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


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.


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


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


Figure 3.1 Compound Interest at 6 Percent Over Time


Time Value of Money Calculator

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


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.


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


Figure 3.3 The Power of Time in Compounding


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.


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%?


Solution

0

100

6

10

-55.84


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.


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.


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


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?


Solution

-1010.36

10000

6

8

0


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%.


Solution

500

0

6

5

-2106.18


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


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?


Solution

0

8

4

16000

-4830.73


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


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.


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’.


Retirement Income Needs

74,598

190,718


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


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


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


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


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.


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.


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.


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.

chapter 03 understanding and appreciating the time value of money with audio sum 14

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