Chapter 3: Time Value of Money: An Introduction

C x [1 + r]^n

(1) Future Value of a Cash Flow: FV_sub_n = C x [1 + r] x [1 + r] x ... x [1 + r] (n times) = _____

C x [1 + r] x [1 + r] x ... x [1 + r] (n times)

(1) Future Value of a Cash Flow: FV_sub_n = _____ = C x [1 + r]^n

FV_sub_n

(1) Future Value of a Cash Flow: _____ = C x [1 + r] x [1 + r] x ... x [1 + r] (n times) = C x [1 + r]^n

Future Value of a Cash Flow

(1) _____: FV_sub_n = C x [1 + r] x [1 + r] x ... x [1 + r] (n times) = C x [1 + r]^n

C / [[1 + r]^n]

(2) Present Value of a Cash Flow: PV = C / [1 + r]^n = _____

C / [1 + r]^n

(2) Present Value of a Cash Flow: PV = _____ = C / [[1 + r]^n]

PV

(2) Present Value of a Cash Flow: _____ = C / [1 + r]^n = C / [[1 + r]^n]

Present Value of a Cash Flow

(2) _____: PV = C / [1 + r]^nxf

Rule 1: Comparing and Combining Values

*_____* *Our first rule is that it is only possible to compare or combine values at the same point in time*.

Identifying Dates on a Timeline

*_____* To track cash flows, we interpret each point on the timeline as a specific date... Note that date 1 signifies *both* the end of year 1 *and* the beginning of year 2, which makes sense since those dates are effectively the same point in time. That is, there is no real time difference between a cash flow paid at 11:59 P.M. on December 31 and one paid at 12:01 A.M. on January 1, although there may be some other differences such as taxation, which we will overlook for now. -Date 0 is today, the beginning of the first year

The Time Value of Money

*_____* Your company has an investment opportunity with the following cash flows: Cost: $100,000 today Benefit: $105,000 in one year Both are expressed in dollar terms, but are the cost and benefit directly comparable? No. Calculating the project's net value as $105,000 - $100,000 = $5,000 is incorrect because it ignores the *timing* of the costs and benefits... In general, a dollar received today is worth *more* than a dollar received in one year: If you have $1 today, you can invest it now and have more money in the future... Figure 1 illustrates how we use competitive market prices and interest rates to convert between dollars today and other goods, or dollars in the future.

Rule 2: Compounding

*_____*... *Our second rule stipulates that to calculate a cash flow's future value, you must compound it*.

Rule 3: Discounting

*_____*... *Our third rule stipulates that to calculate the value of a future cash flow at an earlier point in time, we must discount it*.

Value of $100,000 Investment Today

*_____*... Because... net value is calculated in terms of dollars today (in the present), it is typically called the *net present value*.

Representing Various Time Periods

*_____*... Many of the timelines included in this chapter are simple. Consequently, you may feel that it is not worth the time or trouble to construct them. As you progress to more difficult problems, however, you will find that timelines identify events in a transaction or investment that are easy to overlook. If you fail to recognize these cash flows, you will make flawed financial decisions. Therefore, approach *every* problem by drawing the timeline as we do in this chapter. -Just change the label from "Year" to "Month" if monthly

Role of the Financial Manager

*_____*... Our objective in this book is to explain how to make decisions that increase the value of the firm to its investors. In principle, the idea is simple and intuitive: For good decisions, the benefits exceed the costs. Of course, real-world opportunities are usually complex and the costs and benefits are often difficult to quantify. Quantifying them often means using skills from other management disciplines, as in the following examples: *Marketing*: to determine the increase in revenues resulting from an advertising campaign *Economics*: to determine the increase in demand from lowering the price of a product *Organizational Behavior*: to determine the effect of changes in management structure on productivity *Strategy*: to determine a competitor's response to a price increase *Operations*: to determine production costs after the modernization of a manufacturing plant

The Valuation Principle

*_____*... The best decision makes the firm and its investors wealthier, because the value of its benefits exceeds the value of its costs. We call this idea _____: _____: *The value of a commodity or an asset to the firm or its investors is determined by its competitive market price. The benefits and costs of a decision should be evaluated using those market prices. When the value of the benefits exceeds the value of the costs, the decision will increase the market value of the firm.*

Rule of 72

*_____*... Years to double ≈ 72 / interest rate in percent This simple "_____" is fairly accurate (that is, within one year of the exact doubling time) for interest rates higher than 2%. For example, if the interest rate is 9%, the doubling time should be about 72 / 9 = 8 years. indeed, 1.09⁸ = 1.99! So given a 9% interest rate, your money will approximately double every 8 years.

timeline

A linear representation of the timing of (potential) cash flows.

competitive market

A market in which a good can be bought *and* sold at the same price.

arbitrage opportunity

Any situation in which it is possible to make a profit without taking any risk or making any investment.

compounding

Computing the return on an investment over a long horizon by multiplying the return factors associated with each intervening period.

discounting

Finding the equivalent value today of a future cash flow by multiplying by a discount factor, or equivalently, dividing by 1 plus the discount rate.

Law of One Price

In competitive markets, securities with the same cash flows must have the same price.

simple interest

Interest earned only on the initial deposit (Figure 3.2 shows how compound interest is more profitable than _____).

interest rate factor

One plus the interest rate, it is the rate of exchange between dollars today and dollars in the future. It has units of "$ in the future/$ today."

Inflows, Outflows

Regarding timelines, ___1___ are positive (when money comes in) and ___2___ are negative (use a negative sign when money goes out).

discount rate

The appropriate rate to discount a cash flow to determine its value at an earlier time.

time value of money

The difference in value between money received *today* and money received *in the future*; also, the observation that two cash flows at two different points in time have different values.

compound interest

The effect of earning "interest on interest."

arbitrage

The practice of buying and selling equivalent goods to take advantage of a price difference.

interest rate

The rate at which money can be borrowed or lent over a given period. -Market price of money! -Equilibrium price for supply and demand for money! -Time preference for money -Discount factor -The _____ converts cash across time --By depositing money (lending), we convert money today into money in the future (compounding) --By borrowing money, we exchange money in the future for money today (discounting)

future value (FV)

The value of a cash flow that is moved forward in time.

present value (PV)

The value of a cost or benefit computed in terms of cash today.

discount factor

The value today of a dollar received in the future.

notation

_____: r - interest rate PV - present value FV - future value C = cash flow n = number of periods