6.1. Carbondale Company has the following credit policy 2/10, net 30. Carbondale also charges 1% per month interest on all accounts after 30 days. The following table shows the collection schedule of all sales:

Collection within	10 days	30 days	60 days	90 days
Percentage	10%	30%	40%	20%

To improve the collection rate, Carbondale is thinking of imposing a higher interest rate, 1.5% per month, on all accounts paid after 30 days. Carbondale believes that the new policy will change the collection schedule as follows:

Collection within	10 days	30 days	60 days	90 days
Percentage	10%	50%	30%	10%

There will be no change in the total sales as a result of this new policy. The cost of capital for Carbondale is 12%. Should it try the new policy?

The instructor included the following answer.

NPV(old) = .9899, NPV(new) = .9921, yes. (Answer)

6.2. Dickson City Company has annual sales of $5 million, while the cost of goods sold is $3.2 million. All sales are made on a cash basis. The owner of Dickson has come up with the plan of giving credit to the customers. He believes that this will increase the sales by 25% without increasing any of the fixed costs. He thinks that 20% of the customers will pay within 30 days, 40% within 60 days, 37% within 90 days, and 3% of the customers will default on the sales. The cost of capital to Dickson is 12%.

(A) Should Dickson City introduce the policy of credit sales?

NPV(cash) = $1.8 million, NPV(credit) = 1.941 million, yes. (Answer)

(B) The manager of the firm doubts whether the sales will actually increase by 25% as a result of this strategy. Find the minimum increase in sales to justify introduction of the new credit policy.

15.92% (Answer)

6.3. Ashley Company is considering the credit application of a retail customer who is expected to buy $1000 worth of merchandise every month. The cost of these goods will be $800. The customer is expected to pay after 30 days every month. However, there is a 10% probability of default each month. In case of default, the company will recover 50% of the unpaid bill after 3 months. The cost of capital to the company is 15% per annum. Should Ashley extend credit to the customer?

NPV = 1234, yes (Answer)

6.4. First National Bank of Jermyn has a portfolio of 10,000 credit card accounts. The bank charges $25 annual fee on these cards. There is a 25 day grace period on the accounts, and after that the cardholders pay interest at the rate of 1.25% per month on the unpaid balance. Half of the cardholders pay their balance in full every month, and their monthly bill is $800. The remaining cardholders carry an average balance of $1200 continuously. The operating expenses for the credit card portfolio, including defaults, are $100,000 annually. The cost of capital to the bank is 8%. Mellon Bank has offered to buy Jermyn's credit card portfolio for $5 million, plus the outstanding balance. Should Jermyn accept the offer?

The instructor provided the following answer NPV = $3.971 million, yes

I have included the book if you need a reference. The problems are from chapter 6.

