Mathematical Finance

6.1I nterest and Effective Rates In this section, you will learn about various ways to solve simple and compound interest problems related to bank accounts and calculate the effective rate of interest. Upon completion you will be able to: • Apply the simple interest formula to various financial scenarios. • Apply the continuously compounded interest formula to various financial scenarios. • State the difference between simple interest and compound interest. • Use technology to solve compound interest problems, not involving continuously compound interest. • Compute the effective rate of interest, using technology when possible. • Compare multiple accounts using the effective rates of interest/effective annual yields.

Working with Simple Interest It costs money to borrow money. The rent one pays for the use of money is called interest. The amount of money that is being borrowed or loaned is called the principal or present value.

Interest, in its simplest form, is called simple interest and is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period the money is kept. Although the interest rate is often specified for a year, annual percentage rate, it may be specified for a week, a month, or a quarter, etc. When a person pays back the money owed, they pay back the original amount borrowed plus the interest earned on the loan, which is called the accumulated amount or future value.

Deﬁnition Simple interest is the interest that is paid only on the principal, and is given by

I = Prt

where, I = Interest earned or paid P = Present value or Principal r = Annual percentage rate (APR) changed to a decimal* t = Number of years*

*The units of time for r and t must be the same.

A =P + I =P + Prt A =P(1 + rt)

where, A = Future value or Accumulated value P = Present value or Principal r = Annual percentage rate (APR) changed to a decimal t = Number of years

Example1 You borrowed $2000 from a quick loan business for 3 months at a simple interest rate of 65% per year. a. Find the interest you paid on the money borrowed.

b. Find the total amount you are obligated to pay back at the end of the 3 months.

181 © TAMU Chapter 6: Mathematical Finance Working with Compound Interest

Simple interest is normally charged when the lending period is short and often less than a year. When the money is loaned, borrowed, or invested for a longer time period (mortgages, auto loans, savings), the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded.

Interest on a mortgage or auto loan is usually compounded monthly. Interest on a savings account can be compounded quarterly. Interest on a credit card can be compounded weekly or daily.

Compound Interest Formula r mt A = P 1 + m where, A = Future value or Accumulated value P = Present value or Principal r = Annual percentage rate (APR) changed to a decimal t = Number of years m = Number of compounding periods per year

Compounding Number of Compounding Periods Type per Year, m Annually 1 Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365

The formula for compound interest should look familiar as it was previously discussed as an example of an exponential function.

TVM Solver

When using the TVM Solver, you ﬁll in all but one entry.

N = m ∗ t (the total number of compounding periods or the total number of payments (assume P/Y = C/Y below)) I% = the annual interest rate (as a %) PV = P (Present value or Principal) PMT = regular payment amount per period ($0 in this section) FV = A (Accumulated amount or Future value) P/Y = the number of payments made per year (autoﬁlls as m) C/Y = m (the number of compounding periods per year) PMT: END BEGIN (the payments are made at the end of the period even if no payments are made) Then you move the cursor to the entry you are solving for and press ALPHA ENTER. The answer will appear where the cursor is located.

Caution: When using the TVM Solver all "money" entries (PV, PMT, FV) should be from the perspective of the investor or borrower.

• A negative monetary value indicates money “leaving” the investor or borrower, such as deposits, payments, or investments. • A positive monetary value indicates money “coming to” the investor or borrower, such as receiving a loan or money returned from an investment.

! If “non-money” entries are negative (N, I%, P/Y, C/Y), then some entry was entered incorrectly.

Example2 Suppose that over a six-year period, $5000 accumulated to $6539.96 in an investment certiﬁcate compounded quarterly. What was the interest rate on the investment certiﬁcate?

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN

183 © TAMU Chapter 6: Mathematical Finance

Example3 How much should be invested now into an account paying 7.25% annual interest compounded weekly for it to accumulate to $10000 in 4 years?

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN

Example4 Tuition for a local university, which your child wants to attend in 4 years, is estimated to be $11,000 a year. You have saved up $5000 and plan to invest it in an account earning 9% annual interest, compounded monthly. Will you have enough in 4 years to pay tuition if it remains at $11,000? How long would the money need to be in the account to reach your goal?

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END BEGIN

A = Pert

where, A = Future value or Accumulated value P = Present value or Principal r = Annual percentage rate (APR) changed to a decimal t = Number of years

Example5 How long will an investment need to remain in an account with a 2.4% annual interest rate, compounded continuously, in order for the investment to triple?

For comparison purposes, the government requires the bank to state their interest rate in terms of effective interest rate. This is also know as effective yield or annual percentage yield (APY). The effective interest rate gives the actual percentage by which a balance increases in one year. For a fixed annual interest rate, as the number of compounding periods increases, the annual percentage yield/effective interest rate increases, too. The effective interest rate is often used to compare accounts that are compounded differently and have different stated annual interest rates.

Deﬁnition The eﬀective interest rate or annual percentage yield is given by r m r = 1 + − 1 e f f m when interest is compounded m times a year, and

r re f f = e − 1

when interest is compounded continuously, where r = Annual percentage rate changed to a decimal m = Number of compounding periods per year

re f f is computed as a decimal, but compared as a percent, so you will need to convert back to percentages.

For investment purposes, you want to make/earn the most money, thus you want a higher APY. For loan/borrowing purposes, you want to owe/pay the least amount of money, thus you want a lower APY.

The Eff application on the calculator requires two arguments, Eff(annual interest rate as %, number of compounding periods) = Eff( I%, m) The result of the Eff application is a percentage. The Eff application only works when m is finite. If an account is compounded continuously the formula, r re f f = e − 1, must be used for an accurate result.

Example6 While waiting for your next class you see an advertisement describing the two accounts. Account A: 1.51% APR compounded quarterly Account B: 1.48% APR compounded daily The information about whether these accounts were for investment or borrowing purposes has been torn oﬀ the advertisement. Explain which account is better for each purpose.

Reﬂection:

• Given a financial scenario, can you determine the appropriate formula needed? • When is it appropriate to use the TVM Solver in a financial scenario without payments? • How do you determine the best rate of return for financial gain?