APPENDIX TO CHAPTER XVI 403 
of his income. Such a system would secure justice in the taxa- 
tion of income. It is practically what has usually been called 
the system of taxing « consumption,” 
APPENDIX TO CHAPTER XVI 
§ 1 (ro Cr. XVI, § 6) 
Mathematical Coefficients of Probability, Caution, and Risk 
Let us call the riskless value V, the mathematical value Vv, 
the commercial value V", the coefficient of probability P, the 
coefficient of caution C, and the entire coefficient of risk R. 
“We have, — 
Cal EL 
ral 
V 4 
2) 
Whence it follows that BR = PC. 
That is, the total effect of risk on value is twofold: first, 
through mere probability, which gives mathematical value; and 
secondly, through caution, which gives commercial value. 
Practically, it'is usually impossible to separate P and C. The 
object of this analysis is not so much to introduce the caution 
factor explicitly, as to make the general distinction between R 
and P, and to point out that the actual market value of secu- 
rities is not their actuarial or simple “ mathematical ” value; 
that, in other words, R is not the same thing as P. 
§ 2 (ro Cu. XVI, § 7) 
Formula for Mathematical Value of Risky Bond 
Let us denote by p, the probability of receiving the first 
installment, a,, of income due in one year, and by p. the prob- 
ability of receiving the second installment of income, yy pro- 
vided the first year’s is received, and again by p, the probability 
of receiving a, in three years, provided the previous two have 
been received, and so on for p, . . . p,, where n is the number of 
years to the last payment. The chance of receiving the first 
Payment is p,; hence the “mathematical value” of the first 
payment, when due, is a;p,, the present value of which is 
a. But the chance of receiving the second payment is 
+3 
  
—————— i ee... 
  
    
    
     
   
   
      
  
  
  
  
  
  
  
  
  
  
  
  
   
   
  
  
    
  
TR A LU 
Sas 
EE 
  
ne