The nature of capital and income . 04\-^times a year, the amount is \ n J At the limit, we have the amount of $1 at interest at 4%for 25 years when the rate of interest is payable limit of the above expression, when n is made indefi-nitely great, is the definition of e. The distinction between the different rates of interest maybe shown by a diagram. In Figure 33, let the curve BAB rep-resent a discount curve, any two ordinates of which representexchangeable goods situated at two corresponding points oftime, as a and b, that is, the sum aA of present goods will buythe sum bB o
The nature of capital and income . 04\-^times a year, the amount is \ n J At the limit, we have the amount of $1 at interest at 4%for 25 years when the rate of interest is payable limit of the above expression, when n is made indefi-nitely great, is the definition of e. The distinction between the different rates of interest maybe shown by a diagram. In Figure 33, let the curve BAB rep-resent a discount curve, any two ordinates of which representexchangeable goods situated at two corresponding points oftime, as a and b, that is, the sum aA of present goods will buythe sum bB of goods which lie in the future a time interval ab,beyond the present. If we take these two points, a and b, ayear apart, the rate of interest as reckoned annually is the APPENDIX TO CHAPTER XII 361 slope of the secant AB in relation to the ordinate aA =- aA \. Similarly, the slope of the secant AC drawn through points corresponding to times a half year apart, takenin relation to aA, is the rate of interest reckoned semi-annually,. Fig, 33. and so on. At the limit, the slope of the tangent AT (in re-lation to aA) represents the rate of interest reckoned continu-ously. Similarly, the regressive secant, AB, represents the rate ofdiscount, reckoned annually (if ab represents a year interval), 362 NATURE OF CAPITAL AND INCOME AC, the rate reckoned semi-annually, and so on until the tan-gent ATyOv AT, is again reached, when the distinction betweeninterest and discount disappears. It is easy to prove (by simi-lar triangles) that the reciprocal of the rate of interest continu-ously reckoned, in other words the years purchase, isrepresented by aS the subtangent. § 3 (to Ch. XII, § 6) A Premium Rate of 4% one Year and 3% Each Year after means a PriceRate of the First Year. We have given 4% as the rate of interest (as premium) inthe first year, and 3% as the rate for each year thereafter. Let us suppose that $ 100 to-day is invested for $ 104 nextyear, and that at the end of the
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