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.