# Calculating the future value of annuities in contrast

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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