Corporate Finance s1

CORPORATE FINANCE:

AN INTRODUCTORY COURSE

DISCUSSION NOTES

MODULE #3[1]

FINANCIAL MARKETS AND NET PRESENT VALUE

I) Financial Markets and Net Present Value:

A) Introduction:

Financial markets make it possible for individuals and firms to borrow or lend over time, thereby improving their intertemporal (across time periods) investment and consumption patterns. Think about it! Why do individuals invest? They invest because they are willing to forego present consumption for future, and hopefully larger, consumption. Why do people borrow or sell assets, e.g., common stocks? They want to increase present consumption (spend the money now) at the sacrifice of future consumption (e.g., repay the debt or not receive future dividends). In this context, consumption can include purchasing a new car, a new home, investing in their or their children's education, retirement, charitable contributions, or an inheritance for their heirs. Because of the financial markets, individuals can choose consumption and investment patterns, within the limits of their wealth constraint, so to maximize their utility of consumption over time.

In addition, financial markets provide individuals and firms critical feedback regarding required rates of return on investment, i.e., benchmark rates of return. Accordingly, financial markets are indispensable in a well-functioning market based economy as in the U.S. Investment, particularly in "real assets," is critical in the creation of jobs and increasing the wealth of individuals and, in turn, society. Investment produces future goods and services for our society always at the cost of current consumption. The question is always, is the sacrifice worthwhile?

B) Market Clearing:

Financial intermediaries, for example banks and other financial institutions, perform an important function--they match up borrowers and lenders. This market "clears" when the quantity of money demanded by borrowers equals the amount of money supplied by investors. If a surplus of borrowers exists, interest rates will rise, discouraging some borrowers, money will become too expensive, and encouraging new lenders, the return to lending is higher. If a surplus of lenders exists, interest rates will fall, discouraging lenders and encouraging borrowers. The interest rate that causes markets to clear is referred to as the equilibrium market rate of interest.[2] Interest rates are quite dynamic as the supply and demand of funds varies over time.

C) Preliminary Assumptions:

In the subsequent discussion we will assume the following:

(1) Perfect certainty. Outcomes are known with 100% probability. What does perfect certainty look like on a probability diagram? Without risk, only one interest rate will exist in the market for each maturity. Differential interest rates result from investments with different risk. We (temporarily) assume risk away.

(2) Perfect Capital Markets (PCM).

(a) Information is free and available to all participants that want it;

(b) All participants have equal access to the financial markets, or rb (the borrowing rate) = rl (the lending rate) = r ;

(d) There are no transactions or contracting costs exist; and

(e) No distorting taxes exist.

(3) Investors are Rational in the following sense:

(a) Investors prefer more wealth to less wealth.

(b) Investors prefer cash sooner than cash later.

(4) We have a one-period world, t = 0 (today) and t = 1 (in one time period).

Do you understand why only one interest rate per maturity can exist in the market if we assume away uncertainty, i.e., no risk exists? (In a one-period world there is then only one interest rate since there can only be one maturity.) If more than one rate existed, you could create a "money machine!" What do I mean by a money machine? Borrow low and lend high with the same risk! Buy low and sell high with the same risk! What is "arbitrage" in this context? Arbitrage opportunities exist when the same item (or more generally, perfect substitutes) sell for two different prices. You can buy at the low price and sell at the high price, earning a riskless profit.

Interest rates represent the price of moving money across time. If money sells at more than one price, i.e., interest rate, you would borrow low (low price) and lend high (high price). In a well-functioning economy, arbitrage opportunities should not persist very long; traders will exploit them, driving the prices of the substitute items to equality.

Don’t worry I know these are unrealistic assumptions. We will begin to relax these assumptions in the next and following lectures. However, using these assumptions provides us important insights without complicating factors. It is a common technique in economics and finance to start with the simplest possible world and then add complications one-by-one. This procedure allows us to identify the complications that matter the most and precisely what each one does.

