Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/651



$$\text{N}$$ being the number whereof $$nr$$ is the hyperbolic logarithm, and $$nr \times 0{\cdot}43429448$$ its logarithm in Briggs’ System, and the common tables.

119. From No. 117 and 110, it follows, that the present value of $$s$$ pounds to be received at the end of $$n$$ years, when the interest is convertible into principal at $$\nu$$ equal intervals in each year, is $$s \left ( 1 + \frac{r}{\nu} \right ) ^{-\nu n}$$.

120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at $$\nu$$ equal intervals in the year, while the annuity is payable at $$\pi$$ intervals therein, the amount of each payment being $$\frac{a}{\pi}$$.

121. Case I. $$\mu$$ being any whole number not greater than $$\nu$$, let $$\frac{1}{\pi} = \frac{\mu}{\nu}$$, so that the interest may be convertible into principal $$\mu$$ times in each of the intervals between the payments of the annuity.

Then will the amount of L. 1, at the expiration of the period $$\frac{1}{\pi}$$ be $$\left ( 1 + \frac{r}{\nu} \right ) ^\mu$$ (117), and the interest of L. 1 for the same time will be $$\left ( 1 + \frac{r}{\nu} \right ) ^\mu - 1$$; whence the present value of the perpetuity will be $$\frac{\frac{1}{\pi} a}{\left ( 1 + \frac{r}{\nu} \right ) ^\mu - 1}$$ (8), and the value of the same deferred $$n$$ years, will be $$\frac{a}{\pi} \cdot \frac{\left ( 1 + \frac{r}{\nu} \right ) ^{-\nu n}}{\left ( 1 + \frac{r}{\nu} \right ) ^\mu - 1}$$ (119), therefore the present value of the annuity to be entered upon immediately, and to continue $$n$$ years, will be $$\frac{a}{\pi} \cdot \frac{1 - \left ( 1 + \frac{r}{\nu} \right ) ^{-\nu n}}{\left ( 1 + \frac{r}{\nu} \right ) ^\mu - 1} = \text{V}$$.

122. Case 2. $$\mu$$ being any whole number greater than $$\pi$$, let $$\frac{1}{\nu} = \frac{\mu}{\pi}$$, so that the annuity may be payable $$\mu$$ times in each of the intervals between the payments of interest, or the conversion thereof into principal.

Then, at the expiration of the $$\frac{1}{\nu}$$th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the $$\mu - 1$$ payments of the annuity previously made, will be $$\frac{ar}{\pi \pi} \left [ (\mu-1)+(\mu-2)+(\mu-3)+\cdots\cdots\cdot\cdot+3+2+1 \right ] = \frac{a}{\pi} \cdot \frac{r \mu (\mu - 1)}{2 \pi}$$; to which, adding the $$\mu$$ payments of $$\frac{a}{\pi}$$ each (including the one only then due), the sum, $$\frac{a}{\pi} \left [ \mu + \frac{r \mu (\mu-1)}{2\pi} \right ]$$, is the simple interest which the value of the perpetuity should yield at the expiration of each $$\nu$$th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since $$\frac{r}{\nu} : \frac{a}{\pi} \left [ \mu + \frac{r\mu(\mu-1)}{2\pi} \right ] :: 1 : \frac{a\nu}{r\pi} \left [ \mu+\frac{r\mu(\mu-1)}{2\pi} \right ] = a \left ( \frac{1}{r} + \frac{\mu-1}{2\pi} \right )$$, this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred $$n$$ years, will therefore be $$a \left ( \frac{1}{r} + \frac{\mu-1}{2\pi} \right ) \times \left ( 1 + \frac{r}{\nu} \right ) ^{-\nu n}$$ (119). Whence it appears, that the present value of the annuity to be entered upon immediately, and to continue $$n$$ years, will be $$a \left ( \frac{1}{r} + \frac{\mu-1}{2\pi} \right ) \cdot \left [ 1 - (1 + \frac{r}{\nu})^{-\nu n} \right ] = \text{V}$$.

123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, $$\nu = \pi$$.

Since the interest of L. 1 at the expiration of the period $$\frac{1}{\pi}$$ will be $$\frac{r}{\pi}$$, the value of the perpetuity will be $$\frac{\frac{1}{\pi} a}{\frac{1}{\pi} r} = \frac{a}{r}$$ (8), whence, proceeding as before, we obtain the present value of the annuity, $$\frac{a}{r} \left [ 1 - (1+\frac{r}{\nu})^{\nu n} \right ] = \text{V}$$. When $$\nu = \pi$$, and consequently $$\mu = 1$$, the values of $$\text{V}$$, given in the two preceding cases, will be found to coincide with this.

124. According as $$\nu$$ and $$\pi$$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $$\text{V}$$ be equal to $$y$$, $$h$$, $$q$$, or $$c$$, then will

125. The amount of an annuity is equal to the sum to which the purchase money would amount, if it were put out and improved at interest during the whole term.

For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the grantor, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase money must evident- 1em