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Unformatted text preview: one year 1.06 1.032 = 1.0609 v Annually compounded interest rate 6.000% 6.090 6.136 6.168 6.180 6.183 1.0154 = 1.06136 1.00512 = 1.06168 1.00115452 = 1.06180 1.000164365 = 1.06183 Compound Interest • In general, the formula is
r r $1 × ⎛ 1 + ⎞ = $1 × FF⎛ , k⎞ ⎜ ⎟ ⎜ ⎟ ⎝ k⎠ ⎝k ⎠
k where k is the number of compounding intervals per year. • For n years, this becomes
r $1 × ⎛ 1 + ⎞ ⎟ ⎜ ⎝ k⎠
n× k r = FF⎛ , n × k ⎞ ⎟ ⎜ ⎠ ⎝k Compound Interest
Example Suppose you are offered an automobile loan at an APR of 6% per year compounded monthly. What does that mean, and what is the true rate of interest, given monthly payments? 15 Compound Interest
Example  continued Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Assume $10,000 loan amount. Loan Pmt = 10,000 × (1.005)12 = 10,616.78 ActualRate = 6.1678% Continuous Compounding • It turns out that there is a limit to how much a dollar can grow for a given interest rate. This happens when you pay interest continuously or at every instance. The limit is given by FV = $1 × e (r × n)
where n is the number of years and e = 2.7182. Example: Continuous Compounding
• Suppose a bank offers you an 8% interest rate compounded continuously. If you invest $1 for one year, at the end of the year you will have: If you invest $1 for ten years, at the end of the ten years you will have: 16 Continuous Compounding • The present value for continuous compounding is given by: PV = $1 × e − ( r × n ) = $1 e ( r ×n) • Example: Exxon mastercard charges an APR of 19.34% compounded daily, what is the corresponding EAR? Effective Annual Rates or Equivalent Annually Compounded Rates • The interest rate r which is compounded is known as the Annual Percentage Rate  APR • As we saw, the amount of interest earned from a given APR will depend upon the number of compounding per year. Effective Annual Rates or Equivalent Annually Compounded Rates • Effective Annual Rate or the Equivalent Annually Compounded Rate  EAR. This rate takes into account the compounding and asks what rate is earned over the entire year. • The EAR is given by r EAR = ⎛ 1 + ⎞ − 1 ⎜ ⎟ ⎝ k⎠ k 17 Compounding
• Discrete compounding (k times per period) PV = CT/(1+r/k)Tk • Continuous compounding PV = CT erT 18...
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This note was uploaded on 01/17/2011 for the course MGMT 107 taught by Professor ? during the Winter '08 term at UC Irvine.
 Winter '08
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