282 NATURE OF CAPITAL AND INCOME [Car XV) . 
would have if it were certain to yield the (arbitrarily 
assumed) 5 per cent forever — never more and never less. 
The riskless value is therefore simply the capitalized value 
of a perpetual annuity of $5 per share of $100 face value. 
If the rate of interest is 4 per cent, the result is $5 divided 
by 4 per cent, or $125. 
To obtain the “mathematical” value we simply add to 
the riskless value the value of the chance of getting more, 
and subtract that of the chance of getting less. The 
chance of getting an additional $1 a year is found by expe- 
rience, as set forth above, to be two in twelve, or } each 
year. The present value of the right to this chance has 
therefore a mathematical value } as great as though the $1 
increment were a certainty. But the certainty of $1 a 
year would be worth $25. Hence a chance of 1 in 6 of 
getting $1 a year would be worth mathematically & of $25, 
or $4.16}. In like manner the chance of a second addi- 
tional dollar is one in twelve and is worth (mathematically) 
1 of $25, or $2.08%. These two terms, $4.163 and $2.08%, 
are the additive terms sought. The subtractive terms are 
the mathematical value of the chance of getting $1 less 
than the $5, and of getting still another $1 less. These 
chances, being 3 in 12 and 1 in 12 respectively, are worth 
25 of $25 and 4 of $25 respectively, or $6.25 and $2.08}. 
The whole mathematical value is therefore $1254 (84.163 + 
$2.081) — ($6.25 + $2.083), or $122.913. Applying to this 
the factor of caution, which, let us say, is 1%, we find the 
commercial value to be $110.63. The three values are 
thus, approximately: — 
Wriekloga 7 VAIUE . ss ev ese sie Wow a ls ow $125 
Cathemiatical "valle vo evi ve wie wie a $123 
“oommereial Tt VANE Llc ve a Ee Del a aR le $111 
In this manner we may compute the three values in any 
other case. Usually, however, the chances involved are so 
indefinite that the reckoning is made only by rule of thumb.