Problem Set 2



Solutions: Second Group

Moeller-Finance

1. a. Capital structure refers to the specific mixture of debt and equity the firm uses to finance its operations. The book value of debt and equity simply show the historical cost. The management is more concerned with the market's current perception or valuation of the firm. Specifically, management is concerned with the market value of equity.

b. The primary goal of my management team is to maximize the current market value of the stock. To maximize the current stock value, management should identify and fund investment projects that increase shareholders' value (NPV>0).

c.(i) No. The financial statement reveals that NorTex has severely declining operating cash flows and the stock price has fallen. To maximize the current value of shareholders' equity, you must identify and fund investment projects that increase the shareholders' value. The CEO of NorTex has purchased a corporate condominium and plane which was not productive. He should use these funds for the better investment opportunities.

(ii) Control of the firm ultimately rests with shareholders. They elect the board of directors, who, in turn hire and fire the management. So, shareholders' can replace the CEO by using the board of directors. Additionally, the labor market for upper management will evaluate the performance of these displaced managers and value them accordingly. Another way is by takeover. Those firms that are poorly managed are more attractive as acquisitions than well managed firms because a greater profit potential exists. The shareholders could also give the management some form of managerial compensations based on performances such as stock options and performance shares.

2. In order to see whether we are going to sell the stock or buy more of it, we have to compare the current stock price ($20) with the present value of all future cash flows. Because there are cash flows with different trends, we have to calculate the present values separately. Namely, we have 3 different trends: (1) $2.50 in year 2; (2) $2.75 in year 3 growing 5% for 10 more years; (3) 2% growth thereafter.

Let’s look at (1) first:

PV ( (1) ) = $2.50 / (1+.14)2

PV ( (1) ) = $1.92

Now we can calculate the (2). Starting from year 3 we will receive $2.75 growing 5% annually till year 13. We will use the growing annuity formula to calculate the value of the cash flow at year 2 and then we will discount that amount for two years to bring it to current prices. Thus,

PV ( (2) ) = [1/(1+.14)2] * {[$2.75/(.14-.05)] * [1 – ((1+.05)/(1+.14))11]}

PV ( (2) ) = $14.00

Note: The above formula is the growing annuity formula with 1/(r-g) pulled outside of the brackets.

Finally, we can look at (3). First of all, we have to calculate the cash inflow in year 14 to see the first cash flow of the perpetuity. From year 3 there will be a growth of 5% for 10 years till year 13, then the growth will decrease to 2%. Thus,

Cash Flowt=14 = $2.75 * (1+.05)10 * (1+.02)

Cash Flowt=14 = $4.57

We can now use this value to calculate the present value of the perpetuity. We will use the perpetuity formula to calculate the value of the cash flows at year 13 and then we will discount that amount for 13 years to bring it to current prices. Thus,

PV ( (3) ) = [1/(1+.14)13] * [$4.57/(.14-.02)]

PV ( (3) ) = $6.93.

Therefore, the sum of present values of cash flows will be

PV (Cash Flows) = $1.92 + $14.00 + $6.93 = $22.85.

The current price ($20) is less than the present value of cash flows ($22.85), thus ignoring transaction costs, we should buy more stock of this company because we believe that eventually the stock price will increase to $22.85.

3. Method one

We can look at the problem from the dividends perspective.

The firm is currently paying a dividend amount of

$240,000 / 60,000 shares = $4 per share.

If the firm decides to stay a cash cow, i.e. it pays out all its earnings to shareholders; its stock price will be the perpetuity of $4.00 at 10% required rate of return. Thus,

PriceCash Cow = $4 / .1 = $40.

If the firm undertakes the investments, the dividends will decrease to

$4 * .8 = $3.20

because 20% of the earnings will be spent for investments. Due to the return on the investments, the growth rate in dividends will be (using growth rate = retention ratio * return on equity)

g = .2 * .12 = .024.

Finally, we can use the growing perpetuity formula in order to calculate the stock price of the firm if it undertakes these investments. Thus,

PriceWith Investments = $3.20 / (.1–.024)

PriceWith Investments = $42.11.

Therefore, the difference between the price after investments are undertaken ($42.11) and the initial price ($40) will gives us the net present value of growth opportunities. Thus,

PVGOper Share = $42.11 – $40 = $2.11 per share.

Method two

We can look at the problem from an investments perspective.

In the first year, the firm will invest 20% of $240,000 earnings, which will bring a return of 12% starting from year 2 onwards. Thus,

Costt=1 = $240,000 * .2 = $48,000

Returnt=2,3,… = $48,000 * .12 = $5,760.

If we calculate the value of this investment at year 1, we get

Net Valuet=1 = –$48,000 + $5,760/.1

Net Valuet=1 = $9,600.

