278 NATURE OF CAPITAL AND INCOME [Cmar. XVI

Take, for example, a “5 per cent” bond running for
10 years. Let us assume that the market rate of interest
is 4 per cent, in the sense that $100 at any time will exchange
for $104 certain one year later. Under these conditions,
the bond ought to sell for $108 (see Chapter XIII). That
is, an investor buying this bond at a premium of $8 will
in 10 years “make” 4 per cent provided he receives all
the sums stipulated. The $108 is the “riskless value”
of the bond.
But, while $108 would be the price of the bond were the
investor absolutely sure of the income items to which it
entitles him, he may feel that there is a risk, — for
instance, that the probability of any given interest
payment is only 5%, in which event it is evident that he
will not pay $108. We can easily calculate, on the
basis of the assumed probability, that what has been
called the “mathematical value” of the bond would be
$97. This figure is found by multiplying each income
item by its coefficient of probability and discounting
the result at the market rate of interest, 4 per cent.
We assume here that the risk attached to each individual
 interest and principal payment is independent of
that attached to all the others. The probability of receiving
 each payment is 1%, and the risk of its not being received
isin each case 5. It is evident that the first payment of $5,
due in one year, has at that time a mathematical value of
5 x 4%, and the present value, when discounted at 4 per cent,
would be 2%. If the same expressions be obtained for all
the other items of income, and the sum total of the present
valuations be found, it is evident that in the result the factor
2% will appear for every individual sum, and that the total
will simply be +% as much as though the element of risk were
absent. In other words, the “mathematical value” of
the bond will be +% of its riskless value of $108, or about

$97.
But the actual “commercial value” of the bond will