ERP 513: Treasury Management Assignment Week 6 _____________________________________________________________________________ Assignment Week 6 6.1. Carbondale Company has the following credit policy 2/10, net 30. Carbondale also charges 1% per month interest on all accounts after 30 days. The following table shows the collection schedule of all sales: Collection within Percentage 10 days 10% 30 days 30% 60 days 40% 90 days 20% To improve the collection rate, Carbondale is thinking of imposing a higher interest rate, 1.5% per month, on all accounts paid after 30 days. Carbondale believes that the new policy will change the collection schedule as follows: Collection within Percentage 10 days 10% 30 days 50% 60 days 30% 90 days 10% There will be no change in the total sales as a result of this new policy. The cost of capital for Carbondale is 12%. Should it try the new policy? NPV(old) = .9899, NPV(new) = .9921, yes. 6.2. Dickson City Company has annual sales of $5 million, while the cost of goods sold is $3.2 million. All sales are made on a cash basis. The owner of Dickson has come up with the plan of giving credit to the customers. He believes that this will increase the sales by 25% without increasing any of the fixed costs. He thinks that 20% of the customers will pay within 30 days, 40% within 60 days, 37% within 90 days, and 3% of the customers will default on the sales. The cost of capital to Dickson is 12%. (A) Should Dickson City introduce the policy of credit sales? NPV(cash) = $1.8 million, NPV(credit) = 1.941 million, yes. (B) The manager of the firm doubts whether the sales will actually increase by 25% as a result of this strategy. Find the minimum increase in sales to justify introduction of the new credit policy. 15.92% 6.3. Ashley Company is considering the credit application of a retail customer who is expected to buy $1000 worth of merchandise every month. The cost of these goods will be $800. The customer is expected to pay after 30 days every month. However, there is a 10% probability of default each month. In case of default, the company will recover 50% of the unpaid bill after 3 months. The cost of capital to the company is 15% per annum. Should Ashley extend credit to the customer? NPV = 1234, yes ERP 513: Treasury Management Assignment Week 6 _____________________________________________________________________________ 6.4. First National Bank of Jermyn has a portfolio of 10,000 credit card accounts. The bank charges $25 annual fee on these cards. There is a 25 day grace period on the accounts, and after that the cardholders pay interest at the rate of 1.25% per month on the unpaid balance. Half of the cardholders pay their balance in full every month, and their monthly bill is $800. The remaining cardholders carry an average balance of $1200 continuously. The operating expenses for the credit card portfolio, including defaults, are $100,000 annually. The cost of capital to the bank is 8%. Mellon Bank has offered to buy Jermyn's credit card portfolio for $5 million, plus the outstanding balance. Should Jermyn accept the offer? NPV = $3.971 million, yes TREASURY MANAGEMENT Riaz Hussain Kania School of Management University of Scranton Scranton, Pennsylvania February 2016 This copy of Treasury Management by Riaz Hussain is for your personal use only. You cannot give it, or sell it, to anyone else in any form, printed or electronic. PREFACE I have prepared this textbook for students who are taking a one-semester course on the treasury management of a corporation. This area of finance deals with the short-term financial decision-making at a firm. In day-to-day operations, the financial manager has to work with the current assets and current liabilities of the firm. This course does not consider the long-term decisions at a firm, such as the capital budgeting or the measurement of cost of long-term capital. The main theme of the course is to keep the working capital at its optimal level. This means the inventory should be at the right level, balancing the carrying costs against the shortage costs. This also means that the corporation should keep the right amount of cash on hand, not too little, and not too much. A shortage of cash can cause a serious "cash crunch," and having too much cash is simply wasteful. This workbook should help students learn the basic principles of short-term financial management, and then apply these concepts in solving practical problems at a firm. The book develops well-known mathematical formulas. The problems are set up as modules and they form a cohesive structure that spans the subject. Besides Excel, students are encouraged to learn Maple as a powerful analytical tool and to apply it in solving problems that are more difficult. Some of the topics included in this book, Treasury Management, are the analysis of financial statements, cash management, inventory management, short-term financing, credit management and policy, and short-term investments. For additional reading, the students should use Short-term Financial Management, by Terry S. Maness and John T. Zietlow, third edition, 2004, South-Western. They can develop their understanding of the basic concepts by careful study of a standard textbook. The workbook is a tool that they can apply in solving practical problems. With deep appreciation, I acknowledge the comments and helpful suggestions of my students. Their critical review of the material has greatly improved the quality of this text. Riaz Hussain TABLE OF CONTENTS Preface 1. Analytical Tools 1 2. Financial Principles 24 3. Ratio Analysis 42 4. Financial Planning and Control 61 5. Cash Management 82 6. Accounts Receivable Management 107 7. Short-term Financing 129 8. Inventory Management 141 9. Integrated Problems 161 10. Some Useful Formulas 164 1. Analytical Tools Objectives: After reading this chapter, you will be able to 1. Solve linear and quadratic equations, system of linear equations 