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A Time Value of Money Primer

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A Time Value of Money Primer

By David B. Hamm, MBA, CPA

for Finance and Quantitative Methods

Modules

In its most basic form, interest is calculated by multiplying principal (amount invested) by rate (percent of interest) multiplied by time (number of periods the interest is calculated). This is called simple interest.

I = P r t

Example: A $1,000 deposit at 8% per year for three years' simple interest: I = (1000)(.08)(3) = 240 A $1000 deposit at 8% simple interest for three years earns $240 interest.

The future value (FV) of a simple interest calculation is derived by adding the original principal back to the interest earned.

$1,000 + $240 = $1,240

Expressed as a formula:

FV = P(1 + rt)

FV = (1000)+(1000)(.08)(3) = 1240

Note: usually simple interest is used in financial institutions for interest periods of less than one year. If the rate is expressed as an annual rate (normal practice), then the time period (t) must be a fraction of a year. Example: we invest $10,000 in an 8% , 90-day certificate of deposit. Our total proceeds at the end of the CD period are:

FV = (10000)+(10000)(.08)(90/365) = $10,197.26

Often, if a bank or other financial institution loans a sum for a short term, the lender will prefer to calculate the interest up front and loan out the discounted principal, or principal minus interest to be earned. The interest to be paid up front on a loan is called discount and the discounted principal, or the actual amount loaned is called the present value (PV)

FV

PV = (1+rt)

Repeating the discount basic formula (simple interest):

FV

PV = (1+rt)

Example: If the bank loans out $10,000 for 90 days at 8% simple interest, the PV is:

PV = 10000 / [1 + (.08)(90/365)]

= 10000/ 1.019726

= $9,806.56

However, if interest is left in the account to accumulate for a longer period (usually longer than one year) common practice (and usually state law!) requires that after interest is earned and credited for a given period, the new sum of principal + interest must now earn interest for the next period, etc. This is compound interest. To distinguish from simple interest, we use "n" to refer to the number of "periods" in which the interest is compounded and added to principal.

FV

FV = P(1 + r)n PV = (1+r)n

Suppose we invest our original $1,000 for three years at 8%, compounded quarterly: (The rate per quarterly period is 8% / 4 or 2%. The number of periods (n) is 3 x 4 = 12 quarterly periods.)

FV = (1000)(1.02)12 = $1,268.24

If we wanted to know how much we'd have to invest now (PV) at 8% compounded quarterly to earn $10,000 in three years:

PV = 10000 / (1.02)12 = $7,884.93

Because raising interest factors to an exponent of "n" was a difficult calculation before calculators, some mathematicians used logarithmic functions to calculate the exponent factor. Financial professionals acquired tables of these functions so that either of the above problems could be calculated simply by looking up a FV factor (or to discount, a PV factor) based on the interest rate and number of compounding periods and multiplying the principal by the interest factor.

Now, computerized spreadsheets can build in these financial functions and easily do the work for us. It will be our assumption in this class that you will have a computer present to calculate these financial functions. Our discussion will be based on MS Excel, but Quattro and Lotus and most other major spreadsheets have similar function capability.

In the spreadsheet, it is often advisable to set up and identify cells for your principal, your interest rate, and the number of time periods. Setting up a simple template in this fashion means you can easily update your template for new calculations just by changing amounts in the cells.

Principal-1000

Rate (yr)8%

Yrs3

Quick note: In Excel, present value (PV) is assumed to be a cash outlay, and is thus expressed as a negative value.

The mathematical functions are accessed on the Excel taskbar with the " fx" key.

Select "Financial" functions.

We will most commonly compute =FV (future value) or =PV (present value). Each of the functions in Excel pops up a simple menu to follow to identify data. If you have annual rates or periods that need conversion to semiannual, quarterly, or monthly compounding, the function can multiply the number of periods or divide the rate for you in the menu cell.

1.Try this problem in Excel:

Invest $1,000 (present value) at 8% annual interest compounded quarterly for three years to see how much we can receive (future value) (hint: use the =FV function)

2. Now the reverseâ€”how much would we have to invest now (present value) at 8% compounded quarterly to receive $10,000 (future value) in three years? (use the =PV function)

Again, Excel displays the PV amount as negative.

Not so long ago, banks and finance companies frequently calculated simple interest on consumer loans using the add-on interest method:

Payment = P +I

n

Add principal + interest over the life of the loan and divide by the number of payments.

