Understanding the time value of money (annuity)

Chapter 3: Time
Value of Money (part
2)

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)

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

 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

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.

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.

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.

SOLVING FOR FUTURE VALUE
OF AN ANNUITY (MULTIPLE)
 The future value is the unknown
 CPT FV

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

Financial Calculator
 PMT = -2000
 I/Y = 9
 N = 30
 CPT FV = 272,615

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?

 PMT = -5000
 I/Y = 6
 N = 12
 CPT FV = $84,349.70 at the end of one
year

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?

Yearly
 PMT = -$1,200
 I/Y = 2
 N = 15
 CPT FV= $20,752.10

Extreme Caution!
 Make double sure your time frames are
consistent……..
 Ifthe 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

Monthly
 PMT = $-1,200
 I/Y = .1667 [2/12]
 N = 180 [15*12]
 CPT FV = $251,655.66

Present value (moves backward) &
Future value (moves forward)
 In real life: Winning
the lottery (present
value) or saving for
retirement (future
value)

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.

SOLVING FOR PRESENT VALUE
OF AN ANNUITY (MULTIPLE)
 The Present Value is the unknown
 CPT PV

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!

Financial Calculators
 PMT = -50,000
 N = 25
 I/Y = 5
 CPT PV = $704,697.22

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

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

Solving for the PMT
 No more hypothetical “what ifs”

 You can really use this stuff!

SOLVING FOR PAYMENT
 The Payment is the unknown
 CPT PMT

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

 PV = 6,000
 I/Y = 15
N=4
 CPT PMT = -2,101.59

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

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

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?

Review:
 Future value – the value, in the future, of
a current investment
 Formula?

 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 future
 Formula?

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
investments
 Formula?
 Present value of an annuity – today’s
value of a stream of investments
received in the future
 Formula?

Review (cont’d)
 Amortized loans – loans paid in equal
periodic installments for a specific length
of time

Understanding the time value of money (annuity)

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