understanding the time value of money (annuity)
DESCRIPTION
Introductory lesson on calculating time value of money and annuities for non-finance majors.TRANSCRIPT
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Chapter 3: Time Value of Money (part 2)
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A QUICK DOUBLE CHECK
Calculator set to 4 decimal places Calculator set to END (2nd PMT/BGN key) Calculator is set to 1 payment/yr (P/Y)
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A quick review: a single deposit
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:
(1+i)/(1+r) - 1 Substituting i* for i when controlling for inflation
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What will John’s $100,000 grow to be in 15 years if he leaves it in an account earning an 8% rate of return.
PV = -100,000 I/Y = 8 N = 15 CPT FV = 317,216.91
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Annuities: multiple payments Definition -- a series of equal dollar
payments coming at the end of a certain time period for a specified number of time periods (n).
Examples – mortgages, life insurance benefits, lottery payments, retirement payments.
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Compound Annuities Definition -- depositing an equal sum of
money at the end of each time period for a certain number of periods and allowing the money to grow
Example – having $50 taken out of each paycheck and put in a Christmas account earning 9% Annual Percentage Rate.
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Future Value of an Annuity (FVA) Equation This equation is used to determine the
future value of a stream of deposits/ payments (PMT) invested at a specific interest rate (i), for a specific number of periods (n)
For example: the value of your 401(k) contributions.
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SOLVING FOR FUTURE VALUE OF AN ANNUITY (MULTIPLE) The future value is the unknown CPT FV
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Calculating the Future Value (FVA) of an Annuity:Assuming a $2000 annual contribution with a 9% rate of return, how much will an IRA be worth in 30 years?
FVA = PMT {[(1.09)30 – 1]/.09} FVA = $2000 {[13.27 - 1]/.09} FVA = $2000 {[12.27]/.09} FVA = $2000{136.33} FVA = $272,610
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Financial Calculator
PMT = -2000 I/Y = 9 N = 30 CPT FV = 272,615
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Solving for Future value:
Each month, Anna N. deposits her paycheck ($5,000) in an account offering a monthly interest rate of 6%. How much will Anna have in her account at the end of 1 year?
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Financial Calculator
PMT = -5000 I/Y = 6 N = 12 CPT FV = $84,349.70 at the end of one
year
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Practice Problems If Jenny deposits $1,200 each year into a
savings account earning an Annual Rate of return of 2% for 15 years, how much will she have at the end of the 15 years?
How much will she have if she deposits $1,200 each month? How much will she have if she earns interest monthly?
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Yearly
PMT = -$1,200 I/Y = 2 N = 15 CPT FV= $20,752.10
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Extreme Caution!
Make double sure your time frames are consistent…….. If the payment is a monthly payment; then the
compounding rate of return has to be a monthly rate of return.
Example: A 15% ANNUAL rate of return is equal to a monthly rate of return of 1.25%
15/12 = 1.25
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Monthly
PMT = $-1,200 I/Y = .1667 [2/12] N = 180 [15*12] CPT FV = $251,655.66
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Present value (moves backward) & Future value (moves forward)
In real life: Winning the lottery (present value) or saving for retirement (future value)
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Present Value of an Annuity (PVA) Equation This equation is used to determine the present value of a future stream of payments, such as your pension fund or insurance benefits.
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SOLVING FOR PRESENT VALUE OF AN ANNUITY (MULTIPLE) The Present Value is the unknown CPT PV
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Present Value of an Annuity: An example: Alimony
What is the present value of 25 annual payments of $50,000 offered to a soon-to-be ex-wife, assuming a 5% annual discount rate? (PVA is the only unknown)
PVA = PMT {[1 – (1/(1.05)25)]/.05}PVA = $50,000 {[1 – (1/3.38)]/.05}PVA = $50,000 {[1 – (.295)]/.05}PVA = $50,000 {[.705]/.05}PVA = $50,000 {14.10}PVA = $704,697 lump sum if she takes the pay off today!
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Financial Calculators
PMT = -50,000 N = 25 I/Y = 5 CPT PV = $704,697.22
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Future Value Annuity of that divorce settlement 25 annual payments of $50,000 invested
@ 5% results in
$2,386,354.94 A difference of:
$1,681,354.94
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Amortized Loans Definition -- loans that are repaid in equal
periodic installments With an amortized loan the interest payment
declines as your outstanding principal declines; therefore, with each payment you will be paying an increasing amount towards the principal of the loan.
Examples -- car loans or home mortgages
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Solving for the PMT
No more hypothetical “what ifs”
You can really use this stuff!
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SOLVING FOR PAYMENT
The Payment is the unknown CPT PMT
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Buying a Car With 4 Easy Annual Installments
What are the annual payments to repay $6,000 at 15% APR interest? (the payment is the unknown)
PVA = PMT{[1 – (1/(1.15)4)]/.15}$6,000 = PMT {[1 – (.572)]/.15}$6,000 = PMT {[.4282/.15]}$6,000 = PMT{2.854}$6,000/2.854 = PMT$2,102.31 = Annual PMT
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Financial Calculator
PV = 6,000 I/Y = 15 N = 4 CPT PMT = -2,101.59
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Buying the same car with monthly payments
PVA = PMT{[1 – (1/(1.0125)48)]/.0125}$6,000= PMT {[1 – (.55087)]/.0125}$6,000= PMT {[.44913/.0125]}$6,000 = PMT{35.93}$6,000/{35.93} = PMT$166.99 = monthly PMT http://www.bankrate.com
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Extreme Caution!
Make double sure your time frames are consistent…….. If the payment is a monthly payment; then the
compounding rate of return has to be a monthly rate of return.
Example: A 15% ANNUAL rate of return is equal to a monthly rate of return of 1.25%
15/12 = 1.25
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Buying the same car with monthly payments: Financial Calculator PV = 6,000 I/Y = 1.25 [15/12] N = 48 [4*12] CPT PMT = $-166.98
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Student loan payments
Guestimate your total school loans…..(PVA)
How many years to pay them off? (covert to monthly payments)
At what interest rate? R u consolidating?
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Review: Future value – the value, in the future, of
a current investmentFormula?
Rule of 72 – estimates how long your investment will take to double at a given rate of return
Present value – today’s value of an investment received in the futureFormula?
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Review (cont’d) Annuity – a periodic series of equal
payments for a specific length of time Future value of an annuity – the value, in
the future, of a current stream of investmentsFormula?
Present value of an annuity – today’s value of a stream of investments received in the futureFormula?
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Review (cont’d) Amortized loans – loans paid in equal
periodic installments for a specific length of time