Example: a $5,000 car loan at 8% simple interest for 3 years = $1,200 interest. Therefore ($5,000 +1,200) / 36 months = $172.22 monthly payment

Problem: this was charging interest on the full $5,000 for the whole life of the loan despite the principal being partially paid down each month. The true annual interest rate was therefore much higher than 8%. (Using a financial function, the true APR (annual % rate) would be 14.55% for the full 36 months.)

Current Federal and state consumer law requires that the stated interest rate be reported not only as the simple rate, but also as the true APR. Add-on loans, while still used, are therefore much less popular or common today.

Some â€œrevolving credit" accounts, such as some store credit cards, calculate finance charges monthly based on the unpaid balance from the previous month--the unpaid balance method.

I = Prt but P = previous balance + finance charge + new charges - returns or payments.

- Most bank credit cards use the average daily balance
- method which computes the number of days in each
- month from date of each transaction and divides by
- the number of days in the month to figure an average
- daily balance to be entered into the I = Prt formula.
- Add outstanding balance for account for each day of the previous month
- Divide Step 1 total by number of days in previous month = average daily balance
- Use I= prt to find finance charge, where P is average daily balance in Step 2, r is annual interest rate, and t is no. of days in month/ 365.

An annuity is an interest bearing account into which we make, or we receive, payments of an equal amount each period until the annuity ends.

If the payment is made on the last day of each period, it is an ordinary annuity. (This is most typical and what we will illustrate.)

If the payment is made on the first day of each period, it is an annuity due. (not as common) MS Excel identifies the two types as "0" or blank=ordinary; "1" =annuity due.

Some annuities have no "fixed" ending date, but rather continue for the life of the recipient. These are usually called life annuities and the payment is calculated for a number of periods based on life expectancy.

A perpetuity is an annuity with no ending date. (An example of a perpetuity is an endowed scholarship, where only interest is paid out as scholarship funds and the endowment principal remains invested "forever" or in perpetuity.)

A sinking fund is a fund in which a regular annuity payment is made to accumulate to a future value to be used for some future purpose, such as paying off a bond issue or some other obligation.

Before calculators, polynomials and logarithmic functions were used to calculate annuity tables for financial use. Now, we can simply use spreadsheet financial functions, usually using =PV, =FV, or =PMT in Excel and now inserting payment information where applicable.

Illustration: We need to accumulate a sinking fund of $100,000 in ten years (120 months) to pay off a note payable. If we can invest our funds at 8% compounding monthly, how much must we deposit per month?

Excel functions are available to find any of the above variables, if we have the others.

Illustration (2): When Joe retires on his 65th birthday, his retirement fund carries a balance of $240,000. If Joe transfers this balance into a fund earning 8% to pay him or his heirs $2,000 per month until the fund is exhausted, how long can this annuity last?

Approx 242 monthsâ€”just over 20 years! (Assuming 8% is consistent and there is no risk of loss of principal!)

Finally, if we take out a long term loan, such as a mortgage, or a car loan based on the true APR, the interest expense is calculated for each month based on the unpaid balance of the loan. A fixed monthly payment is computed from which is first deducted the monthly interest, and the balance is applied to reduce principal. The new interest is then recalculated the next month based on the lower principal. This generates a schedule of all loan payments, interest and principal applied, and outstanding balance called an amortization schedule.

In the early months of an amortization schedule, much (perhaps most) of the monthly payment goes toward interest because the unpaid balance is so large. As the principal is paid down, more and more of each payment is applied toward principal.

Example: in a 30 year $100,000 home mortgage at 9%, the

required monthly payment is $804.63 (round up 1 cent)

Of the $804.63 payment, the first month's interest is $750.00 (100,000 x .09/12). Therefore only

804.63-750.00 = $54.63 goes toward principal.

But by the last month of the mortgage, only about $785.22 is left unpaid. Thus only $5.90 goes to interest and the last loan payment is $791.12 to zero out the loan.

In fact it is not until payment #269 (22 years, 5 months into the loan) when the interest portion of the payment is less than the principal portion! Ultimately we would pay $189,653.30 in interest on our $100,000 loan over the 30 years!

We can build an amortization table using an Excel spreadsheet to calculate the principal & interest portion of all our payments:

This spreadsheet can be extended through all 360 monthly payments to total principal and interest paid to the end of the mortgage

End of presentation