D) The Concept of Present Value: a fundamental business concept

Every investor has a utility function defined with respect to consumption, consumption at t = 0 (again, t = 0 represents today) and at t = 1 (again, t = 1 represents one period away). The investor's objective is to maximize the utility of his/her consumption over these two time periods. Some of us would prefer to consume more now; others would prefer to consume more in one period. In other words, it is perfectly rational to have different tastes and preferences for consumption from different individuals or at different periods in our lives. Tastes and preferences are individualistic, a personal choice. Consider examples of how your consumption preferences change over your own life-cycle.

We need a way to relate cash flows that occur at different time periods, i.e., at t = 0 and

t = 1, to solve our consumption preference problem.

For example, which would you prefer?

Case #1--$1.00 at t = 0, or C0 = 1, and zero dollars at t = 1, C1 = 0 or

Case #2--Zero dollars at t = 0, or C0 = 0, and $1.00 at t = 1, C1 = 1?

You can’t directly compare since the cash arrives at different times. Why might you prefer Case #1?

Let "r" = the interest rate, e.g., the rate on T-Bills. (What is a T-Bill?) "r" equals the per period "market interest rate." If r = 10%, how much is Case #1 $1.00 worth at the end of t = 1?

Case #1 Case #2

$1.00(1 + r) $1.00 for all cases of r > 0%.

$1.00(1.10) $1.00 at t = 1

$1.10 at t = 1 $1.00 at t = 1

Since you are "rational," you prefer more to less, and it is okay to compare dollar values that occur at the same point in time; you prefer Case #1 to Case #2.

What if r = 0%?

What if r < 0%?

The above example implies that money has a time value. If r > 0%, a dollar received sooner is more valuable than a dollar received later.

Which of the following cases do you prefer?

Case #3--$1.00 at t = 0, or C0 = 1, and zero dollars at t = 1, so C1 = 0, or

Case #4--Zero dollars at t = 0, or C0 = 0 and $1.15 at t = 1, or C1 = 1.15?

Since the t = 1 payoff for Case #4 is more than $1.00, this example is not as straightforward as was the comparison of Cases #1 and #2. We cannot answer the question without knowing the market interest rate, r.

If r = 10%, Case #3's $1.00 grows to $1.10 at t = 1. Therefore, you prefer Case #4, which pays $1.15 at t = 1.

If r = 20%, Case #3's $1.00 grows to $1.20 at t = 1. You prefer #3 to #4.

In short, we need to know r, the market rate of interest, to make our decision. Supplying this benchmark is an important input to decision making provided by the financial markets.

In our one-period world, we can always compare values of different investments at t = 1 and take the case with the highest t = 1 value. However, for several reasons that will become clear later, in finance we prefer to compare alternatives at t = 0 versus at later time periods, e.g., t = 1. While making comparisons at the same future time period will give us the same answer as comparisons at t = 0, we will discover using t = 0 is far easier when we move beyond our one-period world.

Let C0 = cash at t = 0,

Let C1 = cash at t = 1.

What level of C0 and C1 will make us indifferent? Indifference implies the same dollar amount at the same time period.

We saw that C0 becomes C0(1+r) if we invest at the interest rate r. Indifference between our cases above means that C0(1+r) = C1. A very little algebra gives us the following definitions.

C0(1 + r) = C1; C0 is "compounded" forward for equivalence with C1.

C0 = C1/(1 + r); C1 is "discounted" backward for equivalence with C0.

Compounding means take present amounts forward in time. Discounting means bringing future amounts back to present values at t = 0.

Indifference means that C0 is the t = 0 equivalent of getting C1 at t = 1, or getting cash in one period. Indifference then means C0 is equal to the "Present Value," or PV, of C1, or C0 = PV(C1). Present value is the value today, t = 0, of future cash flow, in this case C1.

If C1 = $1.15 and r = 10%,

PV(C1) = $1.15/(1.10) = $1.045. You don't care if you have $1.045 today or $1.15 in one period. Why don’t you care? It’s not general apathy.

