Simple and Compound Interest (Young: 6.1)

Simple and Compound Interest (Young: 6.1) In this Lecture: 1. 2. 3. 4. 5. 6. 7. 8.

Financial Terminology Simple Interest Compound Interest Important Formulas of Finance From Simple to Compound Interest Examples of Common Compounding Periods Additional Examples Using the TVM Solver on a TI-8X Graphing Calculator (Optional) ________________________________________________ Remark In financial matters we as individual people are often the “borrower” while the bank, as the provider of money, is the “lender”; however, we can also think of ourselves as the “lender” and the bank the “borrower” when we invest money in the bank. The table below summarizes these relationships. Borrower a) Bank or Financial Institution b) Individual or small business

Sample Investment Device • CD (Certificate of Deposit) • Savings Account • Bonds Loans for: • Car • House • Start a business

Lender Individual or small business Bank or Financial Institution

Young 6.1 Survival Guide Notes copyright © 2008 Knobel/Stanley

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Financial Terminology 1. Principal—amount of money borrowed by the borrower NOW (also called the present value of the money borrowed). 2. Interest—a fee paid by the borrower (i.e., the bank in (a) and you in (b)) for the privilege of using the money borrowed for some period of time. 3. Future Value (of the principal borrowed)—total amount (principal plus interest) to be repaid to the lender at the end of the period of time that the money was borrowed. Two Types of Interest: INTEREST

Simple

Compound

1. Simple Interest—a percentage of the amount borrowed (i.e., the principal) is paid to the lender each year (or fraction of a year). 2. Compound Interest—in each compounding period (e.g., day, month, year) a percentage of the amount borrowed PLUS a percentage of the total interest accumulated is paid to the lender at the end of the compounding period. Young 6.1 Survival Guide Notes copyright © 2008 Knobel/Stanley

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What is the difference between the two? In the case of simple interest, the amount of interest paid is based ONLY on the amount borrowed, whereas in a compound interest scenario the amount of interest paid is based on the amount invested PLUS the interest accumulated in the account. Important Formulas of Finance 1. Simple Interest Formula Let P = principal (amount borrowed or lent), or present value r = annual (nominal) interest rate (decimal form) t = length of the loan, measured in years. Then the simple interest I on the loan is,

I = Prt The total accumulated amount A (or future value) paid by the borrow to the lender is,

A = P+ I = P + Pr t = P(1+ rt)


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Class Activity (A) Suppose you invest $1,000 at 8% simple interest. How much money will be in the account after 5 years?

(B) Suppose you invest $1,000 at 8% simple interest. How much money will be in the account after six months?


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(C) Many credit card companies charge simple interest on the unpaid balance of the account for the number of days since your last payment. Suppose that your credit card company charges 9.9% simple interest and you made your last payment 32 days ago at which time the balance was $3,000. How much interest is owed after 32 days?

(D) Continuing with part (C), if your next payment is $200 what will be your new balance?

 Young 6.1 Survival Guide Notes copyright © 2008 Knobel/Stanley

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From Simple to Compound Interest Class Activity (A) Suppose that you invest $1,000 at 8% compound interest, where the interest is compounded quarterly. Complete the table. Let, i = quarterly interest rate = 0.08/ 4 = 0.02 Quarter # m 0 1 2

Amount in account after m quarters 1,000 1000 + 1000 (0.02) = 1000 * (1.02) = 1,020 1020 * (1.02) = [1000 * 1.02] * 1.02 = 1000 * (1.02)2 = 1040.40

Interest paid for quarter N/A 1020 –1000=$20

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4

!

20 (5 yrs = 20 qtrs)

!

!

NOTE: Try to spot the pattern in the table. Young 6.1 Survival Guide Notes copyright © 2008 Knobel/Stanley

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(B) How much would be in the account after 5 years at 8% compound interest, compounded quarterly?

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2. Compound Interest Formula Let P = principal t = number of years m = number of interest periods per year r = annual interest rate i = r = interest rate per period m n = m!t = total number of interest periods (in t years) Then the future value after n interest periods is, ! $ m 't A = P#"1+ r &% m = P(1+ i)n


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Class Activity m 't Identify each quantity in the formula A = P !#" 1+ r $&% using m numbers from the last class activity. Assume a time of 5 years.

