1. What is the order of magnitude difference between an inch and a mile? between an inch and a mile? 2. How many times bigger is a mile compared to an inch? to an inch? 3. Solve for ‘x’ 4. Solve for ‘x’ Bell Quiz 3-5

3.6 Mathematics of Finance

What you’ll learn about  Interest Compounded Annually  Interest Compounded k Times per Year  Interest Compounded Continuously  Annual Percentage Yield  Annuities – Future Value  Loans and Mortgages – Present Value … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!

Vocabulary Principal: The amount of $$ initially invested Compound Interest: interest earned during the 1 st period, earns interest in all subsequent periods. The interest earned in 2 nd period earns interest in all subsequent periods, etc.

Interest Compounded Annually Constant Percentage Rate (from Section 3-2) (referred to populations) Can you see any difference between them? ‘r’ is in a fixed annual percentage rate, and if ‘t’ has the units of years. if ‘t’ has the units of years.

Interest Compounded “k” Times per Year If we compound the annual interest rate multiple times per year, then the rate of increase ‘r’ is divided by the number of times per year it is compounded. The units of the percentage rate now are % per period.

Interest Compounded “k” Times per Year Since the time period is now shorter than a year, then the exponent will now be: then the exponent will now be: The input variable ‘t’ is still in years, but the units of the exponent make the exponent “unit-less” when you plug ‘t’ in years into the equation. ‘t’ in years into the equation.

Interest Compounded “k” Times per Year At the 1 year point: In general terms: You won’t see this in the text book  you better write this down!! the text book  you better write this down!! Where: (1) ‘t’ is in years (2) ‘k’ is number of periods per year

Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years. A(5) = $ P = $400 r = 0.08 k = 12

Paul invests $400 at 8% annual interest compounded monthly. A(5) = $ P = $400 r = 0.08 k = 12 Paul invests $400 at 8% annual interest compounded yearly P = $400 r = 0.08 k = 1 A(5) = $ Not much difference? The extra $8 is also earning interest. Over the course of a long period of time, this makes a huge Over the course of a long period of time, this makes a huge difference. difference.

Over 30 years. A(30) = $ P = $400 r = 0.08 k = 12 Paul invests $400 at 8% annual interest compounded yearly P = $400 r = 0.08 k = 1 A(5) = $ Not much difference? The extra $8 is also earning interest. Over the course of a long period of time, this makes a huge Over the course of a long period of time, this makes a huge difference. difference.

Your Turn: 1. Find the final value of a $5000 investment that earns 6.5% per year compounded monthly. What is the total value of the investment at the end of 3 years? 2. Find the final value of a $5000 investment that earns 6.5% per year compounded daily. What is the total value of the investment at the end of 3 years?

Your Turn: 3. How long did an investment stay in an interest bearing account earning 3% per year compounded quarterly if the principal was $200 and the final value was $400? 4. How long did an investment stay in an interest bearing account earning 5% per year compounded monthly if the principal was $700 and the final value was $1400?

The effect of shorter compounding periods As ‘k’ “Arrow Analysis”   Gets closer and closer to ‘1’ closer to ‘1’ But also  Although gets closer and closer to ‘1’ It is also being raised to a larger and larger exponent. Which effect dominates?

The magic of the number “e” For “continuous compounding” ‘r’ = yearly interest rate t = year invested t = year invested

Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years. t = 5 t = 5 r = 0.08 r = 0.08 A(0) = $400 Remember when it was compounded 12 times per year: per year: P = $400 r = 0.08 k = 12 A(5) = $595.94

Your Turn: 5. Find the final value of the investment: A(0) = $1250 A(0) = $1250 r = 5.4% r = 5.4% t = 6 t = 6 compounded continuously compounded continuously 6. How long was the investment in the account? A(0) = $1250 A(0) = $1250 A(t) = $3000 A(t) = $3000 r = 7% r = 7% compounded continuously compounded continuously

Vocabulary: Annual percentage yield: (APY) The percentage rate that, when compounded annually would yield the same return as the given interest rate with the given compounding period. period. Salesperson A: “I’ll give you 5% interest rate compounded monthly” Salesperson B: “I’ll give you 4.7% interest rate compounded continuously” Which is the better investment?

Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY? 1. Let ‘x’ = APY (we’re trying to find this) 2. Set a simple APY interest rate return (for 1 year’s period) equal to the return from the given interest rate and compounding period. 3. Solve for ‘x’

Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY? Divide by Does the amount invested matter? 4.65% compounded monthly is equivalent to 4.73% compounded once per year.

Your Turn: 7. Find the APY for 8.8% compounded quarterly. 8. Find the APY for 8.7% compounded monthly. 9. Which of the above investments is better?

Vocabulary Annuity: A series of equal periodic payments into an interest bearing account. into an interest bearing account. (Not to be confused with the series of equal (Not to be confused with the series of equal periodic payments to pay off a loan.) periodic payments to pay off a loan.)

Annuities 8% compounded quarterly Equal quarterly payment of $500 (payments made at end of quarter) Value at the end of the first quarter = ? $500 Value at the end of the second quarter = ? $ (1.02) = $1010 Value at the end of the third quarter = ? $500 + $500(1.02) + $500(1.02) 2 Value at the end of the year = ? $500 + $500(1.02) + $500(1.02) + $500(1.02) 23 Annual rate = 8%  interest/period = 8/#periods = 2%/period = $ = $2060.8

Future Value of an Annuity The future value FV of an annuity of: -- n equal periodic payments -- $R payment each period -- interest rate i per compounding period (payment interval) Remember the interest rate per compounding period is the annual interest rate divided by period is the annual interest rate divided by the number of compounding periods in the year. the number of compounding periods in the year.

Future Value Example R = $500 r = 7% r = 7% t = 6 t = 6 k = 4 k = 4 FV = ? FV = $14, These problems involve a formula; plug numbers into the appropriate place based upon the data given in the problem, solve for unknown variable. n (# of payments) = “t” years * “k” periods per year

Vocabulary Present Value: The estimated present worth of an amount of cash to be received or paid in the future. $1000 today is worth how much in 10 years if it is invested in an account earning 5% interest compounded continuously? The present value of $ paid at the end of 10 years at an interest rate of 5% compounded continuously is $1000. at an interest rate of 5% compounded continuously is $1000. Giving you $1000 now is the same as giving you $ in 10 years (interest rate of 5% compounded continuously)

Vocabulary Present Value: The estimated present worth of an amount of cash to be received or paid in the future. How a bank determines the monthly payment for a loan: 1.They determine the final value of an investment equal value of an investment equal to the loan amount. to the loan amount. The present value is the loan amount.

Finding the monthly payment for a loan. 2. They then calculate an annuity (periodic equal payments) so that the future value of the annuity is the same as the final value of the simple interest investment of the original amount loaned.

Present Value of an Annuity Solving for Present value: 3.The PV is the amount being loaned, they then solve for ‘R’ (the monthly payment). solve for ‘R’ (the monthly payment). Do you have to solve this problem? NO, just use the formula. the formula.

Finding the monthly payment amount.

What is the monthly payment for a $250,000 home loan at an APR (annual percentage rate) of 5.75%, over a 30 payoff period? PV = $250,000 i = 5.75% i = 5.75% k = 12 k = 12 t = 30 t = 30 Solve for R: $ /month

Your turn: 10. Plugging this into your calculator can be tricky. How much money did you borrow if your monthly payment is $399 at money did you borrow if your monthly payment is $399 at APR of 6% and the term of the loan is 5 years? APR of 6% and the term of the loan is 5 years?

Your turn: 11. You want to buy a car. The dealer will loan you the $15,000 at 2.9% APR for a five year loan. What is your $15,000 at 2.9% APR for a five year loan. What is your monthly payment? monthly payment?

HOMEWORK P.341: 4-32 even, even (20 problems) (20 problems)