In year 2, the return from the first investment $5,760 will be realized so the earnings will increase to

Earnings = $240,000 + $5,760 = $245,760.

If we use the similar methodology for the first investment, now we will invest

Costt=2 = $245,760 * .2 = $49,152

Returnt=3,4,… = $49,152 * .12 = $5,898.24.

If we calculate the value of this investment at year 2, we get

Net Valuet=2 = –$49,152 + $5,898.24/.1

Net Valuet=2 = $9,830.40.

Net value of investment 2 is

(9,830.40/9,600) – 1 = .024

times greater than net value of investment 1 (which is the same with the growth rate of dividends in method one) and we could avoid calculating the year 2 investment by finding the growth rate as in the previous method.

Therefore, we can create a growing perpetuity at a rate of 2.4% with all these investments starting with the value in year 1 ($9,600). Thus, we will end up with the net present value of growth opportunities, which is

PVGO = $9,600 / (.1–.024) = $126,315.79.

Per share net present value of growth opportunities will be

PVGOper Share = $126,315.79 / 60,000 shares = $2.11 per share.

4.

a. We have to look at the present value of all future cash flows (at 10% required rate of return) in order to calculate and compare the prices of stocks A, B and C.

The dividend trend of stock A is a zero growth perpetuity. Thus,

PriceA = $10 / .1

PriceA = $100.

The dividend trend of stock B is a growing perpetuity (at 4% growth). Thus,

PriceB = $5 / (.1–.04)

PriceB = $83.33.

Finally, the dividend trend of stock C is a mixture of a growing annuity (at 20% growth) and then a perpetuity with no growth. Let’s first calculate the first cash inflow of the perpetuity. $5 next year will grow at 20% for 5 years then the growth rate will be zero. Thus,

Dividendt=7 = $5 * (1+.2)5 * (1+0) = $12.44.

Now if we combine the annuity and the perpetuity, we get

PriceC = [5/(.1–.2)] * [1 – ((1+.2)/(1+.1))6] + (1/(1+.1)6) * ($12.44/.1)

Growing Annuity Part Perpetuity Part

PriceC = $34.28 + $70.23 = $104.51.

We can conclude that the stock C is the most valuable among the three stocks at 10% required rate of return. (Note: The growing annuity formula has 1/(r-g) pulled outside of the brackets.)

b. If we replace the 10% required rate of return in the previous analysis with the 7% required rate of return, we can calculate the new prices.

For stock A, we have

PriceA = $10 / .07

PriceA = $142.86.

For stock B, we have

PriceB = $5 / (.07–.04)

PriceB = $166.67.

Finally, for stock C, we have

PriceC = [5/(.07–.2)] * [1 – ((1+.2)/(1+.07))6] + (1/(1+.07)6) * ($12.44/.07)

Growing Annuity Part Perpetuity Part

PriceC = $38.06 + $118.43 = $156.49.

We can conclude that the stock B is the most valuable among the three stocks at 7% required rate of return.

5. Consider a firm with existing assets that generate EPS of $5. If the firm does not invest except to maintain existing assets, EPS is expected to remain constant at $5 a year. However, starting next year the firm has a chance to invest $1 per share a year in developing a newly discovered source for electricity generation. Each investment is expected to generate a permanent 20% return. However, the source will be fully developed by the fifth year. Assume the investors require a 12% rate of return.

a.        The stock price if the firm chooses to not invest is:

PriceCash Cow = EPS / r

PriceCash Cow = $5 / .12 = $41.67.

b. Calculate the PVGO if the firm has the same investment opportunity for 5 years maintaining a 80% retention ratio.

The PVGO is worth $12.79. The method is similar to part d.

The first investment is represented by a NPV = $-4 + [$0.8 / 0.12] = $2.667

The investment is 80% of $5 earnings = $4 and the return is 20% forever on $4 = $0.80.

Using the short cut of the present value of a 5 year growing annuity, we can simply solve this problem. The growth rate here is represented by 0.8 * 0.2 = 0.16 (However, only for 5 years!). We can therefore use a growing annuity to value all 5 right away:

PVGO(years 1 - 5) = [$2.667 / 0.12 - 0.16] *  [ 1 - (1.16 / 1.12)5] = $12.79.

c. Calculate the PVGO if the firm has the same investment opportunity forever maintaining a 80% retention ratio.

Is this type of investment opportunity possible forever (where g is greater than r)? NO! Essentially this implies that the new investment will be perpetually undervalued by the market. Though it is possible for an investment to be undervalued by the market for a short period, in a competitive, efficient market an investment can not be perpetually undervalued.

d.  Calculate the PVGO if the firm would have had the same investment opportunity for 5 years maintaining an 20% retention ratio.

