INTEREST 67 5. A piece of timber increases in value each year by 4% of the previous year's value for 10 years. If its value at the beginning of the period was $40,000, what would be the value at the end of the 10 years ? 6. At the beginning of the year 1907 the visible coal supply of the United States was estimated to be 3,076,204,000,000 tons, and in that year the consumption was 480,363,424 tons, which was 7.36% in excess of the consumption in 1906. Assuming that consumption will go on increasing at the same rate, how long will it be before the supply is exhausted? —President Van Hise in lectures on conservation. 7. What would be the formula for compound interest when the interest is payable in advance, as in the case of small loans made from brinks ? 8. A rule in common use for finding the time in which a sum of money will double itself at compound interest is, "Divide .69 by the rate of interest and add one third of a year." Prove that the rule is approxi- mately correct. (The Napierian and not the common logarithm of 2 must be used.) 9. Find a rough rule for the time in which any sum of money will quadruple itself at compound interest. 10. A banker charges 7% payable in advance on loans made for 90 days. What is the effective rate ? 11. A. man leaves to a university an estate valued at $2,000,000, on condition that one half the annual income is to be added to the principal until the whole amount reaches $20,000,000, after which one fourth of the income is to be added to the principal until the whole amount reaches $30,000,000. If the funds can be made to yield 5%, when will the value of the estate reach $20,000,000 and when will it reach $30,000,000 ? Solve the problem on the basis of 4%. 12. Construct the graph showing the compound amount of 1 at 4%, payable annually, as the time varies. 13. Construct the graph for the amount of 1 at simple interest at 4% as the time varies, and compare it with the graph in the preceding problem. For what times do the two graphs show the two amounts to be. equal? 26. Discount. The word discount has a variety of meanings in the world of business. To the merchant buying a stock of goods it means a reduction in his bill for the payment of cash; to the same merchant selling his goods it means a reduction from the marked price of an article to secure prompt sale; to the banker it usually means simple interest payable in advance. 68 MATHEMATICAL THEORY OF INVESTMENT The problems that arise in such cases are usually problems in percentage or in simple interest, and for that reason they do not require discussion here. Discount, as we shall use the term, is a consideration for the payment of a sum of money before it is due. It is the difference between the value of a sum of money payable at some future time, with or without interest, at the time when it is due, and its value at some earlier tune, usually the present. We some- times say of a sum of money that it is accumulated to a certain date, meaning thereby that it is put at interest until the date in question; on the other hand, we say that a sum of money is discounted to a certain date, meaning that we seek to find the value of the sum at the date in question as compared with its value given at some assigned later date. The value now of a sum due at some future date, with or without interest, is called the present worth, or present value. The present value of a sum due at some future date may also be denned as that sum which, put at compound interest now, would amount to the sum in question on the given future date. The processes of accumulation and discounting are then exactly the reverse of each other. A clear recognition of this fact will be of the greatest aid in understanding future sections. PROBLEM. To find the present value of 1 due in n years without interest. Let »„ be the required present value, and let i be the rate of interest that might be obtained if the unit principal were in hand to put at interest. By the definition of present value, we have ^(1+0-=1; whence v„=—1—. /i^ (!+<)" (1) If the principal be not 1, it may be denoted, as in the interest formulas, by P. Let F, denote the present value of -P due in n years; then FM=JP?^^=JP(l+()-ra (2) INTEREST 69 ILLUSTRATIVE EXAMPLE. Find the present value of $2365.29 due 8 years hence, if money could be invested at 5%. Solution. By formula (2) Vs= $2365.29 (1.05)-8. The value of the product on the right may be obtained by means of a table of six-place logarithms or by means of a table of present values of $1, from which the value of (1.05)- 8 is taken directly. If neither the table of logarithms nor the table of present values is at hand, the value of (1.05)-3 may be obtained by the binomial expansion. Using the last-named method, we have (1.05)- ' == 1 + (- 3) (.05) + ^^'"g3"'^ (-05)2 (-8) (-3-1) (-3-2) 1.2.3 ' ' Six terms of this series give the result (1.05)-a= .863837 + .... Consequently, TS = $2365.29 x .863837 = $2043.22. PROBLEM. To find the present value of 1 due n years hence with Interest. Let k be the rate of interest borne by the unit principal, i the current rate, and v^ the present value. In this problem the sum due in n years is not 1, but the amount of 1 for n years at rate A, viz. (1 4- A)". Consequently, ^(i+0"=(l+^" , (i+A-y .„. and ^(l+iy- (3) If the principal is -P and the corresponding present value is V,[, ^=j>(l±^. . (4) » (1+0" v ' Formulas (1)-(4) may be expressed in terms of the nominal rates j and V by means of the fundamental relations ^-(^ (V\'w and l+k= 1+^