PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

THE APPLICATION OF FUNDAMENTAL VALUATION PRINCIPLES 325If we assume that both operating earnings and capital grow atconstant rate g, thenandsoOE x =OE x¡1 £ (1 + g)=OE 1 £ (1 + g) x¡1C x = C x¡1 £ (1 + g)=C 0 £ (1 + g) x¢C x = C x ¡ C x¡1 = C x¡1 £ g = C 0 £ (1 + g) x¡1 £ g:Substituting into Equation DCF-2, the DCF value becomes1XValue = FC 0 + OE 1 £ (1 + g) x¡1 £ (1 + h) ¡x¡x=11XC 0 £ g £ (1 + g) x¡1 £ (1 + h) ¡x :x=1(DCF-3)By factoring out the constants, this equation is rewritten asValue = FC 0 + OE 1X1·(1 + g)¸x¡1(1 + h) (1 + h)x=1x=1¡ C 0 £ g1X·(1 + g)(1 + h) (1 + h)¸x¡1:(DCF-4)Note that g, the growth rate, will always be less than h, thehurdle rate. As a result, the sum of the infinite geometric seriescan be solved easily as A ¥ (1 ¡ R), where A is the first term inthe series and R is the multiplicative factor used to generate thenext term in the series. The sum converges to1(1 + g)1 ¡(1 + h)= 1+hh ¡ g :When we substitute this into Equation DCF-4, the (1 + h)terms cancel, so the formula for value based on a DCF approach