Chapter 03 Understanding and Appreciating the Time Value of Money With Audio Sum 14

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Chapter 3

Understanding and Appreciating the

Time Value of Money

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Compound Interest andFuture Values

• Interest paid on interest (Compound interest).

• Reinvestment of interest paid on an investment’s principal

• Principal is the face value of the deposit or debt instrument.

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How Compound Interest Works

• Future value (FV) = Present Value (PV) x Amount it has increased by the end of 1 year (1+i)

• Future value—the value of an investment at some point in the future

• Present value—the current value in today’s dollars of a future sum of money

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How Compound Interest Works

• Annual compounding—reinvesting interest at end of each year for more than 1 year

• FV = PV x Amount Present Value has

increased by the end of n years (1+i)n

• n is equal to the number of years during which compounding occurs

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Figure 3.1 Compound Interest at 6 Percent Over Time

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Time Value of Money Calculator

http://www.zenwealth.com/BusinessFinanceOnline/TVM/TVMCalcIntro.html

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Financial Calculator TVM input keys

Note that some calculators (TI) have the CPT key but others (HP) initiate calculation when you press the key for which you want the solution.

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Calculator Clues

Before solving problem:1. Set to one payment per year2. Set to display at least four decimal places3. Set to “end” mode

Working a problem:1. Positive and negative numbers2. Enter zero for variables not in the problem3. Enter interest rate as a %, 10 not 0.10

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Figure 3.3 The Power of Time in Compounding

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Present Value

• What’s a future amount worth in today’s dollars?

• Inverse of compounding.

• Discount rate is the interest rate used to bring future money back to present.

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Present Value Example

• You’re on vacation in Florida and you see an advertisement stating that you’ll receive $100 simply for taking a tour of a model condominium.

• You discover that the $100 is in the form of a savings bond that will not pay you the $100 for 10 years.

• What is the PV of the $100 to be received 10 years from today if your discount rate is 6%?

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Solution

0

100

6

10

-55.84

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Annuities

• An annuity is a series of equal dollar payments coming at the end of each time period for a specific number of time period.

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Compound Annuities

• A compound annuity involves depositing an equal sum of money at the end of each year for a certain number of years, allowing it to grow.

• You want to know how much your savings will have grown by some point in the future.

• Sum up a number of future values.

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Table 3.4 Illustration of a 5-Year $500 Annuity Compounded at 6%

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Compound Annuities Example

• You’ll need $10,000 for education in 8 years. How much must you put away at the end of each year at 6% interest to have the college money ready?

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Solution

-1010.36

10000

6

8

0

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Present Value of an Annuity

• To compare the relative value of annuities, you need to know the present value of each.

• Need to know what $500 received at the end of the next 5 years is worth given discount rate of 6%.

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Solution

500

0

6

5

-2106.18

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Table 3.6 Illustration of a 5-Year $500 Annuity Discounted Back to the Present at 6%

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Amortized Loans

• Loans paid off in equal installments.

• You borrow $16,000 at 8% interest to buy a car and repay it in 4 equal payments at the end of each of the next 4 years. What are the annual payments?

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Solution

0

8

4

16000

-4830.73

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Figure 3.5 Loan Amortization Schedule Involving a $16,000 Loan at 8% to BeRepaid in 4 Years

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Perpetuities

• A perpetuity is an annuity that continues to pay forever.

• Present value of a perpetuity = annual dollar amount provided by the perpetuity divided by the annual interest (or discount) rate.

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Example of a Perpetuity

A social security retirement payment is the equivalent of a perpetuity.

A $2500 per month payment assuming a 4% rate of return would have a Present Value of $750,000.

2500/.003333=750,000

This would be part of your retirement ‘nest egg’.

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Retirement Income Needs

74,598

190,718

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How much do you need to save each month for 30 years in order to retire on $145,000 a year for 20 years, i = 10%?

0 360 2 201 2

PMT PMT PMT

...

1 19

months before retirement years after retirement

-145k -145k -145k -145k

...Age67

Age37

Age87

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How much must you have in your account on the day you retire if i = 10%?

How much do you need on this date?

2 20...

1 19

years after retirement

-145k -145k -145k -145k

...

0Age67

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You need the present value of a20- year 145k annuity--or $1,234,467.

20 10 -145000 0

N I/YR PV FVPMT

1,234,467

INPUTS

OUTPUT

29

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How much do you need to save each month for 30 years in order to have the $1,234,467 in your account?

You need $1,234,467

on this date.0 3601 2

PMT PMT PMT

...

months before retirement

...Age67

Age37

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You need a payment such that the future value of a 360-period annuity

earning 10%/12 per period is $1,234,467.

360 10/12 0 1234467

N I/YR PV FVPMT

-546.11

INPUTS

OUTPUT

It will take an investment of $546.11 per month to fund your retirement.

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What if you have 40 years in which to accumulate your next egg?

480 10/12 0 1234467

N I/YR PV FVPMT

-195.20

INPUTS

OUTPUT

Now it will only take an investment of $195.20 per month to fund your retirement starting at age 27.

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What if you have 45 years in which to accumulate your nest egg?

540 10/12 0 1234467

N I/YR PV FVPMT

-117.76

INPUTS

OUTPUT

Now it will only take an investment of $117.76 per month to fund your retirement starting at age 22.

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Summary

• The cornerstone of time value of money is compound interest.

• A higher interest rate (higher risk) or the number of years that your money is compounded for increases future values.

• An annuity is a equal dollar periodic payment of investment earnings or paying off installment loans.