The Value of a Bond with Default Probability

1 The Value of a bond with Default ProbabilityStefan Laboratories LLC, Longmont Colorado, 8, 2010 LetXrepresent the present Value of a bond s cash flow stream. When youhave a Default Probability thenXbecomes a random variable with a rangeor (as a simplifying assumption) a finite number of possible values. The wayto Value the bond in this case is to take each possible Value ofX, multiply itby its Probability and sum the results. In other words the Value of the bondshould equal the mathematical expectation illustrate the idea, consider the case of a bond with 4 coupon paymentsuntil maturity. Letpbe the Probability that the bond survives from onecoupon payment to the next and letXi(i= 0,1,2,3,4) be the Value ofXgiven that the bond defaults after making itsithcoupon payment.

2 Thepossible paths this bond can take is illustrated in figure expectation ofXcan then be expressed as:E[X] =X0(1 p) +X1p(1 p) +X2p2(1 p) +X3p3(1 p) +X4p4(1)This formula can also be written asE[X] =X0+ (X1 X0)p+ (X2 X1)p2+ (X3 X2)p3+ (X4 X3)p4(2)10234X0X1X2X3X4p1-pp2p3p4p(1-p)p 2(1-p)p3(1-p)p4 Figure 1: Possible paths of 4 coupon payment bond with constant probabilityof surviving between paymentsTo come up with expressions for theXi, the following variables need to bedefined:F= face Value of the bondC= coupon paymentI=C/F= coupon interestR= recovery rate = [0,1]f= risk free interest rated= 1/(1 +f) = discount factorI define the recovery rate as the fraction of the face Value that is payed if thebond defaults so it must be in the range of 0 to 1.

3 Assuming thatRremainsconstant, the values ofXiareX0=RFX1= (C+RF)/dX2=C/d+ (C+RF)/d2X3=C/d+C/d2+ (C+RF)/d3X4=C/d+C/d2+C/d3+ (C+F)/d4If you put these into the formula forE[X], you get2E[X] =C((p/d) + (p/d)2+ (p/d)3+ (p/d)4)+(3)RF(1 p)(1 + (p/d) + (p/d)2+ (p/d)3)+(4)F(p/d)4(5)Now it should be obvious how to extend this to the case ofNcoupons insteadof just 4. The formula in more compact form isE[X] =CN k=1(p/d)k+RF(1 p)N 1 k=0(p/d)k+F(p/d)N(6)Each of the sums in this formula is a geometric series that can be collapsedinto a single term. The formula for collapsing a general geometric series isN k=0ak= 1 +a+a2+..+aN=1 aN+11 a(7)Using this result in the formula forE[X] gives youE[X] =C(p/d)1 (p/d)N1 (p/d)+RF(1 p)1 (p/d)N1 (p/d)+F(p/d)N(8)This expression can be simplified a little by dividing through byFand mak-ing the substitutionsz=p/d,I=C/F.

4 with a little algebra, you getE[X]F= (Iz+R(1 dz))1 zN1 z+zN(9)WithE[X] equal to the price of the bond , this equation can be solved nu-merically forzwhich then gives the survival Probability asp=dz. Given anassumption of what the survival Probability is, the equation can be used tocalculateE[X] which is the fair price for the get the Default Probability for the bond , simply subtract the survivalprobability from 1, Default Probability = 1 p. The cumulative defaultprobability, or the Probability that the bond defaults anytime within thenextncoupon periods is 1 are several ways to test the formula for logical consistency. First lookat the case where the survival Probability is zero so that withz= 0 theformula reduces toE[X]F=R(10)This is logical since when Default is immanent the price should just equal therecovery the case where survival is certain and the risk free rate is zero you havez= 1 andE[X]F=N I+ 1(11)The price here is equal to the total of the coupon payments plus the facevalue, as you would