This preview shows page 1. Sign up to view the full content.
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 e-rT 18...
View Full Document
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