The first investment is represented by a NPV = $-1 + [$0.20 / 0.12] = $0.667

The investment is 20% of $5 earnings = $1 and the return is 20% forever on $1 = $0.20

The second investment is represented by NPV = $-1.04 + [$0.208 / 0.12] = $0.693

The earnings have grown in this year by $0.20 to $5.20, of which 20% is reinvested, so $1.04 is reinvested now generating a 0.2 * $1.04 = $0.693 perpetual return.

This we can continue for the remaining 3 years and calculate the net present value of each of the 5 investments. Alternatively, realizing that the growth rate here is represented by 0.2 * 0.2 = 0.04 (However, only for 5 years!), we can see that each year the net present value is increasing by 4% as well. We can therefore use a growing annuity to value all 5 right away:

PVGO(years 1 - 5) = [$0.667 / 0.12 - 0.04] *  [ 1 - (1.04 / 1.12)5] = $2.58

e.  ADDITIONAL PART: Calculate the PVGO if the firm would have had the same investment opportunity forever maintaining an 20% retention ratio.

If the firm could do this investment scenario for every year, it would imply we could use the perpetuity formula pricing dividends:

Dividend approach:

P0 without growth = $5 / 0.12 = 41.67

P0 with growth = $4 / (0.12 -0.04) = $50

Hence, PVGO = $50 - $41.67 = $8.33

Investments approach:

Alternatively, the PVGO from part d becomes a growing perpetuity:

PVGO(years 1 -5 ) = [$0.667 / 0.12 - 0.04] = $8.33

6.

[pic]

[pic][pic]

The first part in the bracket is to calculate the PV at the end of year one by using the first two years dividends value with an increase rate of 5%.

The second part in the bracket is to calculate the PV at the end of year one by using the second four years dividends value with an increase rate of 1%.

The third part in the bracket is to calculate the PV at the end of year one by using the dividends value with no increase after 6 years.

The last part of this formula, which is outside the bracket, is to calculate the PV at time zero.

7. Face Value of the Bond = $1,000; Coupon rate = 6 %; Period (t) = 5 years; and Yield-to-Maturity (YTM) = 8 %

The annual interest payment is 6% of the $1,000 face value, or $600.

The current bond price is the present value of the coupon payments plus the present value of the face value. Because the coupon payments are a series of constant payments, we can use the present value annuity formula to calculate the present value of the coupon payments.

Bond Value = PV(annuity) + PV(face value)

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8.

EPS=5 Million / 1 Million = 5

[pic]

Without the investments, there is no increase opportunity for this firm, so we can compute its value by using:

[pic]

Value of total assets [pic] Million shares = $ 41.67 Million

If adopts the first investment plan, we get:

Investment[pic],

which generates [pic]

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[pic]Investment rate [pic]

[pic]

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Total value = 42.86 * 1Million shares =$ 42.86 Million

If adopts the second investment plan, we get:

Investment [pic], which generate [pic]

We can convert these numbers to per share which would be a $5 investment and $0.75 cash flow.

So we need to calculate the PV of the investment:

[pic]

We also need to discount this amount to time zero:

[pic]

Value of the firm = 41.67+1.00 = $ 42.66 per share or $42.66 million total value

Since the first investment plan can generate more value to the company, which means generate more value to shareholder’s wealth, this plan should be recommended to CEO.

9a. Given the available information, you can estimate the cost of equity for XYZ by using the dividend growth model. Remember we know that analysts are typically optimists so I’m going to divide the predicted growth rate by 2 to get something that is more historically accurate so g equals 10% (0.2/2).

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b. Given the available information, you can estimate the return on debt by using the CAPM (basically as soon as you see a beta, you should think of the CAPM as a possibility). The CAPM shows that the Rd=0.03+0.5(0.1)=0.08 or 8%.

10. Equivalent annual cost is the present value of a project’s costs on an annual basis. To compare equipment with different economic lives, the costs need to be put on an annual basis such that the two machines may be compared. To do this we need to compute equivalent annual costs (EAC). First calculate the present value of the costs using the discount rate of 15%. Since equal operating costs occur each year, the present value of these costs can be determined using the present value of annuity formula for an eight year annuity. The initial costs can be added to the present value of the annuity to obtain the total present value of the costs.

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[pic]

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Thus, the equivalent annual cost of Machine A is $49,573.57.

b. The EAC of Machine B can be determined in a similar manner to Machine A. Here we need to find the present value of the operating costs of $6000 per year for 10 years.

[pic]

The EAC is computed using an annuity factor for a ten year annuity.

[pic]

[pic]

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The equivalent annual cost of Machine B is $ 65,772.58.

c. Machine A would be the better purchase because it effectively costs $49,573.66 per year versus $65,772.58 for Machine B. By choosing Machine A, you are effectively saving $16,198.98 per year.

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