P=

t=

m=

r=

i=

n= 

Examples of Common Compounding Periods

Compounding Period Annually Semiannually Quarterly Monthly Daily

Number of Interest Periods per Year, m 1 2 4 12 365

Length of Each compounding Period 1 year 6 months 3 months 1 month 1 day


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Class Activity What happens if we continue to compound the interest more and more times each year (i.e., so that the compounding period is getting smaller and smaller) in the last example? Recall that the principal was 1,000, the rate was 8%, and we were investing the principal for 5 years. m = # of periods per year 4 (quarterly)

0.08$ 5 * m & m % 1000(1 + 0.08/ 4)4 • 5 !1485.95

! A =1000#1+ "

365 (daily)

8760 (hourly)

m! " Compounding “continuously”

NOTE: Your instructor will fill in what goes in the last box. 


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3. Continuous Compounding Interest Formula Let P = principal (amount borrowed or lent), or present value r = rate of interest compounded continuously t = time measured in years Then the accumulated amount, A, after t years is,

A = Pert ; e ! 2.71828... Remark Students often get confused in deciding whether to the continuous compounding formula above or the compound interest formula that we discussed earlier in these notes. Remember that we use the formula above only in the case that the problem states that interest is compounded “continuously.”


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Class Activity How much money will be in account after 8 years, if $500 is invested at a nominal annual rate of 6% and the interest is compounded: (A) semiannually?

(B) continuously?

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Class Activity (Optional) Solve each equation for x, using algebraic methods. (A) 300=100(1+ x)4

(B) 300 = 100(1.1)5x [Hint: Recall that ln ab =b ln a ]


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(C) 300 = 100e0.2x [Hint: ln ea =a ]

Answers: (A) 31/4 ! 1 ! 0.316074013 ; (B)

ln 3 ! 2.305340921 ; (C) 5 ln1.1

ln 3 ! 5.493061443 0.2

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Additional Examples We now use the algebraic methods of the previous class activity to solve additional problems involving compound interest.


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Class Activity Find the annual rate of interest, r, needed for $2,500 to grow to $7,500 in 6 years if the interest is to be compounded semiannually.


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Class Activity (A) How many quarters, n, are needed for $4800 to grow to $30,000 if the annual interest rate is 15%, compounded quarterly?

(B) To how many years does your answer in part (A) correspond?

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Class Activity How long in years will it take for an investment to double (i.e., the “doubling time”) if it is invested at 6%, compounded continuously?

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Using the TVM Solver on a TI-8X Graphing Calculator (Optional) The TI-83 Plus and TI-84 series calculators have the capability to solve equations involving compound interest. The next activities show how to use this tool [Note: Be sure to check with your instructor regarding his or her policies on using the TVM Solver on tests and quizzes.] To access the TVM Solver, press APPS, select “1:Finance”, and “1:TVM Solver…” as shown below:


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A screen similar to the following screen will appear:

The table below summarizes what each quantity in this last window represents. Quantity in TVM Explanation Solver N Number of interest periods (in t years) Annual interest rate I% PV Principal, P, entered as a NEGATIVE NUMBER PMT Payment—put the payment equal to zero when solving compound interest problems FV Future value (or accumulated value) for the account. Enter this quantity as a POSITIVE number. C/Y Number of interest periods per year (this is the same as m in the compound interest formula) P/Y Number of payments per year—set this value to be the same as the value you enter for C/Y. PMT: END BEGIN Be sure that END is selected (i.e., payments are made at the END of the interest period)


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Class Activity Use the TVM Solver on your graphing calculator to find the annual rate of interest, r, needed for $2,500 to grow to $7,500 in 6 years if the interest is to be compounded semiannually. (A) Fill in the missing values below. N= PV = PMT = FV = C/Y = P/Y =


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(B) Input your values to coincide with the values entered in the screen capture below. Note that we have entered a value of 0 for the quantity to be solved for and have positioned the cursor next to the value we want to calculate (i.e., I%).

(C) Next, press ALPHA, SOLVE (above the ENTER key) and the following screen should appear:

Based on the screen, what is the annual interest rate?


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Class Activity Use the TVM Solver to determine how many quarters are needed for $4800 to grow to $30,000 if the annual interest rate is 15%, compounded quarterly?

Answer: N = 49.77950784 quarters


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Simple and Compound Interest (Young: 6.1)

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