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.
