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?

Financial Calculator

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…….. 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

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

Financial Calculator

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

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

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 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?

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?

Review (cont’d) Amortized loans – loans paid in equal

periodic installments for a specific length of time