Sec. 8] THE RISK ELEMENT 279 
ordinarily be less than this “mathematical value” of $97. 
We may suppose it to be $92.50, indicating a coefficient, of 
caution of 22. Here, as in the case of the lottery ticket, 
we have regarded the actual value of the bond as obtained 
from its riskless value by applying first the probability 
factor, and second the caution factor, %*. 
If the probabilities of receiving the individual interest 
payments were not regarded as independent, the calcula- 
tions of the mathematical value would differ somewhat 
from the preceding. Thus, if we suppose that default in 
one interest payment carried with it, by the terms of the 
contract, the default in all subsequent interest payments, 
we should have to apply the theory of probability some- 
what differently! but the principle would be the same. 
§8 
There is another way, and one which conforms more to 
ordinary usage, in which the commercial value of the bond 
may be derived from the riskless value. While the price of 
the bond will vary inversely with the risk, the rate of 
interest varies directly with the risk; so that as the value 
of the bond descends, the corresponding rate of interest 
will ascend. Thus we have riskless, mathematical, and 
commercial rates of interest — 4 per cent, 5.4 per cent, and 
6 per cent — corresponding respectively with the riskless, 
mathematical, and commercial values of the bond — $108, 
397, $92.50. 
The question sometimes arises, where the element of 
risk thus raises the basis on which the bond is sold, whether 
the 6 per cent is a true “rate of interest.” The ques- 
tion is purely one of definition. Were it possible, it would 
be simpler to confine the application of the phrase rate of 
interest” to an exchange between present and future risk- 
less income. But in this case, it is always exceedingly diffi- 
! For the consideration of this case, see Appendix to Chap. XVI, § 2.