Value+of+the+firm

=**Definition & Explanation:**=

According to the text, Managerial Economics and Business Strategy by Michael R. Baye, the value of a firm is the present value of the firm’s current and future profits.

The value of a firm is linked to profit maximization. A firm looking to maximize their profits is actually concerned with maximizing its value. As such, it is important for a firm to be able to determine its present value accurately.

The value of a firm can be simplified using time value of money principles. Thus, the value of a firm is defined as the present value of expected future cash flows plus current cash flows. In this case, we will assume the expected cash flows to be equal to the expected profits for the firm. In order to calculate the value of the firm most companies discount the expected future profits to today using a given interest rate, i, and then add in the current profits.

The equations below can be used to determine the PV of a firm based on current and future profits:

=**Equation #1: General Equation**=


 * PV(firm)= p(0) + [ p(1) / (1+i) ]+ [ p(2) / (1+i)^2 ]+ [ p(3) / (1+i)^3 ]…….[p(x) / (1+i)^x]**

p(0) = profit p(x) = profit for X years out i = interest rate

The equation above gives the best estimate of a firm’s value. However, the firm must have estimates of future profits to use the equation. If the firm does not have future profit estimates, they can determine its value using the constant growth equation (see equation #2 below). Because firms do not have a "maturity" like other profit streams may, we can consider this to be a perpituity (extending forever - or at least for an indefinite time period).

In order to understand how the value of a firm can change and what variables are linked to value of a firm, we must make a few assumptions. First, we assume that the company's profits will grow at a constant rate indefinitely (g). Also, we must assume that profit growth is less than the interest rate ( g < i ). We can then use with the constant growth equation.

=**Equation #2: Constant Growth Equation**=


 * PV(firm) = p(0) + [ p(0) (1+g) / (1+i) ]+ [ p(0) (1+g)^2 / (1+i)^2 ]+ [ p(0) (1+g)^3 / (1+i)^3] …. [ p(0) (1+g)^x / (1+i)^x**


 * __Simplified:__**
 * PV(firm) = p(0) * [ (1+i) / (i-g) ]**

p(0) = profit for the current year i = interest rate g = growth rate

One final scenario in which the value of a firm can be estimated is when current profits have already been paid out to the shareholders in the form of a dividend. In this case, the value of a firm is calculated as follows:

=**Equation #3: Dividend Equation**=


 * PV(exdividendfirm) = PV(firm) - p(0)**

PV(exdividendfirm) = p(0) * [ (1+g) / (i-g) ]**
 * __Simplified:__

Therefore, as long as our assumption that the interest rate and growth rate are both constant holds, maximizing profits will also maximize the value of the firm.

=**Real World Examples:**=

Stock Valuation - stocks are valued in the exact same way as seen above. The only difference is that it is the value of a firm on a per share basis.

NPV Analysis - Many managers are faced with important decisions on whether to accept or reject costly projects or which alternative may be better for the company. They do a very similar analysis pertaining to cash flows associated with the project alone.

=**Questions:**=

1) If a firm’s current profits are $10,000 and the firm is expected to earn $10,500 in profits in each of the next 3 years, What is the value of the firm in present term. The interest rate is 8%.

A) 41,500 B) 39,170 C) 37,060 D) 40,000

The answer is C. The following equation is used to determine the firm’s value: PV(firm)=p(0) + [p(1)/(1+i)]+ [p(2)/(1+i)^2]+ [p(3)/(1+i)^3], where p=10,00 p(1), p(2) and p(3)=10,500, and i=.08. PV(firm)=10,000+[10,500/(1+.08)]+ [10,500/(1+.08)^2]+ [10,500/(1+.08)^3] PV(firm)=10,000+9722+9002+8335 PV(firm)=37,060

2 )Determine the maximum value of the a firm when the firm’s current profit is $20,000, the constant growth rate is 6%, and the interest rate is 7%.

A) 3,000,000 B) 20,000 C) 2,500,000 D) 2,140,000

The answer is D. The following equation is used to determine the firm’s value: PV(firm)=p(0)[(1+i)/(i-g)], where P(0)=20,000, i=.07, and g=.06. PV(firm)=20,000[(1+.07)/(.07-.06)] PV(firm)=20,000[1.07/.01] PV(firm)=20,000*107 PV(firm)=2,140,000

3) Which of the following statements is true regarding a firm’s value?

A) The value of a firm is the sum of its expected profits. B) The value of a firm is the sum of the PV of its current and future profits. C) The value of a firm is its current profit.

The answer is B. The present value of a firm takes into consideration both the current and expected profits of a firm.

4) A company expects to earn $1,000,000 per year indefinitely. The associated interest rate is 7%. What is the value of this firm?

A) $934,579.44 B) $14,285,714.29 C) $1,070,000.00 D) $1,075,268.82

The answer is B. (1,000,000 / .07 = 14,285,714.29)

5) Put firms A, B, and C in descending order based on thier firm values.
 * || Firm A || Firm B || Firm C ||
 * growth rate || 5% || 3% || 4% ||
 * interest rate || 12% || 10% || 8% ||
 * current year profit || $400,000 || $500,000 || $450,000 ||

A) C A B B) C B A C) B A C D) B C A

The answer is B. Using the simplified version of Equation #2 above, PV(firm) = p(0)[(1+i)/(i-g)], The value for Firms A, B, and C are $6.4 million, $7.86 million, and $12.15 million respectively. Therefore in descending order that would be C, B, A.

6) A company is valued at $13.6 billion. It's interest rate is 9%. Current profits are expected to grow at 5% indefinitely. What are the company's current profits?

A) $680,000,000 B) $499,000,000 C) $1,224,000,000 D) $544,000,000

The answer is B. The equation to use is Equation #2 above, PV(firm) p(0)[(1+i)/(i-g)]. Except this time we know PV(firm) and are asked to solve fro p(0). Rearranging algebraically, p(0) PV(firm) / [ (1+i) / (i - g) ]. After doing the math, the current years profits can be estimated to be $499 million.

7) John Buck, Inc. earned $420,000 in profits this year which they paid out as a dividend. Their interest rate is 5% and their growth rate is 4.5%. There are currently 1,000,000 shares of common stock outstanding. What is the stock price for John Buck, Inc.?

A) $88.20 B) $88.00 C) $87.78 D) $84.00

The answer is C. Using Equation #3, PV(exdividendfirm) = p(0)[(1+g)/(i-g)], the value of the firm can be estimated at $87.78 million. You then must divide the value of the firm by the number of shares outstanding to get the stock price for the firm.

Sources: Managerial Economics & Business Strategy, Bay, Michael R., 5th Ed, McGraw Hill 2006, p16-19