406 NATURE OF CAPITAL AND INCOME 
case of such high risk we cannot, therefore, apply the simple 
rule of adding to the rate of interest the rate of risk to obtain 
the “mathematical value”; and the “commercial value” 
would, of course, be even less than the mathematical value. 
In other words, it is practically impossible to compensate for a 
risky investment by increasing the rate of interest as though 
it were an insurance premium. In actual practice such a 
“bond” would be absolutely worthless; for, while the above 
calculations are correct on the basis of a chance of payment of 
one in ten, practically this chance of payment would be zero. 
The high risk not only makes the terms of the loan onerous, 
but these onerous terms make the uncertainty of repayment 
greater, and so on in a vicious circle. A lender who fancies he 
can offset a risk as high as 1% by lending only 50 cents instead of 
$100 for a returnable principal of $100; will find that he has 
not offset that risk, but merely increased it. 
In the previous calculations, we assumed that a default in 
one payment carried with it a default in all subsequent pay- 
ments. We may, however, easily extend our formula to the 
general case by designating the chances of payment in succes- 
sive years, whether interdependent or not, by p, for the first 
year, p, for the second (instead of by p, p, as before), p; for the 
third, etc., and changing the first equation on page 404 accord- 
ingly. 
§ 3 (To Cu. XVI, § 10) 
Variability about a Mean, as measured by the ‘‘Standard Deviation * 
For a more minute analysis of the bearing of chance it is 
preferable to measure the variability with reference to the 
mean. Thus, in the case mentioned, where the dividends are 
successively 5%, 5%, 6%, 5%, 5%, 4%; 5%; T%; 5%, 3%; 
49%, 5%, instead of measuring the variability of dividends with 
reference to 59, we should measure it with reference to the 
mean rate, which is 4.99,. The deviations from this mean 
during the twelve successive years were therefore: + 0.1, 
+01, +11, +01, +01, —09, 40.1, +21, +01, —1.9, 
—0.9, +0.1.