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Chapter 3

Understanding The Time Value of Money

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Time Value of Money A dollar received

today is worth more than a dollar received in the future.

The sooner your money can earn interest, the faster the interest can earn interest.

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Interest and Compound Interest

Interest -- is the return you receive for investing your money.

Compound interest -- is the interest that your investment earns on the interest that your investment previously earned.

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Future Value Equation

FVn = PV(1 + i)n

– FV = the future value of the investment at the end of n year

– i = the annual interest (or discount) rate– PV = the present value, in today’s dollars, of a

sum of money

This equation is used to determine the value of an investment at some point in the future.

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Compounding Period

Definition -- is the frequency that interest is applied to the investment

Examples -- daily, monthly, or annually

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Reinvesting -- How to Earn Interest on Interest

Future-value interest factor (FVIFi,n) is a value used as a multiplier to calculate an amount’s future value, and substitutes for the (1 + i)n part of the equation.

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The Future Value of a Wedding

In 1998 the average wedding cost $19,104. Assuming 4% inflation, what will it cost in 2028?

FVn = PV (FVIFi,n)

FVn = PV (1 + i)n

FV30 = PV (1 + 0.04)30

FV30 = $19,104 (3.243)

FV30 = $61,954.27

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The Rule of 72

Estimates how many years an investment will take to double in value

Number of years to double =

72 / annual compound growth rateExample -- 72 / 8 = 9 therefore, it will

take 9 years for an investment to double in value if it earns 8% annually

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Compound Interest With Nonannual Periods

The length of the compounding period and the effective annual interest rate are inversely related; therefore, the shorter the compounding period, the quicker the investment grows.

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Compound Interest With Nonannual Periods (cont’d)

Effective annual interest rate =

amount of annual interest earned

amount of money invested

Examples -- daily, weekly, monthly, and semi-annually

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The Time Value of a Financial Calculator

The TI BAII Plus financial calculator keys– N = stores the total number of payments– I/Y = stores the interest or discount rate– PV = stores the present value– FV = stores the future value– PMT = stores the dollar amount of each annuity

payment– CPT = is the compute key

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The Time Value of a Financial Calculator (cont’d)

Step 1 -- input the values of the known variables.

Step 2 -- calculate the value of the remaining unknown variable.

Note: be sure to set your calculator to “end of year” and “one payment per year” modes unless otherwise directed.

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Tables Versus Calculator

REMEMBER -- The tables have a discrepancy due to rounding error; therefore, the calculator is more accurate.

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Compounding and the Power of Time

In the long run, money saved now is much more valuable than money saved later.

Don’t ignore the bottom line, but also consider the average annual return.

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The Power of Time in Compounding Over 35 Years

$0

$50,000

$100,000

$150,000

$200,000

Selma Patty

Selma contributed $2,000 per year in years 1 – 10, or 10 years.

Patty contributed $2,000 per year in years 11 – 35, or 25 years.

Both earned 8% average annual return.

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The Importance of the Interest Rate in Compounding

From 1926-1998 the compound growth rate of stocks was approximately 11.2%, whereas long-term corporate bonds only returned 5.8%.

The “Daily Double” -- states that you are earning a 100% return compounded on a daily basis.

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Present Value

Is also know as the discount rate, or the interest rate used to bring future dollars back to the present.

Present-value interest factor (PVIFi,n) is a value used to calculate the present value of a given amount.

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Present Value Equation

PV = FVn (PVIFi,n)– PV = the present value, in today’s dollars, of a

sum of money

– FVn = the future value of the investment at the end of n years

– PVIFi,n = the present value interest factor

This equation is used to determine today’s value of some future sum of money.

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Calculating Present Value for the “Prodigal Son”

If promised $500,000 in 40 years, assuming 6% interest, what is the value today?

PV = FVn (PVIFi,n)

PV = $500,000 (PVIF6%, 40 yr)

PV = $500,000 (.097)

PV = $48,500

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Annuities

Definition -- a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods.

Examples -- 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 -- saving $50 a month to buy a new stereo two years in the future– By allowing the money to gain interest and

compound interest, the first $50, at the end of two years is worth $50 (1 + 0.08)2 = $58.32

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Future Value of an Annuity Equation

FVn = PMT (FVIFAi,n)

– FVn = the future value, in today’s dollars, of a sum of money

– PMT = the payment made at the end of each time period

– FVIFAi,n = the future-value interest factor for an annuity

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Future Value of an Annuity Equation (cont’d)

This equation is used to determine the future value of a stream of payments invested in the present, such as the value of your 401(k) contributions.

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Calculating the Future Value of an Annuity: An IRA

Assuming $2000 annual contributions with 9% return, how much will an IRA be worth in 30 years?

FVn = PMT (FVIFA i, n)

FV30 = $2000 (FVIFA 9%,30 yr)

FV30 = $2000 (136.305)

FV30 = $272,610

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Present Value of an Annuity Equation

PVn = PMT (PVIFAi,n)

– PVn = the present value, in today’s dollars, of a sum of money

– PMT = the payment to be made at the end of each time period

– PVIFAi,n = the present-value interest factor for an annuity

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Present Value of an Annuity Equation (cont’d)

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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Calculating Present Value of an Annuity: Now or Wait?

What is the present value of the 25 annual payments of $50,000 offered to the soon-to-be ex-wife, assuming a 5% discount rate?

PV = PMT (PVIFA i,n)

PV = $50,000 (PVIFA 5%, 25)

PV = $50,000 (14.094)

PV = $704,700

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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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Buying a Car With Four Easy Annual Installments

What are the annual payments to repay $6,000 at 15% interest?

PV = PMT(PVIFA i%,n yr)

$6,000 = PMT (PVIFA 15%, 4 yr)

$6,000 = PMT (2.855)

$2,101.58 = PMT

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Perpetuities

Definition – an annuity that lasts foreverPV = PP / i

– PV = the present value of the perpetuity– PP = the annual dollar amount provided by

the perpetuity– i = the annual interest (or discount) rate

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Summary

Future value – the value, in the future, of a current investment

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

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Summary (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

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

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Summary (cont’d)

Amortized loans – loans paid in equal periodic installments for a specific length of time

Perpetuities – annuities that continue forever