2. Use geometric series in financial calculations 3. Understand the basic concepts of statistics 4. Use WolframAlpha, Excel, or Maple to solve mathematical problems 5. Understand the concept of optimization Before we actually start studying finance and the financial management as a discipline, it is worthwhile to review some of the fundamental concepts in mathematics first. This will help us appreciate the usefulness of analytical techniques as powerful tools in financial decision-making. We shall briefly review elementary algebra, basic concepts in statistics, and finally learn Excel or Maple as a handy way to cut through the mathematical details. Our approach toward learning finance is to translate a word problem into a mathematical equation with some unknown quantity, solve the equation, and get the answer. This will help us determine an exact answer, rather than just an approximation. This will lead to a better decision. 1.1 WolframAlpha The mathematical software, Mathematica, has the ability to solve a wide range of mathematical problems. Mathematica has a website at WolframAlpha, which is free to use. You should explore it and use for performing many of the mathematical operations. If you click on the button WRA , you will go to the website for WolframAlpha. The next sections will show a variety of problems that you can do at WolframAlpha. 1.2 Linear Equations To review the basic concepts of algebra, we look at the simplest equations first, the linear equations. These equations do not have any squares, square roots, or trigonometric or other complicated mathematical functions. Example 1.0. Suppose John buys 300 shares of AT&T stock at $26 a share and pays a commission of $10. When he sells the stock, he will have to pay $10 in commission again. Find the selling price of the stock, so that after paying all transaction costs, John's profit is $200. Let us define profit as the difference between the final payoff F, after commissions, and the initial investment I0, including commissions. We can write it as a linear equation as follows = F I0 1 Treasury Management 1. Analytical Tools _____________________________________________________________________________ We require a profit of $200, thus, = 200. Suppose the final selling price of the stock per share is x, the number we want to calculate. Selling 300 shares at x dollars each, and paying a commission of $10, gives the final payoff as, F = 300x 10. The initial investment in the stock, including commission, is I0 = 300(26) + 10 = $7810. Make these substitutions in the above equation to obtain 200 = 300x 10 7810 Moving things around, we get 200 + 10 + 7810 = 300x Or, 8020 = 300x Or, x = 8020/300 $26.73 This means that the stock price should rise to $26.73 to get the desired profit. Note that the answer has a dollar sign and it is truncated to a reasonable degree of accuracy, namely, to the nearest penny. Consider another problem involving dollars, doughnuts, and coffee. 1.1. Jane works in a coffee shop. During the first half-hour, she sold 12 cups of coffee and 6 doughnuts, and collected $33 in sales. In the next hour, she served 17 cups of coffee and sold 8 doughnuts, for which she received $46. Find the price of a cup of coffee and that of a doughnut. This is an example where we have to find the value of two unknown quantities. The general rule is that you need two equations to find two unknowns. We have to develop two equations by looking at the sales in the first half-hour and in the second hour. Suppose the price of a cup of coffee is x dollars, and that of a doughnut y dollars. First half-hour, 12 cups and 6 doughnuts for $33, gives Second hour, 17 cups and 8 doughnuts for $46, gives 12 x + 6 y = 33 17 x + 8 y = 46 Now we have to solve the above equations for x and y. First, try to eliminate one of the variables, say y. You can do this by multiplying the first equation by 8 and the second one by 6, and then subtracting the second equation from the first. This gives 8*12 x + 8*6 y = 8*33 6*17 x + 6*8 y = 6*46 Subtracting second from first, (8*12 - 6*17) x = 8*33 - 6*46 Simplifying it, - 6 x = - 12, or x=2 Substituting this value of x in the first equation, we have 12*2 + 6 y = 33 2 Treasury Management 1. Analytical Tools _____________________________________________________________________________ 6y = 33 - 24 = 9 Or, y = 9/6 = 3/2 This gives The answer is that a cup of coffee sells for $2 and a doughnut for $1.50. To verify the result, use the following instruction at WolframAlpha. WRA 12*x+6*y=33,17*x+8*y=46 1.3 Non-linear Equations Non-linear equations contain higher powers of the unknown variable, or the variable itself may show up in the power of a number. For instance, a quadratic equation is a nonlinear equation. The general form of a quadratic equation is ax2 + bx + c = 0 (1.1) b b2 4ac x= 2a (1.2) The roots of this equation are To verify equation (1.2), use the following instruction at WolframAlpha. WRA a*x^2+b*x+c=0 Consider the following examples of non-linear equations. Examples 1.113x = 2.678 1.2. Solve for x: First, we recall the basic property of logarithm functions, namely, ln(ax) = x ln a Taking the logarithm on both sides of the given equation, we obtain x ln(1.113) = ln(2.678) Or, x= ln(2.678) 0.9850702 = = 9.201 ln(1.113) 0.1070591 To verify the result, use the following instruction at WolframAlpha. WRA 1.113^x=2.678 3 Treasury Management 1. Analytical Tools _____________________________________________________________________________ (2 + x)2.11 = 16.55 1.3. Solve for x, 2 + x = (16.55)1/2.11 This gives x = (16.55)1/2.11 - 2 = 1.781 Or, The keystrokes needed to perform the calculation on a TI-30X calculator are as follows: 16.55 1 2.11 2 To verify the result, use the following instruction at WolframAlpha. WRA (2+x)^2.11=16.55 1.4. Find the roots of 5x2 + 6x 11 = 0 Here a = 5, b = 6, and c = - 11. This gives us x= -6 36 - 4(5)(-11) -6 256 -6 16 11 = = 10 = 5 or 1 10 10 