Calculating the Future Value of Annuities
In contrast to the calculating present value (e.g. of lottery winnings, mortgages etc),
what if we needed to calculate the future value of savings? If we are making regular
savings deposits to meet a nest-egg target, we need to calculate not the present value,
but rather the future value of accumulated savings. Simple! We simply calculate the
future value based on the present
value. so the PV is now,
PV = $C [(1/r) –(1/r(1 + r)t)], and
future value of PV is PV x (1 + r)t
FV = $C [(1/r) –(1/r(1 + r)t)] x (1 +
r)t which works out to
FV = $C x [(1 + r)t–1]/r
Annuities with a constant growth cash
flow
While we assumed that the cash
flows ($C) will be the same for

both annuities and perpetuities, usually these cash flows grow at a constant rate (to
account for inflation, depreciation, etc.)
Simple case: an infinite stream of cash flows constantly growing (growing perpetuity)
Typical case: a finite stream of cash flows that grow at a constant rate, and terminate
after a certain period
Simple Case
A growing perpetuity with a constant growth rate of ‘g’ has a PV that can be shown as
PV of a growing perpetuity = C1/r-g
Note that C1 is the cash flow at the end of the first period
Buying a condo as an investment (Part 1)
A condominium apartment as an investment generates $12,000 (rent minus expenses)
annually. This cash flow grows 3% per year indefinitely. If the interest rate is 8%, what iss
the present value of the cash flows?
PV = C1/ (r-g) = $12,000/(0.08-0.03) = $240,000
So, investing in a condo with $12,000 annual cash flows, assuming a growth of
3%, and interest rate of 8%, gives a present value of it future cash flows at
$240,000
Typical Case
A growing annuity with a growth rate of ‘g’ has a PV that can be shown as, which
termination at some time t:
Buying a condo as an investment (part 2)
For the same condo generating $12,000 annually and growing at 3% a year, what is the
present value of cash flows if it is expected to have a life of 20 years (i.e. will be torn
down after 20 years). Assume interest of 8%.
PV = [$12,000 / (0.08-0.03)] x [1 – (1.03/1.08) ^ 20] = $147,000.50
So, investing in a condo with $12,000 annual cash flows, assuming a growth of 3% and
an interest rate of 8% over the 20-year life of the condo, gives a present value of it future
cash flows at $147,000.50
Future value of the growing annuity
As before, the FV can be calculated, by simply multiplying the PV by (1+r) ^t

So, for the condo example, the future value of the condo at the end of the 20 years will
be
$147,000,50 x (1.08) ^20
= $685,163
EAR
We have used annual interest rates, and compounded the cash flows annually, but rates
are sometimes compounded semi-annually, monthly, or even daily, so the compounding
comes into effect, and actual rate over a year is actually higher.
The quoted rate (usually called the annual percentage rate, or APR), is a simple interest
rate. However, the effective rate is actually a compounded rate, which is the true rate.
This is called the Effective Annual Rate (EAR)
In general, APR rates (usually quoted), need to be converted back into the actual interest rate