AN ACTUARIAL ANALYSIS OF THE LOAN SCHEMES 

 OF CT^RTALN RAND RUH.DING SOCIETIES. 



H\ I'n.f. jon.x I'atku K Dalton, M.A., D.Sc. 



§1. Rand Biiildiiii;- Societies cater for three groiii)s (if 

 clients : — 



( /') Soi'iiigs Bank Dcfusitors, who are content with a comparatively 

 low rate of interest (4 to 44 per cent.) in return for the greater 

 security offered by lirst claim on the assets ; 



iii) hivcstors. who take sliarcs in the Society and run greater risks 

 in the Iione of realisino- larger profits (8 to Q per cent.) ; and 



(in) Bonxnti'crs. tt) whom the Society lends the capital provided l)y 

 depositors and investors, and who, having to provide the inter- 

 est thereon as well as the expenses of management and ordi- 

 nary profits (if any) of the Society, are called upon to pay 

 interest at still higher rates (10 per cent.). 



§2. Before entering ti])on detailed criticism, .'i little theory 

 might well be reviewed. Whenever money is advanced on mort- 

 gage of depreciating securities, as is the case in Building Society 

 operations, the loan is generally redeemed by a ])rocess of amor- 

 tisation. The theory of the operation is sim])le. 'Jdie loan repay- 

 ments constitute a terminable annuity, of which the borrower is 

 the vendor, and the Building Society the jntrchaser. The jnir- 

 chaser expects to obtain interest on his loan at a definite rate, 

 and, furthermore, to have his cajjital still intact at the end of the 

 term. Each payment, therefore, made by the borrower must be 

 regarded as consisting of two jjarts : one, the interest element, 

 providing interest on the loan at the rate agreed ujjon ; and the 

 other, the capital elemeiU, or sinking fund, constititiing a repay- 

 ment of capital. The sinking fund may be treated in either of 

 two ways ; it may be directly applied to the reduction of the capi- 

 tal debt, thereby reducing the interest element and increasing 

 the capital element contained in Ihe succeeding instalment; or, 

 it may be carried to a separate capital redemj^tion account, there 

 to accumulate at the given rate until at the termination of the 

 annuity, it amounts to the original value of the loan. 



As long as a single rate of interest convertible with the 

 same frequency is involved, there is no difference between these 

 methods of treating the annuity i^ayments ; it is merely a matter 

 of accounting. Thus, a loan of ii is repaid by /; equal periodical 

 instalments. iiUerest being at tlie rate of £i per £ per period. The 



n "' +'-*" 



instalment payable is X . Of the first such jiayment 



*^( T -f i)"~~ I 



interest absorbs £{, and the rest, say £f, goes to the reduction of 

 capital. If the debt is diminished at once by this amount, the 

 capital element contained in. the next instalment is equal to / 

 plus the interest saved by the reduction of the debt — that is to 

 say, the cajiital element in the second payment is / ( i + i) ; this 



