336 On the Value of Leases. [Nov. 



(/i) years on the payment of a fine of (b). What is the lease 

 worth, considering it as a perpetuity subject to the ground rent 

 and renewal fines, and allowing interest at r per cent. ? 



Solution. MuhuOit: 



Let,^^5=,,the amount of U, at the end of ^orte year/iift&^\)per 



> -"■. ^ ^ I • , '^'/w^v-xsnrtN^ 



ax 1 — :j7- ^ 



cent.; then will ^ ^ — be the present value of an'lannuity (a) 

 to continue for (m) years. But if (m) be infinite^ ithen will 

 =^ be = 0, and the present value of a perpetual annttity d ^ill 



be 



R-l' 



At the end of m years, the amount of the fines and the inte- 

 rest thereon will be 6 x R"' + 6 x R'"-" + /:> x R'"-'^", &c. 

 = 6 X (R'" + R—" + R'"-^", &c.) = F. Let p represent 

 the number of times //. is contained in m without remainder, and 

 put R*" + R""-" + R"-'"- ...... R— "» = S. Then dividing 



this equation by R* we' have R'"-'* 4- R™-*'' + R"*-»* , ; , . . 



j^m -y + • . » _ _^, ; ignd^^^ b>y subtracting the latter e(^ualti<in < frbm 

 the former>^ Rn^^^R--^^^^'* = S -. |-, and S = ^"'"^"".^^ ^ 



; ' t -K" , i 



~R^ 



. F, the amount of the fines and interest thereon will therefore 



, , Rm— Rm-p + l-n. , !. 



be, = o X — — , and smce 1/. present mon^y is equal 



to R* to be paid at the end of m years, R* : 1 :: — — — —— : 



■ , the present value ot the nne^j,^ j / 



will b6''te'd too, -and the quantity^ ;.L^ ' ! i, .Ua! /l l^f ffigiJ9?'^jf?}^®^y 



^^ R" ^ ^'^ "jdiiijHdbiiuui 

 small will disappear from the equation, and p Wl\belthe 



present value of the finWs oh a perpetuity. The present value of 

 the lease will, therefore, be r^-^ — — — r. W. B. 



