1. Understanding The Time Value of Money
Chapter 3
Understanding The Time Value of Money

2. 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.

3. 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.

4. 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.

5. Compounding Period
Definition -- is the frequency that interest is applied to the investment
Examples -- daily, monthly, or annually

6. 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.

7. 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 ( )30
FV30 = $19,104 (3.243)
FV30 = $61,954.27

8. 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 rate
Example / 8 = 9 therefore, it will take 9 years for an investment to double in value if it earns 8% annually

9. 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.

10. 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

11. 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

12. 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.

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

14. 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.

15. The Power of Time in Compounding Over 35 Years
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.

16. The Importance of the Interest Rate in Compounding
From 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.

17. 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.

18. 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.

19. 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

20. 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.

21. 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 ( )2 = $58.32

22. 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

23. 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.

24. 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 ( )
FV30 = $272,610

25. 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

26. 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.

27. 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

28. 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

29. 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, = PMT

30. Perpetuities Definition – an annuity that lasts forever PV = 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

31. 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

32. 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

33. Summary (cont’d)
Amortized loans – loans paid in equal periodic installments for a specific length of time
Perpetuities – annuities that continue forever