The cost of failure

Quantitative Finance Level 1

10-year treasuries are currently yielding 3%. ABC issued a new 10-year bond with a A+ rating, and it has a 6% yield. What is the implied probability of ABC defaulting on the bond?

3.00% 2.83% 2.91% 2.00%

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Calvin Lin by Calvin Lin

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1 solution

Alex Li
Mar 11, 2015

Note that the expected value of the second bond is 1.06 ( 1 − p ) 1.06(1-p) , and that of the first bond is 1.03 1.03 , where p p is the probability of defaulting. Equating the two, 1.06 ( 1 − p ) = 1.03 1.06(1-p) = 1.03 , so p = 1 − 1.03 1.06 = 2.83 % p = 1-\frac{1.03}{1.06}=\boxed{2.83\%}

16 Helpful 1 Interesting 5 Brilliant 8 Confused

Can you explain how you arrived at this calculation? What is the logic behind that?

Calvin Lin Staff - 6 years, 3 months ago

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The expected value is what you would probably get buying this type of bond over and over again a large number of times. The treasury is assumed to have 0% probability of default, so in that case (1-p) = 1

So 1.03(1-p) = 1.03 simply.

We are assuming the market values the expectation values of these 2 bonds equally, which is often a good approximation.

So we have 1.06(1-p) = 1.03

Craig Brownell - 5 years, 4 months ago

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I (and probably other readers), would tend to consider the default on the entire 10 year span (including default of principal), maybe make all that clearer in the question :)

Pacino Al - 4 years, 9 months ago

So the problem was requesting annual likelihood of default not total likelihood?

Harry Pottash - 2 years, 4 months ago

Can you explain this Plesse better! And what means: 1.03 so p?

Felix Lindner - 1 year, 8 months ago

Where is the wiki for this problem? Is it available for non-premium members? I couldn't find it.

Félix Pérez Haoñie - 1 year, 7 months ago

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Not all problems have wikis attached.

This problem has a wiki attached, but not all wikis have been contributed to by the community. In this case, this wiki is still incomplete.

Calvin Lin Staff - 1 year, 7 months ago

This is assuming interest compounded annually and default accrued annually.

If this is on a continuous basis, the rate spread would be the default rate.

Leon Ki - 1 year, 4 months ago

Ok, yeah. That makes sense, actually. 🧐

ViralTaco Capobianco - 2 days, 21 hours ago

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