To verify the result, use the following instruction at WolframAlpha. WRA 5*x^2+6*x-11=0 1.4 Geometric Series In certain problems in financial management, we have to deal with a series of cash flows. When we look at the present value, or the future value, of these cash flows, the resulting series is a geometric series. Thus, geometric series will play an important role in managing money. Let us consider a series of numbers represented by the following sequence a , ax , ax2 , ax3 , ... , axn1 The sequence has the property that each number is multiplied by x to generate the next number in the list. There are altogether n terms in this series, the first one has no x, the second one has an x, and the third one has x2. By this reasoning, we know that the nth term must have xn1 in it. This type of series is called a geometric series. Our concern is to find the sum of such a series having n terms with the general form S = a + ax + ax2 + ax3 + ... + axn1 (1.3) To evaluate the sum, proceed as follows. Multiply each term by x and write the terms on the right side of the equation one-step to the right of their original position. We can set up the original and the new series as follows: 4 Treasury Management 1. Analytical Tools _____________________________________________________________________________ S = a + ax + ax2 + ax3 + ... + axn1 xS = ax + ax2 + ax3 + ... + axn1 + axn If we subtract the second equation from the first one, most of the terms will cancel out, and we get S xS = a axn S(1 x) = a(1 xn) Or, or, Sn = a (1 xn) 1x (1.4) This is the general expression for the summation of a geometric series with n terms, the first term being a, and the ratio between the terms being x. This is a useful formula, which we can use for the summation of an annuity. To verify equation (1.4), use the following instruction at WolframAlpha. WRA sum(a*x^i,i=0..n-1) If the number of terms in an annuity is infinite, it becomes a perpetuity. To find the sum of an infinite series, we note that when n approaches infinity, xn = 0 for x prompt, end each line with a semicolon, and strike the return key. 10 Treasury Management 1. Analytical Tools _____________________________________________________________________________ 2^64; 18446744073709551616 50!; 30414093201713378043612608166064768844377641568960512000000000000 evalf(Pi,50); 3.1415926535897932384626433832795028841971693993751 Here evalf calculates the result in floating point with 50 significant figures. Maple can also do algebraic calculations. For instance, to solve the equations 5x + 6y = 7 6x + 7y = 8 for x and y, enter the instructions as follows: eq1:=5*x+6*y=7; eq1 := 5 x + 6 y = 7 eq2:=6*x+7*y=8; eq2 := 6 x + 7 y = 8 solve({eq1,eq2},{x,y}); {y = 2, x = -1} The symbol := is used specifically to define objects in Maple. In other words, if we type eq1; then the computer will recall the equation defined as eq1 and display it as 5x+6y=7 Maple can also do differentiation and integration. Consider the function ln x x3 + x To differentiate this function with respect to x, we type in diff(x^3+ln(x)/x,x); 1 ln(x) 3 x2 + x2 x2 11 Treasury Management 1. Analytical Tools _____________________________________________________________________________ To integrate the result with respect to x, recreating the original function, we enter int(%,x); ln x x3 + x Here we use % as a symbol to designate the previous expression. We can also use Maple to plot functions. For instance, if we want to see the visual representation of the well-known sine wave, we write plot(sin(x),x=0..2*Pi); which gives the diagram shown in Figure 1.1. Fig. 1.1: Plot of sin x for 0 8 where x is the amount invested in the working capital, in millions of dollars. Find the optimal level of working capital for Carter, and the minimum total annual cost of working capital management. The financing cost of money invested in the working capital is F = .12x The total cost is the sum of shortage cost and financing cost. In symbols, it is 5 T = S + F = x 8 + .12x 17 Treasury Management 1. Analytical Tools _____________________________________________________________________________ To find the maximum or minimum of this function, we differentiate it with respect to x, and set the derivative equal to zero. This gives us dT 5 = (x 8)2 + .12 = 0 dx Multiplying throughout by (x 8)2, we get 5 + .12(x 8)2 = 0 Or, (x 8)2 = 5/.12 Or, x 8 = 5/.12 x = 8 5/.12 = 14.455 or 1.545 Only the first value agrees with the requirement that x > 8. Therefore, from a practical point of view, we have just one optimal value of the working capital, namely, $14.455 million. The total cost of working capital management will be 5 T = .12(14.455) + 14.455 8 = $2.509 million To verify, you can do the problem in Maple as T:=5/(x-8)+.12*x; diff(T,x); solve(%); subs(x=max(%),T); 1.13. Cleveland Company's cost of capital is 15%. It has invested x dollars in current assets. The following function gives the shortage cost of current assets S = 3000 ex/5000 Find the optimal level of its current assets. Draw a diagram that shows the financing costs, shortage costs, total costs, and the optimal level of current assets. The total cost, financing plus shortage cost, is represented by the function T = .15x + 3000 ex/5000 Differentiating it with respect to x, and setting the result equal to zero, we get dT x/5000 =0 dx = .15 3000/5000 e 18 Treasury Management 1. Analytical Tools _____________________________________________________________________________ Canceling terms, we have ex/5000 = .15(5/3) = .25 Taking natural logarithm on both sides, we get x/5000 = ln(.25) = 1.3862944 x = 5000*1.3862944 = $6931 Or, The shortage-cost and the financing-cost curves intersect at the point where the two costs are equal. This takes place when .15x = 3000 ex/5000 Fig. 1.2: The costs are on the scale at left, and the working capital on the scale at bottom. We can calculate the value of x by using the following Maple commands solve(.15*x=3000*exp(-x/5000)); and the result is $6011. We can see this in Fig. 1.2. Problems Solve the following equations: 1.14. 