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Time Value of Money

Who would prefer $30,000 today versus $30,000 two years from now? Why?

Interest = amount charged for use of money.

Simple versus compound interest

8th wonder of the world: compound interest.

Rule of “72” works as a close approximation with

reasonable interest rates. (How long will it take $100 to double at 8% annual interest? (Rule of 72 = 72 / 8 = 9 years!) FVF(9,8%) = 1.993

Future value (FV) of a single amount (P) = P (1 + i)n

Present value (PV) = FV / (1 + i)n .

Please note that P = PV (at time 0, any amount received today has a PV equal to that amount).

When you look in a FV single-amount table, you will find numbers all greater than 1. These are the “table factors” and represent the (1 + i) n above.

The numbers in PV single-amount tables represent 1 / (1 + i)n above. These numbers are all less than 1.

Present value calculations are frequently used in accounting. Present and future values are common inputs for everyday decisions in your life.

Annuities

Annuities (equal cash flows for a number of periods) come in two basic types:

(1) ordinary (regular, in arrears), cash flows at end of period.

Interest payments

Car/house payments

(2) due (in advance), cash flows at the beginning of the period.

Insurance payments

Rent payments

Future value of an ordinary annuity is calculated for the end of the period in which the last payment occurs (i.e. on day of last payment).

Future value of an annuity due is calculated for the end of the period one period after the last payment occurs (i.e. one period after the last payment).

Present value of annuities is equally confusing. With ordinary annuities, you calculate the present value (beginning of 1st period) of cash flows that do not start until the end of the first period and stop at the end of the nth period. With annuities due, you calculate the PV (beginning of 1st period) of cash flows that start at the beginning of the first period and end at the beginning of the nth period (same as the end of the n-1 period).

It is usually very helpful to draw a time line with payments indicated.

REMEMBER that PV is the most useful concept in both accounting and your life (than anything that you will ever learn).

The beauty of PV is that it is an equivalent amount for future cash flows—it is a way to equate multiple future cash flows to one single present amount.

You can obtain the “table factors” from tables, financial calculators, computation, Excel spreadsheets.

Deferred Annuity: the first cash flow occurs more than one period after the agreement. Simple to calculate the PV of this kind of annuity: 2 steps. First calculate the PVA of the future annuity as of the beginning of the annuity and then take the PV(as of the agreement date) of this PVA.

Double-back-to-back reverse annuity: (tricky)

How much do I have to save at the end of the next 10 years in order to be able to have $72,000 (starting with the middle of year 11) each year for 20 years of retirement? Assume that my savings will grow at 5% (after tax) until I retire and 4% after I retire.

Rates

APR (annual percentage rate) = periodic rate*number of periods/yr. E.g., 1% monthly return = 12% APR

APY (annual percentage yield) = effective or true rate, takes into account compounding. E.g., 1% / month = 12.68% APY [((1.01)^12) – 1].

ARR (annual rate of return) can be calculated different ways and therefore, could be either APR or APY or something else.

Credit card companies advertise 18% APR, this equals 1.5%/month which equals 19.56% APY.

Interperiod interest is prorated (e.g., 4% APR compounded annually means that 6 month rate is .04/2 = 2%.) If annual compounding (rare these days), then APR = APY!

How is an APR calculated? Annual Percentage Rate (APR) Formula. The APR is calculated using the following formula. APR Formula. APR = (Periodic Interest Rate * 365 Days) * 100; Where: Periodic Interest Rate = [(Interest Expense + Total Fees) / Loan Principal] / Number of Days in Loan Term; To express the APR as a percentage, the amount must be multiplied by 100.