3. 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
6. 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)
7. 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
8.
9.
10.
11. 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
12.
13. 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
14.
15.
16.
17.
18. • End of Chapter 4 PART 3
• See you next lecture with a very important
revision !!!
• Don’t, miss it !!!!!
