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I am trying to build out a probability of default model for a bond. Given the current price of a bond and the current risk free rate, I am trying to calculate the probability of default.

So assume a bullet bond with a 5 year maturity. Let say the coupons pay annually for simplicity and they pay out a 5% on a 100 par. 

Y1   Y2  Y3  Y4  Y5
 $5 $5 $5 $5 $105

Let say there is a risk this bond default and we estimate the recoveries on the bond are 40 cents on the dollar.

So here are all the possible scenarios for this bond

    $40  Default Yr1
    $5 $40 Default Yr2
    $5 $5 $40 Default Yr3
    $5 $5 $5 $40 Default Yr4
    $5 $5 $5 $5 $40 Default Yr5
    $5 $5 $5 $5 $105 No Default Yr5

Lets call the value of the bond is V. So Given all these payoffs. The value of the bond should be the E(V), the expected value of all the possible payoffs. But what probability density should I use to weight the possible scenarios? I could assume that on any given year the probability Firm XYZ will default is 2%. Hence the probability the firm survives year 1 is 98%, the firm surivevs to year 2 is (98%)^2, year 3 (98%)^3, survives to year 4 (98%)^4, survives year 5 (98%)^2

(.02)
(.98)(.02)
(.98) (.98)(.02)
(.98) (.98) (.98)(.02)
(.98) (.98) (.98) (.98)(.02)

But obviously these probability do not produce a pdf because they do not sum to 1. Obviously the probabilities listed above do not include all the real of possibilities and that is why? What should I be weighing by? Thank you for your help.

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    $\begingroup$ I think I understand what you want to do is describing a distribution of defaulting on your bond, there are two things I want to say. 1. You treat the default event as i.i.d on each year, so there is no probability event happen on the five year bond as whole, and so the probability adding up is not equal to one. 2. My knowledge of DP bond is first it's a conditional probability, second there is proportional hazard model(stochastic), third bootstrapping probability from hazard rate, and recovery rate. $\endgroup$
    – TomHan
    May 18, 2014 at 10:31

3 Answers 3

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This does not sum to 1 because you have forgotten to add the 6th scenario, the NonDefault (ND).

If Ps is the probability of survival and Pd the probability of default, the ND has the probability Ps^5.

This makes: Pd+ Ps*Pd+ ... Ps^4*Pd+ Ps^5= Pd*(1+Ps+...+Ps^4)+Ps^5= Pd*(1-Ps^5)/(1-Ps)+Ps^5= (1-Ps)**(1-Ps^5)/(1-Ps)+Ps^5=1.

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  • $\begingroup$ What's the difference between "survival" and "non-default?" $\endgroup$
    – Tom Au
    Jun 4, 2014 at 16:57
  • $\begingroup$ I understand survival as a function of delta t (you are non dead between the moments t0 and T) while I understand non-default as a function of t (you are not dead at moment T). Survival is the investment point of view and non-default is the liquidation approach. $\endgroup$
    – user7056
    Jun 5, 2014 at 18:38
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As you have the market price of the bond given, you may infer an unconditional default probability as follows:

  1. Calculate the price of the bond under riskfree rate, say its 110$.

  2. Divide the Market Price by the Riskfree Price, say $105/110 = 0.95$.

Hence the implied default probability is expectedly $5\%$.

The other way you are describing seems to me not what was asked, as you could specify any distribution for the hazard rates, but it was asked for the 'implied default probability'. You would then need to fit the parameters of the distribution-implied price to the market price using some numerical optimizer (e.g. Least Squares).

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You missed counting event that no default happens.

You can check out my blog on this topic: http://rickyzhang.asuscomm.com/blog/?p=29

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