16x - 54 = 15x - 32 x = 22 1.15. (x +1) (x 2) = (x - 1) (x + 2) x=0 1.16. (10 x + 3) (3 x + 4) = (5 x + 6) (6 x + 7) 19 x = 15/11 Treasury Management 1. Analytical Tools _____________________________________________________________________________ x-2 x-7 1.17. x - 3 = x - 9 x = -3 x+4 x+6 = x+5 x+8 x = -2 1.18. Solve the following equations for x and y: 1.19. 1.20. 2x + 6y = 32 5x + 8y = 45 x = 1, y = 5 3x + 4y = 15 5x + 8y = 45 x = -15, y = 15 Solve for x, 1.21. (1 + x)3.2 = 8.4 x = 0.9446 1.22. 1.767x = 3.876 x = 2.38 1.23. 3.909x = 15.99 x = 2.033 1.24. 2x2 + 7x - 9 = 0 x = 1, -9/2 1.25. 3x2 + 4x - 7 = 0 x = 1, -7/3 1.26. 2.5 + (2.5)(.3) + (2.5)(.3)(.3) ..., infinite terms 3.571 1 1 1 1.27. 1.1 + 1.12 + 1.13 + ... 9 terms 5.759 Find the roots of Find the sum of the following series: 30 30(1.05) 30(1.05)2 1.28. 1.12 + 1.122 + 1.123 + ... 36 terms 386.60 10 500 1.29. 1.12i i=1 2825.11 100 25 1.30. 1.12i i=1 208.33 1.31. The cash flows from two projects under different states of the economy are as follows: 20 Treasury Management 1. Analytical Tools _____________________________________________________________________________ State of the economy Probability Project A Project B Poor 20% $3000 $5000 Average 30% $4000 $7000 Good 50% $6000 $15,000 Find the coefficient of correlation between the two projects. .9922 1.32. The expected return from two stocks, Microsoft and Boeing, under different states of the economy are as follows: State of the economy Probability Microsoft Boeing Poor 10% -5% -40% Average 40% 10% -10% Good 50% 20% 30% Find the coefficient of correlation between the two stocks. .9471 1.33. Clinton Company has cost of capital 12%. The following function represents the shortage cost for its net working capital 2 S = x 8 for x > 8 Here S is the shortage cost in thousands of dollars, and x is the level of the net working capital, also in thousands of dollars. Find the following: (A) The optimum level of net working capital. $12,082 (B) The financing cost, shortage cost, and total cost at the optimal point. $1450, $490, $1940 per year 1.34. Coolidge Corporation has estimated its cost of capital to be 12%, and the shortage cost of working capital as 125 S = x 25 for x > 25 where S is the shortage cost per year in $million, and x is the level of working capital in $million. Find the optimal level of working capital. What is the total cost of working capital per year? $57.27 million, $10.75 million 1.35. Eisenhower Corporation has cost of capital 10%. The shortage cost of current assets may be represented by the function S = .3/x, where S is the shortage cost per year (in $million), and x is the amount invested in current assets (in $million). Find the optimal level of, and the minimum total annual cost of, the current assets for this firm. $1.732 million, $346,400 21 Treasury Management 1. Analytical Tools _____________________________________________________________________________ 1.36. Fillmore Company has estimated its cost of capital to be 12%. The following function represents the shortage cost of its current assets, 1 S = 5(x 10) for x > 10 where S is the shortage cost per year (in $million), and x is the amount invested in current assets (in $million). Find the optimal level of the current assets for this firm. What is the minimum annual cost of these assets? $11.291 million, $1.510 million 1.37. Garfield Company's cost of capital is 12%. It has invested x (million dollars) in current assets. The shortage cost of current assets is represented by the function S = 7 ex/4 Find the following: (a) The optimal level of current assets. (b) Total annual cost of these assets at the optimal level. $10.72 million $1.766 million 1.38. Grant Company has cost of capital 11%. It has estimated that the shortage costs are given by S = 4.2 ex/10 where S is the shortage cost in $million/year, and x is the amount of current assets, in $million. Find the optimal level of current assets. What is the minimum total annual cost of maintaining current assets? $13.40 million, $2.574 million 1.39. Harding Company's cost of capital is 12%. It has invested x (million dollars) in current assets. The following function represents the shortage cost of current assets S = 8 ex/4 Find the following: (a) The optimal level of current assets. (b) Total annual cost, shortage and financing, of these assets. $11.25 million $1.83 million 1.40. (Advanced) Harrison Corporation has 11% cost of capital. It has estimated that the shortage cost of inventory is ex/5 S= ,x>0 x where x is the capital invested in inventory. Find the optimal level of inventory that Harrison should maintain. Find the total annual cost of inventory at the optimal level. $2.8419 million, $511,927 1.41. (Advanced) Hayes Corporation has 12% cost of capital. It has estimated that the shortage cost of cash in a checking account is 22 Treasury Management 1. Analytical Tools _____________________________________________________________________________ x2 2 S = 100 + x , x > 0 where x is the balance in the checking account in millions of dollars. Find the optimal checking account balance and the total cost of the checking account. $3.282 million, $1.111 million Key terms annuity, 4 correlation coefficient, 7, 8 covariance, 7, 8 Excel, 1, 9 expected value, 6, 7, 8 function, 12 geometric series, 1, 4 linear equations, 1, 3 long-term capital, 15 long-term capital, 15 Maple, 1, 9, 10, 11 normal probability distribution, 6 optimization, 12 perpetuity, 4 probability distribution, 6 quadratic equation, 1, 3 short-term capital, 15 23 standard deviation, 7 statistics, 1, 6 tax shield, 12 variable, 12 weighted average cost of capital, 12 working capital, 15 2. Financial Principles Objectives: After reading this chapter, you will be able to 1. Calculate the cost of capital of a firm. 2. Apply the concepts