ch05

(1)

CHAPTER 5

The Time Value of Money

(2)

Time Value of Money

Future Value and Compounding Present Value and Discounting Finding the Interest Rate

Rule of 72

(3)

 TVM is based on the belief that people prefer to

consume goods today rather than wait to consume similar goods tomorrow.

 _{People have a positive time preference.}

The Time Value of Money

Consuming Today or Tomorrow

 Money has a time value because a dollar today

(4)

 Today’s dollar can be invested to earn interest

or spent.

 _{Value of a dollar invested (positive interest}

rate) grows over time.

 _{Rate of interest determines trade-off between}

spending today versus saving.

 _{Example: Inflation 18%, saving 14%, which is}

better, spend now or save now, spend later?

The Time Value of Money

(5)

Timelines as Aids to Problem Solving

 _{Timelines are an easy way to visualize cash}

flows.

(6)

Exhibit 5.1: Five-year

Timeline for $10,000

(7)

Future Value versus Present Value

 _{Future value measures what one or more cash}

flows are worth at the end of a specified period.

 _{Present value measures what one or more}

cash flows that are to be received in the future will be worth today (at t=0).

 _{Financial decisions are evaluated either on a}

(8)

The Time Value of Money

 Discounting is the process of converting future

cash flows to their present values.

 Discounting rate

 _{Compounding is the process of earning}

interest over time.

(9)

Exhibit 5.2: Future Value &

(10)

Future Value and

Compounding

Single Period Investment

 _{We can determine the value of an investment}

at the end of one period if we know the interest rate to be earned by the investment.

 _{If you invest for one period at an interest rate}

of i, your investment, or principle, will grow by (1 + i) per dollar invested.

 _{The term (1+}_i_{) is the future value interest factor,}

(11)

Future Value and

Compounding

Two-Period Investing

 _{After the first period, interest accrues on}

original investment (principle) and interest earned in preceding periods.

 _{A two-period investment is simply two}

(12)

 _{The principal is the amount of money}

on which interest is paid.

 _{Simple interest is the amount of interest paid}

on the original principal amount only.

 _{Compounding interest consists of both simple}

interest and interest-on-interest.

Future Value and

Compounding

(13)

Future Value and

Compounding

 _{General equation to find the future value after}

any number of periods. The Future Value Equation

 _{We can use financial calculators or future}

value tables to find the future value factor at

(14)

where:

FV_n = future value of investment at the end of period n PV = original principle (P₀) or present value

i = the rate of interest per period, which is often a year n = the number of periods

(5.1) n i 1 PV

FV_n  (  )

The general equation to find the future value is:

Future Value and

(15)

You leave your $100 invested in the bank savings account at 10 percent interest for five years. How much would you have in the bank at the end of five years?

Future value example

Future Value and

Compounding

5 5

5

FV $100 (1 0.10)

= $100 (1.10)

  

(16)

Exhibit 5.4: How Compound

(17)

Compounding More Frequently Than Once a Year

The more frequently the interest payments are compounded, the larger the future value of $1 for a given time period.

Future Value and

Compounding

m×n n

(18)

Non-annual compounding example

Future Value and

Compounding

2 2 2

4

FV $100 (1+0.05 / 2)

= $100 (1+0.025) = $100 (11038) = $110.38



 

 

You invest $100 in a bank account that pays a 5 percent interest rate semiannually for two years.

(19)

When interest is compounded on a continuous basis, we can use the equation below.

where: e = exponential function which is about 2.71828

Future Value and

Compounding

(20)

Continuous compounding example

Future Value and

Compounding

0.05 5

0.25

FV = $10,000

= $10,000 2.71828 = $10,000 1.284025 = $12,840.25

e 

 

 

Your grandmother wants to put $10,000 in a

(21)

Using Excel: Time Value of

(22)

Using Excel: Time Value of

Money

_{Excel also has the following functions for time}

value of money problems.

PV: PV(RATE, NPER, PMT, FV)

FV: FV(RATE, NPER, PMT, PV)

Discount Rate: RATE(NPER, PMT, PV, FV)

Payment: PMT(RATE, NPER, PV, FV)

(23)

Present Value and

Discounting

Present value calculations state the current value of a dollar in the future.

 _{This process is called discounting}_,_{and the}

interest rate i is known as the discount rate.

 _{The present value (PV)} _{is often called the}

discounted value of future cash payments.

(24)

The equation below gives us the general equation to find the present value after any number of periods.

(5.4) n i) (1 n FV PV  

Present Value and

(25)

Suppose you are interested in buying a new BMW 330 Sports Coupe a year from now. You estimate that the car will cost $40,000. If your local bank pays 5 percent interest on savings deposits, how much will you need to save to buy the car?

Present value example

Present Value and

Discounting

(26)

Present Value and

Discounting

_{The further in the future a dollar will be received, the less}

it is worth today.

_{The higher the discount rate, the lower the}

present value of a dollar.

(27)

Finding the Interest Rate

A number of situations will require you to determine

the interest rate (or discount rate) for a given stream of future cash flows.

 _{to determine the interest rate on a loan.}  _{to determine the return on an investment.}

(28)

Examples

 Buy government bond price $90 each, in next

(29)

The Rule of 72

 _{Rule of 72 is used to determine the amount of}

time it takes to double an investment.

 _{It says that the time to double your money}

(TDM) approximately equals 72/i, where i is expressed as a percentage.

(30)

Compound Growth Rates

Compound growth occurs when the initial value of a number increases or decreases each period by the factor (1 + growth rate).

(5.6)

n

g)

(1

PV

n

FV







_{Examples include population growth, earnings}

(31)

Compound Growth Rates

Compound growth rate example

Because of an advertising campaign, a firm’s sales increased from $20 million in 2007 to more than $35 million three years later. What has been the average annual growth rate in sales?

3