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.