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Financial Mathematics

Financial Mathematics. Savings Adapted from “Compound Interest” Powerpoint by Patrick Callahan, Ph.d. First, a review. $1.00 in 2007 does NOT equal $1.00 in 2008 Why? B/c $1.00 in 2007 buys MORE than $1.00 in ’08 CPI – Method for converting $ from year to year, to make comparisons.

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Financial Mathematics

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  1. Financial Mathematics Savings Adapted from “Compound Interest” Powerpoint by Patrick Callahan, Ph.d

  2. First, a review • $1.00 in 2007 does NOT equal $1.00 in 2008 • Why? • B/c $1.00 in 2007 buys MORE than $1.00 in ’08 • CPI – Method for converting $ from year to year, to make comparisons

  3. For example… • You buy a bike in 2007 and spend $200. • Your sister bought a similar bike in 1998 for $175. • Who’s bike cost more? • In nominal dollars, yours did: $200 > $175 • But in real (or constant) dollars, hers did • B/c $175 (in 1998) = $223 (in 2007)

  4. Another example • Wages are also affected by inflation • What you can buy with your earnings in 2008 is less than what you could buy with the same wages in 1998. • Need your wages to increase at a rate at least equal to inflation.

  5. What is central is the idea that CPI helps to give us CONTEXT for understanding the value of money across time

  6. Inflation • The change in the CPI from year to year is the inflation rate • So, one goal is to have your income keep pace with inflation • Another goal would be to have your income outpace inflation • which would give you some leftover…to spend or to save

  7. Why save money? • To get enough to purchase a big ticket item (car, house, pay for graduate school) • To have money to live on after you retire and no longer have a steady income from your work

  8. Why not stuff it in your mattress? • You’ll dig it out and spend it • Someone will break into your home and steal it • Your house will burn down and it will go up in smoke • Inflation cuts into the value • $20,000 stuffed away in 1970 = $3743 in 2007 (Carmela Soprano knew this!)

  9. Savings Accounts • When you put money into a savings account, it earns interest • This means the amount GROWS over time • (The bank pays you money to lend them your money, which they then lend out to others at a slightly higher rate. – More on this later!)

  10. Two Kinds of Savings Accounts • Basic Savings Account • Usually no minimum balance required • Pays a very low interest • Can withdrawal money whenever you want • Money Market Account • Usually has a minimum balance • Pays a higher interest rate • Often limits the number of withdrawals/month

  11. Two types of interest • Simple interest: Fixed percentage of original amount invested or deposited. • Compound interest: Fixed percentage of original amount plus accumulated interest. • You earn interest on your interest.

  12. Example: $1000 invested at 10%

  13. Simple v. compound • Simple interest = linear growth • Compound interest = exponential growth • Which is better?

  14. Formula for compound growth Balance=Principle(1+r/n)yn Balance = How much in your account Principle = What you started with (originally) r = annual interest rate n = compounding frequency y = number of years

  15. Money can compound at different time periods Balance=Principle (1+r/n)yn This changes the value of n: Annually: n=1 Quarterly (every 3 months): n=4 Monthly: n=12

  16. Different Compounding • Basic Formula: Balance=Principle(1+r/n)yn • Various Versions: Yearly: Balance = Principle (1+r)y Quarterly: Balance=Principle (1+r/4)4y Monthly: Balance=Principle (1+r/12)12y

  17. An Example: • 5% APR, • compounded quarterly, for 7 years Balance=Principle(1+r/n)yn Balance=Principle (1+.05/4)7*4 =1,000 (1.0125)28 = $1415.99

  18. Another example: • 5% APR • Compounded monthly, for 7 years Balance=Principle(1+r/n)yn Balance=Principle (1+.05/12)7*12 = 1,000 (1.0041667)84 = $1418.04

  19. Excel Example

  20. Annual percentage yield [APY] • In formulas, r was annual percentage rate or APR • When interest compounded more often than once per year, actual interest earned in a year is greater than APR

  21. Example: $10,000 invested for 10 years at 8% APR Annually: $21,589.25 Quarterly: $22,080.40 Monthly: $22,196.40

  22. Computing APY • Compute the balance for one period. • Calculate percentage change from two consecutive periods (new balance-old balance)/old balance

  23. Computing APY • Another version • APY = (1 + r/n )n – 1 where r is the stated annual interest rate and n is the number of times you’ll compound per year. • Example: 8% rate, compounded monthly • APY = (1+.08/12)12 – 1 • APY = 8.29

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