Finance-Theory · Unit 8 · Video 3 · Interactive Practice

What the Credit Spread Pays For: Default Premium, Risk Premium and the Road to Securitization

IKey Formulas

Formula	Name	What it needs
$\text{Spread} = y_{\text{Baa}} - y_{\text{Treasury (10yr)}}$	Measured credit spread	Moody's Baa yield, the lowest grade that still counts as investment grade, against the very safe ten-year Treasury
$y_{\text{prom}} = \left(\dfrac{F}{P}\right)^{1/T} - 1, \qquad y_{\text{exp}} = \left(\dfrac{E[\text{payoff}]}{P}\right)^{1/T} - 1$	Promised and expected YTM (zero-coupon bond)	Promised: the yield if the bond does not default, from the full face value $F$ . Expected: the probability-weighted average of the possible yields; the video's zero-coupon example computes it from the expected payoff
$\text{DP} = y_{\text{prom}} - y_{\text{exp}}, \qquad \text{RP} = y_{\text{exp}} - y_f$	Default premium and risk premium	$y_f$ is the yield on a default-free bond of similar maturity and coupon
$y_{\text{prom}} - y_f = (y_{\text{prom}} - y_{\text{exp}}) + (y_{\text{exp}} - y_f) = \text{DP} + \text{RP}$	Yield spread decomposition	Add the two premiums and the expected yield cancels

Key Insight: In the video's example the default-free rate is 8%, the risk premium lifts the expected yield to 9%, and the default premium lifts the promised yield to 12%: the 4-point spread is 1 point of risk premium plus 3 points of default premium.

IIStacking the Spread

The price and the expected payoff of a risky zero fix both premiums, and together they always add up to the spread.

XYZ price (P) 321.97 Expected payoff 762.22

💡 In this example default is 3 of the 4 points. The studies the video cites find default explains well under half of actual spreads: 17.8% for ten-year A-rated industrials (Elton et al., 2001), 5–22% (Delianedis & Geske, 2001), 20–30% (Huang & Huang, 2002).

IIIHow Far the Spread Moves

The Baa–Treasury spread swings from about 7.5 points in the 1930s to about 1.43 in 2026.

Episode 1

IVWho Holds the Loan

Securitization takes the bank out of the middle: borrowers' debt becomes securities that investors buy directly.

Structure diversification accurate risk measurement 'normal' markets sophisticated investors

VQuiz Questions

Problem 1 · Split the Spread

Given: A corporate bond has a promised YTM of $11.0\%$ and an expected YTM of $8.5\%$ ; a default-free bond of similar maturity and coupon yields $7.0\%$ . Find the default premium and the risk premium.

✅ Correct!

\text{DP} = 11.0 - 8.5 = 2.5\%

and

\text{RP} = 8.5 - 7.0 = 1.5\%

; together they make the

4.0\%

spread.

❌ The labels are swapped. The default premium is the top slice, promised minus expected (

11.0 - 8.5

); the risk premium is the lower slice, expected minus default-free (

8.5 - 7.0

❌ Not quite.

4.0\%

is the whole spread,

11.0 - 7.0

. The default premium is only the part above the expected yield:

11.0 - 8.5

❌ Not quite.

4.0\%

is promised minus default-free, the whole spread. The risk premium starts from the expected yield:

8.5 - 7.0

Show solution

\text{DP} = y_{\text{prom}} - y_{\text{exp}} = 11.0\% - 8.5\% = 2.5\%

\text{RP} = y_{\text{exp}} - y_f = 8.5\% - 7.0\% = 1.5\%

Check: $\text{DP} + \text{RP} = 2.5\% + 1.5\% = 4.0\% = 11.0\% - 7.0\%$ , the whole yield spread.

Problem 2 · Which Payoff Goes in the Expected Yield?

Given: A five-year zero-coupon bond issued by ABC has face value $1,000 and sells for $620.92. Allowing for default, its expected payoff is $870.87. Find its expected yield to maturity.

✅ Correct!

(870.87/620.92)^{1/5} - 1 = 0.07

: the expected yield uses the expected payoff, not the face value.

❌ That is the promised yield.

