Art. LX.—On Sinking Funds.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 2, 1869, Page 236

Art. LX.—On Sinking Funds. By Captain F. W. Hutton, F. G. S. [Read before the Auckland Institute, September 7, 1868.] The subject of Sinking Funds is one of much importance to this and other countries, but I have not been able to find any book that treats of it, and I therefore think that an investigation of its principles may prove both useful and interesting. By “Sinking Fund” is meant a sum of money put away annually in order to pay off a loan. There are two principal ways in which this money is applied: either it may be invested year by year until, with the interest accruing on it, it amounts to a sum sufficient to pay off the original loan; or else it may be used to take up yearly a portion of the loan until the whole has vanished. I propose to investigate both these methods, and then compare them together. The second case, where the fund is applied yearly to buy up the loan, is very simple. Let a equal the amount of the loan. — p the amount put by as Sinking Fund each year, and — T equal the number of years it will take to pay off the loan. Then

it is evident that as 1/p of the loan is brought up every year, T=a/p   (1) P=a/T   (2) a=pT   (3) With regard to the first case: let a and p be as before, but let t be the number of years it will take to pay off the loan by this method, and let v equal 1+ the interest on one pound at the rate at which the Sinking Fund is invested, so that if it is invested at 5 per cent. it will equal 1·05. Now at the end of the 1st year the fund will amount to p v 2nd" (p+p v) v=pv + p v2 3rd" (p v+p v2 +p) v = pv + p v2 + p v3 tth" p v + p v2 +p v3 + &c … … . p v t but at the end of the last or tth year, the fund must equal a ∴ p v + p v2 +p v3 + &c. . p v t multiply by v and subtracting pv2+pv3+4+&c… …pvt+1=av/p vt+1 − pv = av −a ∴ a(v−1)=p v(vt −1)   (4) ∴ a=pv(vt −1)/v−1   (5) p=a(v−1)/v(vt − 1)   (6) When p is known the per centage required for forming a Sinking Fund equal to p can be found by multiplying p by 100 and dividing by a. From (4) we get vt −1=a(v−1)/pv ∴vt=a(v−1)+pv/pv t log v=log a(v−1) + pv − log p v ∴ t=log a(v−1) + pv log pv/log v   (7) From (4) we also get pvt +1−(a+p) v+a=0   (8) From which v can be found by the following rule, known as Bernoulli's. 1. Find by trial two numbers nearly equal to t. 2. Substitute these assumed numbers for t and mark the error that arises from each with + if too great, and—if too small. 3. Multiply the difference of the assumed numbers by the least error, and divide the product by the difference of the errors when they have like signs, but by their sum when they have unlike. 4. Add the quotient to the assumed number belonging to the least error when that number is too little, or subtract if too great.

5. This operation may be repeated until t is found sufficiently near. I will now take the total amount of interest that has to be paid on the loan until it is all taken up. This on the first system will evidently be a t(v′—1), where v′ is 1+ the interest that has to be paid on one pound of the loan for a year. On the second system the interest payable at the end of the 1st year would be (a—p) (v′—1) 2nd "(a—2p) (v′—1) 3rd " (a—3p) (v′—1) (T—1)" (a—(T—1)p)(v′—1). But the year before the whole loan was taken up only 1/pth of it would be left, it is evident that a—(T—1) p=p So that we have an equidifferent series of which (a—p) (v′—1) is the first term, p (v′—1) the last, and T 1 the number of terms. The sum of them therefore, or the whole interest to be paid on the loan T−½ (a−p) (v′−1)+p(v′−1) =a(v′−1)/2(T−1) Therefore a(v′−1)t:a(v′−1)/2(T−1):: amount of interest by first method : amount of interest by second method Or 2t:T−1 :: …. .: … … But besides the interest on the loan there has also to be paid for the Sinking Fund by the first method p t pounds, and by the second method p T pounds. So that pt : p T :: {amount paid for Sinking Fund by first method} : {amount paid for Sinking Fund by second method And combining the two we get 2t+ pt: T—1 +p T :: {whole amount paid by first method} : {whole amount paid by second method Or (p+2) t: (p+1) T—1 :: … … . Now the limits of p are o and a, and as it gets small both T and t increase, but t will increase slower that T for it also depends upon the value of v which remains stationary. On the contrary as p gets large t will decrease more slowly than T for the same reason, and the position of equality will of course depend upon the values of v and a. If however we take a>1000; p<a/14, and v=1·05—which in practice will include all cases—it will be found that (p+2) t<(p+1) T—1. The actual amount that would have to be spent by either method can be easily found by substituting in the following formulæ the different values for a, p, v, and v′.

By the first method {a(v′−1)+p}.log{a(v−1)+pv}/logv. By the second method a(v′−1)(a−p)/2p + a From this comparison it follows that when money can be invested at 5 per cent., and the Sinking Fund is less than 7 per cent. of the loan, the first is the more economical method; and the smaller the Sinking Fund, and the higher the rate of interest, the greater will be the saving effected by investing the fund in other securities, than by using it to buy up annually part of the loan. This however is only the mathematical or pecuniary view of the question; from the political point of view many reasons can be given why the second method should be preferred, and the difference pecuniarily is not sufficiently great to override them.