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 negligible
 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 approximate
 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 coefficient
 of caution. In other words, the bond is worthless. In