(1000/620.92)^{1/5} - 1 = 10\%

uses the full face value, which is what you earn only if ABC does not default. The expected yield uses the expected payoff of 870.87.

❌ Not annualised.

870.87/620.92 - 1 \approx 40.3\%

is the growth over all five years. Take the fifth root first:

(870.87/620.92)^{1/5} - 1

❌ Compounding ignored.

40.3\% / 5 \approx 8.05\%

spreads the five-year growth evenly without compounding. A yield to maturity compounds, so take the fifth root:

(870.87/620.92)^{1/5} - 1

Show solution

The expected yield replaces the face value with the payoff you expect after allowing for default:

y_{\text{exp}} = \left(\frac{870.87}{620.92}\right)^{1/5} - 1 = (1.40255)^{0.2} - 1 = 1.0700 - 1 = 7.00\%

For comparison, the promised yield is $\left(\frac{1000}{620.92}\right)^{1/5} - 1 = (1.61051)^{0.2} - 1 = 10.00\%$ .

Problem 3 · Both Premiums from Prices

Given: The ABC bond of Problem 2 (face $1,000, price $620.92, expected payoff $870.87, five years, zero coupon), and a five-year Treasury strip, face $1,000, selling for $783.53. Find the default premium and the risk premium.

Default premium

Risk premium

✅ Correct! The stack reads

5\%

default-free,

7\%

expected,

10\%

promised:

\text{RP} = 2\%

and

\text{DP} = 3\%

make up the

5\%

spread.

❌ That is the whole spread.

10\% - 5\%

runs from the default-free yield to the promised yield. The default premium is only the part above the expected yield:

10\% - 7\%

❌ That is the risk premium.

7\% - 5\% = 2\%

is expected minus default-free. The default premium is promised minus expected.

❌ That is a yield, not a premium.

7\%

is ABC's expected yield itself. A premium is a gap between two yields: here promised minus expected.

❌ That is the whole spread.

10\% - 5\%

includes the default premium. The risk premium starts from the expected yield:

7\% - 5\%

❌ That is the default premium.

10\% - 7\% = 3\%

is promised minus expected. The risk premium is expected minus default-free.

❌ That is a yield, not a premium.

7\%

is the expected yield; subtract the default-free

5\%

to get the extra that pays for bearing risk.

Show solution

Step 1: the three yields.

y_f = \left(\frac{1000}{783.53}\right)^{1/5} - 1 = (1.27628)^{0.2} - 1 = 5.00\%

y_{\text{exp}} = \left(\frac{870.87}{620.92}\right)^{1/5} - 1 = 7.00\%, \qquad y_{\text{prom}} = \left(\frac{1000}{620.92}\right)^{1/5} - 1 = 10.00\%

Step 2: the premiums.

\text{DP} = 10\% - 7\% = 3\%, \qquad \text{RP} = 7\% - 5\% = 2\%

Check: $\text{DP} + \text{RP} = 5\% = 10\% - 5\%$ , the whole yield spread.

Problem 4 · How Much of a Real Spread Is Default?

Given: In mid-September 2026 the Baa–Treasury spread was about $1.43$ percentage points. Suppose default accounted for $17.8\%$ of it, the share Elton et al. (2001) found for ten-year A-rated industrial bonds. About how many percentage points of the spread would default explain?

✅ Correct!

1.43 \times 0.178 \approx 0.25

points; the other

1.18

points pay for tax effects and the systematic risk premium.

❌ That is the other part.

1.43 - 0.25 = 1.18

is what default does not explain: the tax effects and the systematic risk premium.

❌ Not quite.

17.8\%

is a share of the spread, not a number of points. Multiply:

1.43 \times 0.178

❌ Wrong share.

1.07

75\%

1.43

, the default share in the video's XYZ example (

3

4

points). That example was built for arithmetic; the studies put default well under half.

Show solution

\text{default part} = 0.178 \times 1.43 = 0.25454 \approx 0.25 \text{ percentage points}

\text{rest} = 1.43 - 0.25 = 1.18 \text{ percentage points}

In the three studies that estimate it (17.8%, 5–22%, 20–30%), default explains well under half of the spread; most of it pays for other things, such as taxes and the systematic risk premium.

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