APPENDIX TO CHAPTER XVI 405 
which is the product of two very small quantities. Thus, if q 
is t}g and ¢ is 144, the value of the fractional term becomes 
approximately .0005, which is a negligible quantity (compared 
with 14+7i4+¢=1+4.04 + .01). Hence when ¢ is small, the 
formula for mathematical value becomes approximately, — 
o a gy 3 : An 
Tritg A+itof Arita’ Taritor 
In other words, when the risk of default is small, its effect is 
substantially the same as that which follows from a rise in the 
rate of interest. If the rate of interest when risk is absent is 
4%, a risk of 19, will therefore merely increase the “basis” on 
which the loan can be contracted to about 5%. Thus, if we 
recur to the so-called 59) ten-year bond, and suppose that 
the probability of each successive payment is Tos; and the 
risk of default, ¢, is 114, then the mathematical present value 
of the bond, when interest is 49, is approximately, — 
a a4 y hi 
“Ttite (A+itoP tative 
5 5 
+ (1.05) t 
m 
  
m 
+, ete., 
5 
~1.05 Hogg He 
In other words, the present value is approximately the same 
as the present value of a 59, bond on a 59, basis, which is of 
course par, or 100. 
But if the risk is great, the approximate formula given will 
no longer apply. Thus, if the chance of default is 1%; or, in 
other words, if the chance of payment is only i, the formula 
for the mathematical value of the property becomes, — 
ay) | al)’ | a)’ 
Vi= re ++ ot A +iy +, ete. 
In this case it is evident that all terms after the first are neg- 
ligible compared with the first (unless the successive items a, 
ag, ete., increase with sufficient rapidity to offset the decreasing 
fractions 11, 1445p, ete.). In the case of a 59 ten-year $100 bond, 
in which the risk of default is at any moment %, the approxi- 
mate value of the bond, obtained by omitting all terms after 
  
  
the first, would be ih, or approximately 50 cents! This 
“ mathematical value ” might be still further reduced by a co- 
efficient of caution. In other words, the bond is worthless. In