of value creation and net present value in making investment decision at a firm. 2.1 Cost of Capital Firms need financial capital to invest in plant and equipment. What is capital? Capital is simply the money that companies need to do their business. They need cash to pay their workers and to buy raw materials. Capital is the lifeblood of any company. The investors supply the capital to the corporations. Capital is not free; there is a cost of capital. Figure 2.1 represents the relationship between the corporations and investors. The cost of capital to the corporation is equal to the required rate of return by the investors. Investors Capital Return on investment Corporation Fig. 2.1: The relationship between the investors and a corporation. The cost of capital to a corporation is equal to the required rate of return for the investors. This is because there is a free trade, of equal value, between two parties. Let us represent this fundamental principle by the following diagram. The cost of capital to a corporation = The required rate of return for the investors There are two forms of capital: equity capital and debt capital. Firms can raise equity capital by selling shares of their stock. A corporation can raise debt capital by selling its bonds. It is also possible for a corporation to issue a hybrid security, such as a convertible bond. Suppose a corporation, say, Ford Motor Company, needs $100 million to expand its manufacturing facilities. It can raise the capital by first contacting an underwriter, such as Merrill Lynch. After some negotiations, Merrill Lynch may buy the entire stock issue by paying Ford, perhaps, $94 million. Ford gets $94 million and is now out of the picture. Merrill Lynch gets $6 million for selling the stock to the public. Since they have an extensive network of retail brokerage outlets, they are able to sell the stock to their customers. If they have some unsold stock, they will just keep it in their own investment portfolio. 24 Treasury Management 2. Financial Principles _____________________________________________________________________________ 2.2 Cost of Debt Suppose a company, such as The Home Depot, wants to borrow $300 million for expansion of its business. It can do so by selling its bonds. It has to go through an underwriter once again. The underwriting fees on bonds are somewhat lower, perhaps around 3%. The bonds must carry an attractive coupon rate, say 6%, depending on the market conditions. The cost of debt capital, kd, for a corporation is equal to the yield to maturity of its bonds. The cost of debt will depend upon several factors. The most important ones are: (1) The prevailing interest rates in the financial markets; for instance, it depends on the 10-year Treasury bond yield. In December 2012, the 10-year Treasury bonds have a yield of 1.77%. (Google Finance). (2) The financial condition of the firm, whether it has a steady income stream or not. If a company has strong cash flows, a history of consistent earnings, and dominant position in the market, its cost of debt capital will be lower. (3) The amount of debt that the company already has; if it has very little debt, it can borrow money easily and at lower cost. (4) The time to maturity of the bonds; short-term bonds will have a lower interest rate compared to longer-term bonds. Consider the yield of Treasury securities on December 24, 2012. Note that as the time to maturity for the securities increases, so does the yield to maturity. Corporate bonds have the same property. Time to maturity Yield to maturity 3 Month 0.03% 6 Month 0.09% 2 Year 0.26% 5 Year 0.77% 10 Year 1.77% 30 Year 2.92% To calculate the yield to maturity of a bond, one can use the following expression. Yield to maturity = Annual interest payment + Annual price change Average price of the bond during its remaining life Writing in symbols, it becomes kd = YTM = cF + (F B) (F + B)/2 where kd = pre-tax cost of debt of the company 25 (2.1) Treasury Management 2. Financial Principles _____________________________________________________________________________ c = coupon rate of the bond F = face value of bond, normally, $1000 B = market price of the bond n = time to maturity, in years This relationship is only approximate. Because interest is tax deductible, the after-tax cost of debt is (1 t)kd. Or, After-tax cost of debt = (1 t)kd (2.2) 2.3 Cost of Equity We can find the cost of equity capital by using either Gordon growth model, or by using the Capital Asset Pricing Model, or CAPM, for short. Suppose we know the expected long-term growth rate g of the dividends of a firm, its current price per share, P0, and its dividend for next year, D1. According to Gordon's growth model, D1 P0 = R g (2.3) In equation (2.3), R is the required rate of return for the stockholders. The required rate of return is the same as the cost of equity for the company, ke. Moving things around, we get the cost of equity for the firm, ke, as D1 ke = P + g (2.4) 0 A very important result in finance is the capital asset pricing model. We may write it as E(R) = r + [E(Rm) - r] (2.5) where E(R) is the expected return, by the stockholders, of a stock. In the above equation, is the beta of a stock, r is the risk-free rate, r, and E(Rm) is the expected return on the market. The expected return by the stockholders is equal to the cost of equity for the firm, that is, E(R) = ke. If we know , r, and E(Rm), then we can find the cost of equity capital for the firm by using the relationship representing capital asset pricing model, ke = r + [E(Rm) - r] (2.6) Consider a company that has just paid its annual dividend of $2.00. The investors expect that the dividend will grow at the rate of 6% for the long haul. Its dividend next year will be 2(1.06) = $2.12. Suppose the price of the stock is $25. Then its cost of equity, by (2.4), is 2.12 ke = 25 + .06 = .1448 = 14.48% 26 Treasury Management 2. Financial Principles _____________________________________________________________________________ Let us assume that the of this stock is 1.45, the risk-free rate is 6%, and the expected return of the market is 12%. Putting these numbers in (2.6), we find ke = .06 + 1.45(.12 .06) = .1470 = 14.70% We have to keep in mind that neither of these two methods of calculating the cost of equity is accurate. They give only approximate results. 