# The Pamphlet Collection of Sir Robert Stout: Volume 8

# Appendix

### Appendix.

*To construct a table of the logarithms of the present value of* £1 *due at the end of any number of years*.

The present value of £1 due a year hence is , the logarithm of which is the arithmetical complement of log (1+*i*). When the rate is 3 per cent, log (l+*i*)=log 1.03=0.01283,72247, the arithmetical complement of which is 1.98716,27753. By taking the first nine multiples of this logarithm may be constructed a table of the logarithms of the present value of £1 due at the end of any number of years from 1 to 100 which will be true to the last figure to six places of decimals, thus:

Logarithm of the present value of £1.

Years.

1 | 1.987102 775 |

2 | .974325 550 |

3 | .961488 325 |

4 | .948651 100 |

5 | .935813 875 |

* *

*To construct columns* D *and* N.

Since D* _{x}*—

*l*

_{x}v^{x};therefore log D* _{x}*=log

*l*+log

_{x}*v*.

^{x}Log D* _{x}* is formed in reverse order to facilitate the formation of column N, thus:

*To construct columns* C *and* M.

Since C* _{x}*=

*d*

_{x}v^{x+1};

*To construct columns* C′*and* M′.

Since *C′ _{x}*=

*s′*D

_{x}^{w}*, where*

_{x}*s′*denotes the average number of weeks' sickness to each person in the year following the age

_{x}*x*, and

*w*the present value of £1 due half a year hence;

therefore log C′* _{x}*=log

*s′*+log

_{x}*w*+log D

*.*

_{x}The present value of £1 due half a year hence at 3 per cent. is the logarithm of which is the arithmetical complement of log 1.015. Log 1.015=0.00646,60422, the arithmetical complement of which is 1.99353,39578. Writing this logarithm to six places of decimals at the bottom of a card to be added to the other two logarithms at each age, the formation is as follows: