Review of Present Value Calculations
1.
Future value of an amount P collecting interest at a rate r for n periods
Future Value Interest Factor =
Example:
Future value of $1000 after 6 years, r=10% (Annual Compounding) =
Alternatively:
= 1000[] =1000[1.7716]
2. Present value of a single amount, F payable n years from now:
Present value interest factor is tabulated.
Example:
Present value of $1000 receivable after 10 years, if the interest rate is 12% is:
Alternatively:
= 1000*0.3220 = $322
3. Future value of an annuity of $A at the end of each year:
For a three year annuity,
For an “n” year annuity,
Using the sum of a geometric series formula,
The term in square brackets is tabulated.
Example:
If a person saves $1000 per year for 10 years, and deposits the money at the end of each year in a bank which pays 8%interest, he will have, after 10 years
n=10 years; r=0.08; A=$1000
Alternatively,
1. Present value of an annuity
for a three year annuity
for an n year annuity
Using the sum of a geometric series formula,
The term in square brackets is tabulated.
If , where an infinite annuity is called a “Perpetuity.”
Example:
Find the present value of the following ordinary annuity:
$200 per year for 10 years at 10%
A = $200
n=10 years; r=10%
Example:
The present value of $200 per year forever with r=10% annually is:
(A=200; r=0.1)
2. Compounding and Discounting more than once a year
If compounding is done “m” times a year, at a nominal interest rate, R per year, for T years:
n= number of periods = mT
r= interest rate (effective) per period = R/m
Future value,
Present value of an amount F received T years from now,
Example:
How much does a deposit of $5000 grow at the end of 6 years if nominal rate of interest is 12% and compounding frequently is 4 times per year?
R=0.12;
, number of periods
3. Continuous compounding and discounting
We know that,
When
Thus, as .
(Future value under continuous compounding)
Conversely, the present value of an amount F received T years from now an annual interest rate (nominal) R, if discounting is done continuously is:
Example:
Under continuous compounding $5000 grows to (R=12%), after 6 years:
Example:
If someone offers to “double” your money in 6 years, what is the interest rate you are getting (under annual compounding)?
4. Present value of a growing annuity:
g = annual growth rate
A is the first payment, second payment is A(1+g), third is … when g = 0, we get:
which is the standard PV of an annuity formula.