2.4 Weighted Average Cost of Capital Since the corporations mix the two forms of capital when they apply it to their business, it is necessary to find the weighted average cost of capital. We can find this from the following result B S WACC = (1 t)kd V + ke V (2.7) In this equation, WACC is the weighted average cost of capital, t is the income tax rate of the firm, B is the market value of the debt, S is the market value of the equity of the firm, and V is the total (market) value of the company. Suppose a company has $30 million in debt and $70 million in equity capital. The total value of the company is thus $100 million. Further, the debt/assets ratio, or B/V = .3, and likewise S/V = .7. Assume that the pretax cost of debt is 10%, the cost of equity is 15%, and the tax rate of the company is 30%. Putting t = .3, kd = .1, ke = .15, B/V = .3, and S/V = .7, in (2.7), we get WACC = (1 .3)(.1)(.3) + .15(.7) = .126 = 12.6% 2.5 Value Creation Corporations are constantly trying to upgrade their facilities, incorporate new technology, and improve their existing methods and procedures. The basic reason for this ongoing effort is to create value for the firm. Suppose a firm is streamlining its operations, and it is thus able to save C dollars per year. Suppose further that these savings will last for the next n years. Assume the cost of capital for the firm is r. Did the company create value by adopting this procedure? Yes. The additional value is equal to the present value of all the savings. We may write this value added, V, to be n C (2.8) V = (1 + r)i i=1 We can write this summation as n C C[1 - (1 + r)n] V = (1 + r)i = r i=1 27 (2.9) Treasury Management 2. Financial Principles _____________________________________________________________________________ Next, we assume that the firm is able to incorporate a permanent change in its operations that will save it C dollars per year forever. The value created in this case is C V = (1 + r)i (2.10) i=1 We can write the infinite summation as follows, C C (1 + r)i = r i=1 (2.11) Let us consider a couple of examples to illustrate these ideas. Suppose you sign a lease with the owner of an apartment. You agree to pay $300 a month, in advance each month. The property owner uses a risk-adjusted discount rate of 12% in the valuation of this lease. What is the value of this lease for him? In valuation models, we frequently use the concept of risk-adjusted discount rate. Suppose the owner in this example has borrowed the money at the rate of 8% to finance the purchase of the apartment house. He has to make at least 8% to pay for the financing of this business. However, he is also taking additional risk. Perhaps the tenants will not pay the rent and leave. Perhaps they will cause damage to the property, and not pay for it. Perhaps he cannot rent the apartment for some time. In order to take into account all these unforeseen events, he simply adds a risk premium to this original cost of capital. The risk-adjusted discount rate will then become 12%. The present value of the first month's rent is $300, because you are paying it in advance. The remaining 11 installments will have a present value given by (2.4). The proper discount rate is the monthly rate, namely, 1% per month. Thus 11 300 300[1 1.0111] Value of lease = 300 + 1.01i = 300 + = $3410.29 .01 i=1 To verify the result, use the following instruction at WolframAlpha. WRA sum(300/1.01^i,i=0..11) Let us look at another example. A company has decided to switch to a new method for the production that will save it $5 million annually forever. The company uses a discount rate of 8% to evaluate such innovations. This improvement will increase the value of the company by the following amount, given by (2.7). Increase in value, V = 5/.08 = $62.5 million. 28 Treasury Management 2. Financial Principles _____________________________________________________________________________ 2.6 Net Present Value One of the most important tools in financial analysis is the concept of net present value. We can use this concept in the evaluation of projects, or investments. We also use it in the working capital management of a company. The concept of net present value is based on the cost-benefit analysis. However, it looks at the costs and benefits in terms of their present value. By using proper risk-adjusted discount rate, we can convert the future cash flows to their present values. Thus, we are able to capture the risk of the project in its NPV. This is a major advantage of this methodology. In general, a project requires several cash inflows and outflows at different times. As a particular case, we can consider a project that requires a single investment I0 at the beginning, but it generates regular cash flows C in the future. If the proper risk-adjusted discount rate in this case is r, then we may define the net present value as follows, n C (1 + r)i i=1 NPV = - I0 + (2.12) Here I0 = the initial investment in the project C = the cash flow from the project r = risk-adjusted discount rate n = life of the project in years We may use (2.9) to carry out the summation in (2.12) Suppose we define the after-tax cash flow C in (2.12) as earnings after taxes, then C = Earnings - taxes = Earnings - (income tax rate) (taxable income) = Earnings - (income tax rate) (earnings depreciation) = E - t(E - D) = E - tE + tD = E(1 - t) + tD The after-tax cash flow, C, is thus C = E(1 - t) + tD (2.13) We may write equation (2.9) as n E(1 - t) + tD (1 + r)i i=1 NPV = - I0 + (2.14) Let us consider a few examples from corporate finance where we can apply these concepts. 