
  • S&P BBB: the last step above the line, investment grade (medium grade).
  • Moody's Caa: junk (highly speculative).

Only S&P BBB is investment grade.

Problem 2 · Already in Default

Given: A company has already failed to make a promised coupon payment. Which set of ratings describes it at the three agencies?

✅ Correct! S&P and Fitch keep a separate grade, D, for an issuer already in default; Moody's uses C both for that and for imminent default.
❌ Not quite. Moody's scale has no D. Its lowest grade, C, covers both imminent default and default.
❌ Close. Moody's C is right, but at S&P and Fitch C means imminent default. A missed payment has already happened, so they grade it D.
❌ Not quite. Ca and CC are the "most speculative" grades, two steps above default and one above imminent default. A missed promised payment is a default itself.
Show solution

A default is a failure to make a promised payment of interest or principal, so a missed coupon puts the issuer in the bottom row of the table.

In that row S&P and Fitch write D. Moody's has no separate grade: its scale ends at C, used for both imminent default and default. The set is Moody's C, S&P D, Fitch D.

Problem 3 · Bonds vs Stocks, 2025

Given: As of 2025, US fixed-income securities outstanding totaled about $61.2 trillion against $68.9 trillion of US stock-market value; worldwide, $160.7 trillion of fixed income against $157.8 trillion of equities. Find the US bond market as a percentage of the US stock market, and the worldwide gap between the two markets.

US fixed income as a percentage of US stock-market value

Worldwide gap

✅ Correct! In the US bonds are just under nine-tenths of the stock market, and worldwide they are slightly larger: the two markets are on the same scale.
❌ Upside down. 68.9/61.2≈1.12668.9 / 61.2 \approx 1.126 is stocks as a multiple of bonds. The question puts bonds on top: 61.2/68.961.2 / 68.9.
❌ Not quite. 7.7 is the gap in trillions, 68.9−61.268.9 - 61.2, not a percentage. Divide the bond figure by the stock figure instead.
❌ Wrong figures. 160.7/157.8≈1.018160.7 / 157.8 \approx 1.018 uses the worldwide totals. The question asks about the United States.
❌ Check the direction. Worldwide the fixed-income figure, 160.7, is the larger one; equities are 157.8.
❌ Wrong pair. 7.7 trillion is the US gap, and there stocks are the larger market. The worldwide gap is 160.7−157.8160.7 - 157.8.
❌ Mixed markets. 160.7−68.9=91.8160.7 - 68.9 = 91.8 subtracts the US stock market from the world bond market. Compare world with world.
Show solution

United States (trillions of dollars):

61.268.9=0.888244…≈88.8%\frac{61.2}{68.9} = 0.888244\ldots \approx 88.8\%

Worldwide:

160.7−157.8=2.9160.7 - 157.8 = 2.9

so fixed income is larger by about $2.9 trillion, under 2% of either market. Bonds and stocks are on the same scale, and the bond market is that large because most of its participants use it as a safe haven.

Problem 4 · When the Bondholder Becomes the Owner

Given: A hypothetical firm has one bond outstanding and owes its holder $100 at maturity. At maturity its assets will be worth $60, $90 or $150, each equally likely. If the firm cannot pay in full it goes bankrupt and the bondholder takes over the assets. Find the bondholder's payoff in each outcome and the average payoff.

Payoff when the assets are worth 60, 90, 150

Average payoff

✅ Correct! In two of the three outcomes the payoff is simply the value of the firm's assets, so what you receive is random, the position of a stockholder.
❌ Not quite. That treats the bond as default-free. With assets worth $60 or $90 the firm cannot pay $100, so it defaults.
❌ Too generous. When the assets are worth $150 the firm pays its promised $100 and keeps the rest; the bondholder takes over the assets only when the firm cannot pay.
❌ Not quite. Default does not leave the bondholder with nothing: the firm must hand over control of its assets, so the bondholder receives what they are worth.
❌ Not quite. $100 is the promised payment, and also the average of the asset values (60+90+150)/3(60 + 90 + 150)/3. Average the payoffs instead, which are capped at 100.
❌ Missing an outcome. (60+90)/2=75(60 + 90)/2 = 75 averages only the two default outcomes. All three outcomes are equally likely.
❌ Not quite. 100/3≈33.33100/3 \approx 33.33 counts nothing in the default outcomes, but there the bondholder receives the firm's assets.
Show solution

The bondholder receives the promised 100 when the assets cover it and the assets themselves when they do not: payoff=min⁡(100, V)\text{payoff} = \min(100,\ V).

min⁡(100,60)=60,min⁡(100,90)=90,min⁡(100,150)=100\min(100, 60) = 60, \qquad \min(100, 90) = 90, \qquad \min(100, 150) = 100

Each outcome has probability 13\tfrac{1}{3}:

60+90+1003=2503≈83.33\frac{60 + 90 + 100}{3} = \frac{250}{3} \approx 83.33

So the average payoff is $83.33 against $100 promised. In the two default outcomes the payoff moves one-for-one with the firm's assets: the bond is behaving like equity.

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