Question

Determining PB Ratio for Companies with Different Returns and Growth

Assume that the present value of expected ROPI follows a perpetuity with growth g (Value = Amount/ [r - g]). Determine the theoretically correct PB ratio for each of the following companies A and B.

Company	Net Operating Assets	Equity	RNOA	ROE	Weighted Avg. Cost of Capital	Growth Rate in ROPI
A	$100	$100	18%	18%	10%	2%
B	$100	$100	11%	11%	10%	4%

Round answers to two decimal places.

	PB Ratio
Company A	Answer
Company B	Answer

Answer 1

Solution :

ROPI for company A = $100 ( 18% - 10%) = $8

ROPI for company B = $100 * (11% - 10%) = $1

Growth rate of ROPI for company A = 2%

Growth rate of ROPI for company B= 4%

Present value of ROPI for company A = [$8 / 1.10] + [$8 * (1.02) / (1.10)^2] + [ [$8 * (1.02)^2 / (1.10)^3]...............upto infinity terms

This is sum of infinity terms of GP, therefore

a (first term) =  $8 / 1.10

r (Common ratio) = 1.02 / 1.10

Sum of infinity terms of GP = a / (1-r)

= ($8 / 1.10) / [1 - (1.02/1.10)] = $100

Therefore market value of equity for Company A = $100 + $100 = $200

Present value of ROPI for company B = [$1 / 1.10] + [$8 * (1.04) / (1.10)^2] + [ [$8 * (1.04)^2 / (1.10)^3]...............upto infinity terms

This is sum of infinity terms of GP, therefore

a (first term) =  $1 / 1.10

r (Common ratio) = 1.04 / 1.10

Sum of infinity terms of GP = a / (1-r)

= ($1 / 1.10) / [1 - (1.04/1.10)] = $16.67

Market value of equity for company B = $100 + $16.67 = $116.67

PB ratio - Company A = Market value / Book value = $200 / $100 = 2

Company B = $116.67 / $100 = 1.17