29 Treasury Management 2. Financial Principles _____________________________________________________________________________ Examples 2.1. Delta Corporation wants to raise $40 million in new capital by selling bonds at a discount. Delta will sell the bonds at $800 each, although their face value is $1000 each. They will have a coupon of 8% and will mature after 5 years. The tax rate of Delta is 35%. Find the pre-tax and after-tax cost of new capital to the firm. The cost of capital to the firm is equal to the expected return by the investors. This is just the yield to maturity of new bonds. The annual interest is $80 per bond. The price appreciation of the bonds is (1000 800) = $200 over a five year period. The average price of the bond during the five-year period is $900. Using YTM = cF + (F B) (F + B)/2 (2.1) we get kd = Y = 80 + (1000 800)/5 (1000 + 800)/2 = 0.1333 = 13.33% (pre-tax) After-tax cost of debt = (1 t)kd = (1 .35)(.1333) = .08667 = 8.667% (after-tax) 2.2. Madison Company wants to issue new bonds with a coupon of 8.5%, maturing after 5 years, and a face value of $10 million. The bonds will sell at par. The company is in the 40% tax bracket. The flotation cost of this issue will be $500,000. Calculate the after-tax cost of capital for this issue. The flotation costs are the fees that the company pays to the underwriters to sell the bonds. This is $500,000 and so the company will acquire only $9.5 million in new capital. This is equivalent to selling the bonds at 95% of their face value. When a company sells the bonds at a discount, it should consider their yield-to-maturity as the cost of debt capital. The interest cost is 0.085(10) = $0.85 million annually. The price appreciation of the bonds is (10 9.5)/5 = $0.1 million annually. The average price of the bonds is $9.75 million. Using (2.1), kd = Y = (.85 + .1)/9.75 = .09744 and (1 t)kd = (1 0.4)(0.09744) = .05846 = 5.846% 2.3. The WACC of General Telecom is 11% and its debt/assets ratio is 45%. The pre-tax cost of debt is 9%, and its tax rate is 40%. Find the cost of equity for General Telecom. In B S WACC = (1 t) kd V + ke V substitute WACC = .11, t = .4, kd = .09, B/V = .45, S/V = .55. This gives 0.11 = (1 0.4)(0.09)(0.45) + ke (0.55) 30 (2.7) Treasury Management 2. Financial Principles _____________________________________________________________________________ Solving for ke, ke = 0.11 (1 0.4)(0.09)(0.45) = 0.1558 = 15.58% 0.55 To verify the result, use the following instruction at WolframAlpha. WRA .11=(1-.4)*.09*.45+x*.55 2.4. Webster Corporation has a beta of 1.21 and tax rate of 30%. The expected return from the market is 15% and the riskless rate is 10%. The company wants to issue $50 million in new stock. Find the after tax cost of capital to the firm. Find the cost of equity by using CAPM, ke = r + [E(Rm) r] We get (2.6) ke = 0.1 + 1.21[0.15 0.1] = 0.1605 = 16.05% The cost of equity capital does not depend upon the tax rate of the corporation. Thus, the answer is the same for both before and after tax. 2.5. Jefferson Company has tax rate of 30% and growth rate of 4%. The company just paid the annual dividend of $1.25 on its common stock, which sells for $17 per share. Calculate the pretax and after tax cost of equity to the firm. The next year dividend will be 4% higher than this year's dividend, $1.25. D1 ke = P + g 0 Use (2.4) ke = 1.25(1.04)/17 + 0.04 = 0.1165 = 11.65% both before and after tax. 2.6. Basic: Aachen Company plans to make an investment that requires an initial outlay of $10,000, but it will pay back $1,000 annually for 20 years with the first payment after one year. The proper discount rate is 12%. Should Aachen accept the investment? Here I0 = 10,000, C = 1000, r = .12, n = 20. Using (2.9) and (2.12), we get 20 1000 1000[1 1.1220] = $2530.56. NPV = 10,000 + 1.12i = 10,000 + 0.12 i=1 Since the NPV of the project is negative, we should reject it. Another way to look at the problem is that its annual rate of return is $1000/$10,000 = .1 = 10%. It will never be profitable at a discount rate of 12%. If the discount rate were, say, 7%, it would become profitable. We may see this as 31 Treasury Management 2. Financial Principles _____________________________________________________________________________ 1000 1000[1 1.0720] NPV = 10,000 + 1.07i = 10,000 + = $594.01. 0.07 i=1 20 To verify the calculation with Maple, type in NPV:=-10000+sum(1000/(1+r)^i,i=1..20); eval(subs(r=.12,NPV)); eval(subs(r=.07,NPV)); To verify the answer with WolframAlpha, try the following: WRA WRA -10000+sum(1000/1.12^i,i=1..20) -10000+sum(1000/1.07^i,i=1..20) 2.7. Uneven cash flows: Augsburg Corporation intends to invest in a project whose initial cost is $100,000 and the proper discount rate is 12%. The project will generate $15,000 annually for years 1 through 5 and then $10,000 annually for the years 6 through 10. Should Augsburg accept the project? Because the cash flows are different for different periods, we have to set the problem up as two summations, one for years 1-5, and the second for years 6-10. Thus 10 10000 5 15000 NPV = 100,000 + 1.12i + 1.12i i=6 i=1 Another way of looking at the cash flows is as follows: there is $10,000 coming in every year for years 1-10, plus an additional $5,000 for the years 1-5. We may represent this as follows, 10 10000 5 5000 NPV = 100,000 + 1.12i + 1.12i i=1 i=1 We can perform the summation by using (2.5), 10000(1 1.1210) 5000(1 1.125) NPV = 100,000 + + = $25,474, reject it 0.12 0.12 The Maple instruction for this problem is NPV=-100000+sum(15000/1.12^i,i=1..5)+sum(10000/1.12^i,i=6..10); To do it on WolframAlpha, copy and paste the following expression, WRA -100000+sum(15000/1.12^i,i=1..5)+sum(10000/1.12^i,i=6..10) 32 Treasury Management 2. Financial Principles _____________________________________________________________________________ 2.8. Uncertain cash flow: Bamberg Corpora