380 NATURE OF CAPITAL AND INCOME
where V represents the value the bond will have after the next
«interest ” payment, and ¢'the time elapsing to that payment.
Or, it is V' (1+41)*", where V represents the value after the
Zast “interest” payment, and ¢'' the time since said payment.
Practically this last formula reduces to V + Vit", which in turn
is practically the same as V+ at"; for a and Vi are practically
equal, each being nearly the true interest for one installment
period. This is the formula usually employed by brokers, at"
being called the “interest earned ”’ since the last coupon.
§ 9 (ro Cu. XIII § 7)
Alternative Method, whereby the « Premium *’ in the Price of the Bond
is compounded separately
The so-called 59, bond running for 10 years, which is sold
on a basis of 4 9%, may be considered as consisting of the follow-
ing two property rights: (1) the right to four dollars a year
for 10 years and $100 at maturity, and (2) the right to one
dollar a year for 10 years. It is evident that the present
value of the first property is $100, to which, therefore, we need
only to add the value of the second property, namely, the an-
nuity of $1 a year for 10 years. It is therefore the present
value of this small annuity, consisting, we may say, of the
difference between the real and nominal interest on $100,
which constitutes the “premium” on the price of the bond.
This present value is $8, and is found in the manner already
explained for terminable annuities, being the total discount on
$25 at the end of 10 years, $25 being the capital-value of a
perpetual annuity of $1 a year, when interest is reckoned at
49,. Consequently, the bond is worth in all $108. This value
is represented diagrammatically in Figure 42.
Let AA' represent the 10-year period, with the $5 interest
payments shown by the ten vertical lines at unit intervals.
A'B' represents the $100 « principal ” due in 10 years, and
AB represents what the bond would be worth ($100) if the
interest payments were $4 instead of $5. To this must
therefore be added the present value of $1 a year for 10 years.
This is the total discount on B'C" (drawn equal to $25), the
capitalization of a perpetual annuity of $1 a year. The total
