understanding the time value of money; single payment

Calculators must be set to 4 decimal places

Calculators must be set to 1 payment per year

Chapter 3

Understanding The Time Value of Money:

Time Value of Money A dollar received today is worth

more than a dollar received in the future.

The sooner your money can earn interest, the faster the interest can earn interest.

Interest and Compound Interest Interest (i) -- is the return you receive for

investing your money. Compound interest -- is the interest that

your investment earns on the interest that your investment previously earned.

Inflation (r) – is when rising prices reduce the purchase power of money.

The effect of 3% interest on a one time deposit of $100 Reminders……3% = .03 in decimal form On your calculator hit the 3 key and then

hit the % key (4th row from the top left hand side)

The effect of 3% interest on a one time deposit of $100 Deposit ($X) + Deposit ($X) times interest rate (i) = new account balance For example: $100 ($X) + $ 3 [100 (.03) = $X(.03)] $103

Or in one easy step:

Your deposit ($X) multiplied by (1 + the interest rate—in decimal form) = new acct. balance

= $X(1 + i) = new account balance = $100(1.03) = $103

$X + $X(i) = $X(1 + i)

The effect of compounding interest over time (long form)

= $X1 + [$X1(i)] = $X2

= $X2 + [$X2(i)] = $X3

= $X3 + [$X3(i)] = $X4 ………. = $100x1 + [$100x1(.03)] = $103x2

= $103x2 + [$103x2(.03)] = $106.09x3

= $106.09x3 + [$106.09x3(.03)] = $109.27x4

In other words (short form)

X(1 + i)3 = X4 OR

X(1 + i)n

X = $$ deposit I = % interest rate Where n = number of periods you are

compounding

Practice problems

How much will you have in savings if you deposit $10 and leave it in an account earning 5% interest compounded annually for 10 years?

How much will you have in savings if you deposit $100 in an account earning 12% compounded annually for 20 years?

No financial calculator

10 * 1.05 to the 10th 10(1.05)10

With your financial calculator you enter the number and then tell it where to go…..

Key in -10 then hit the PV key Key in 5 then hit the i/y key (don’t change

to decimal it does it for you) Key in 10 hit the N key Hit CPT FV key to show answer

Financial Calculator….

PV = -10 (DOLLARS) I/Y = 5 (PERCENT) N = 10 (YEARS/PERIODS)

COMPUTE FV (FUTURE VALUE)

$-16.28 ***your answer will show up as a negative

number. That is expected because the $10 was an outflow of cash from one’s current consumption to one’s retirement account. If you don’t want your answer to show up as negative then you have to remember to make the PV negative.

Practice problems

How much will you have in savings if you deposit $100 in an account earning 12% compounded annually for 20 years?

No financial calculator

100 * 1.12 to the 20th

100(1.12)20

With your financial calculator you enter the number and then tell it where to go…..

Key in -100 then hit the PV key Key in 12 then hit the i/y key (don’t change

to decimal it does it for you) Key in 20 hit the N key Hit CPT FV key to show answer

Financial Calculator….

PV = -100 (DOLLARS) I/Y = 12 (PERCENT) N = 20 (YEARS/PERIODS)

COMPUTE FV (FUTURE VALUE)

$-964.63 ***your answer will show up as a negative

number. That is expected because the $10 was an outflow of cash from one’s current consumption to one’s retirement account. If you don’t want your answer to show up as negative then you have to remember to make the PV negative.

The Rule of 72 Estimates how many years an investment

will take to double in value Number of years to double =

72 / annual compound growth rate (%) Example -- 72 / 8 = 9 therefore, it will

take 9 years for an investment to double in value if it earns 8% annually

Determining the Future Value of your Money over time Future value (FV) is the value to which

your money will grow at a specific compounding interest rate (i).

Future value is hypothetically moving your money forward (n) numbers of periods (days, months, years).

Future Value Equation FV = PV(1 + i)n

FV = the future value of the investment at the end of (n) numbers of periods

i = the annual percentage (interest) rate (APR) PV = the present value, in today’s dollars, of a sum of

money This equation is used to determine the value of

an investment at some point in the future.

Future Value Equation

FV = PV(1 + i)n

Practice Problems:

What is the future value (FV) of $1,000 at the end of 15 years if it is invested in an account bearing 11% annually (APR)?

Financial Calculators

PV = -1000, N=15, I/Y = 11; CPT FV = $4,784.58

Practice Problems:

What is the future value (FV) of $1,500 after 20 years if it is invested in an account earning 8% annually (APR)?

Financial Calculators

PV = -1500, N = 20, I/Y = 8; CPT FV = $6,991.44

Bringing your money back from the future.

Determining the Present Value of your money Present Value (PV) Is hypothetically

moving dollars from the future back into the present at a specific interest rate (i) for a specific number of periods (n)

“inverse compounding”

Present Value Equation PV = FV(1/(1 + i)n)

PV = the present value, in today’s dollars, of a sum of money

FV = the future value of the investment at the end of n years

i = annual interest rate (%) n = number of periods

This equation is used to determine today’s value of some future sum of money.

Present Value Equation

PV = FV[1/(1 + i)n]

Practice Problems

Josh is due to receive his inheritance ($100,000) in 5 years. It is in an account earning 10% annually. Josh wants his money now. If Josh withdraws his money today, how much will he receive

Solution

PV = FV[1/(1 + i)n]PV = -100,000[1/(1.10)5]PV = $62,092

FINANCIAL CALCULATORS

FV = -100,000 N = 5 I/Y = 10 CPT PV = 62,092

Interest’s enemy: Inflation

An economic condition in which rising prices reduce the purchasing power of money.

Inflation adjusted interest rate (i*)

Substitute i* for i during PV and FV formulas

I = the interest rate R = inflation rate

Inflation adjusted interest rate (i*)

FV = PV(1 + i*)n Controlling for inflation

PV = FV[1/(1 + i*)n]Controlling for inflation

In your financial calculator the “I” in I/Y is now replaced with I* (the inflation adjusted interest rate)

You MUST calculate the I* first!!!!

Summary

FV = PV (1 + i)n

What your money will grow to be PV = FV (1/(1 + i)n

What your future money is worth today Inflation adjusted interest rate: (i*)

Substituting i* for i when controlling for inflation

Next class…

Chapter 3 part 2