Here’s why: ($1.045)(1.10) = $1.15, i.e., if we had $1.045 at t = 0 we could invest it at 10% and it would become $1.15 at t = 1. Alternatively, if you were getting $1.15 at t = 1 and the interest rate was 10%, a bank would loan you $1.045 at t = 0 if you promised to pay $1.15 at t = 1.

Let's take another example. If r = 15%, which case would you prefer?

Case #5--$1.10 today and zero at t = 1.

Case #6--$1.25 at t = 1 and zero at t = 0.

The PV of Case #5 (PV(C0)) = $1.10. Why? The present value (PV) of a dollar today (at t = 0) is a dollar.

PV of Case #6 = $1.25/(1.15) = $1.087 = PV(C1).

Therefore, you prefer Case #5; its PV is higher. Comparing present values is comparing dollars today. Thus, very simply, which ever is higher is more preferable for rational investors.

What is the most that you'd pay for Case #6 (getting $1.15 in one period)? $1.087! If you paid this amount at t = 0, you would earn 15% on your investment. Why?

What market interest rate, r, would make you indifferent between Case #5 and Case #6?

$1.10 = $1.25/(1 + r). Solve for r. (1 + r) = $1.25/$1.10 = 1.1364. r = 0.1364, or 13.64%.

If r < 13.64%, you prefer Case #6. If r > 13.64%, you prefer Case #5. If r = 13.64%, you are indifferent; the future (t=1) and present (t=0) values of the two cases are equal. Test yourself to make sure that you understand these statements!

E) The Concept of Net Present Value (NPV):

An investment costs $100 at t = 0 and pays off, with certainty, $125 at t = 1. Illustrate this situation using a cash flow diagram as shown in the RWJ text.

Is this investment a good deal? You can't tell yet; the answer depends on r!

Where do we get an r? From observing the market! The market interest rate (r) provides a benchmark rate by which we can judge an investment as good (accept) or bad (reject).

Our general rule can be stated as: We want to accept investments that are worth more than they cost. We reject investments that are worth less than they cost.

To evaluate the investment just described we must compare the PV of $125 at received at t = 1 to $100, which is the (PV of the) cost (of generating the future payout) at t = 0. If the PV of $125 is greater than $100, the PV of what you get is greater than what you pay we should accept the investment.

Net Present Value (NPV) = PV(inflows) - PV(outflows)

If NPV > 0, accept the project. Our wealth is increased by making the investment.

If NPV < 0, reject the project. Our wealth is decreased.

If NPV = 0, we are indifferent. Our wealth is unchanged.

NPV = PV(C1) - PV(C0) = PV(inflows) - PV(outflows)

NPV = $125/(1 + r) - $100.

Say that r = 12%.

NPV = $125/(1.12) - $100 = $111.61 - $100 = $11.61. Since NPV is positive, accept the project.

What does $11.61 represent? It represents the wealth increase from undertaking the project. Your current wealth increases from $100 to $111.61. Since we prefer more wealth to less, the investment makes us better off and so we accept the project.

An alternative way to look at this same decision is that if we invested $100 at t = 0 at the market rate, r = 12%, we'd have

$100(1.12) = $112 at t = 1. We can obtain this t = 1 amount by investing at the market rate, e.g., buying a government security. Thus this $112 is what you lose if you invest in the project; the future opportunity cost of investing the $100 in the project. If you invest in the project you can’t spend the same $100 to buy the government bond.

Contrast this amount to what you have if you invested the $100 at t = 0 in the project.

t = 0 t = 1

If you invest in the market and not the project -$100 $112

If you do invest in the project and not the market -$100 $125

With the project, we are $13 better off at t = 1 than without the project. What is the PV of this $13 difference at t = 1?

PV = $13/(1.12) = $11.61. Therefore, the project increases our wealth by $11.61 at t = 0. This amount is the NPV of the project.

NPV accounts for the time value sacrifice of your investment. It considers the opportunity cost of passing up alternative investments at the market rate. We can always choose to invest at the market rate of interest instead of investing in the project, e.g., go out and buy a U.S. government security. Thus it is a meaningful comparison for a riskless investment project.