Interest Rates That Vary With Time
• Find the present equivalent value given a future value and a varying interest rate over the period of the loan ik : the interest rate of the kth period
• FN P = ----------------- eq.(4-31)
N (1 + ik)
• (P/F,i%,N)= 1/(1+i)N eq.(4-2)
k + 1
Example:
If F4 = $1,000 and i1 = 10% , i2 = 12%, i3= 13% and i4=10% then
P = $1,000* (P/F, 10%, 1)* (P/F, 12%, 1)* (P/F, 13%, 1)* (P/F, 10%, 1)
P = $1,000 (0.9091)(0.8929)(0.8850)(0.9091)
Nominal And Effective Interest Rates
• Nominal Interest Rate (r ) For rates compounded more frequently than one time a year, the stated annual interest rate.
• Effective Interest Rate ( i ) For rates compounded more frequently than one year, the actual amount of interest paid.
• i = ( 1 + r / M )M - 1 M : the number of compounding periods per year
• Annual Percentage Rate ( APR ) percentage rate per period times number of periods. APR = (r/M) x M
Method1
Method2
This kind of problems can be solved with two ways :
1. get the effective interest rate which is compatible with the period for
example i = 16.14 per year compounded yearly and N = 10 years
OR
2. use the effective interest rate per quarter = 6%(nominal per year) / 4
(compounded times per year) = 1.5% per quarter compounded quarterly
and N = 10 years * 4 = 40 quarter periods
Continuous Compounding and Discrete Cash Flows
• Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval.
• Given nominal per year interest rate r, compounding per year M one unit of principal = [ 1 + (r / M ) ] M
• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp
• Given lim (p ∞ ) = [ 1 + (1 / p) ] rp = er
• ( F / P, r%, N ) = (1+i)N = e rN
Then i = e r - 1
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