present value and accumulated value
Money sitting at different dates is like objects at different altitudes — you cannot compare them until you bring them to the same level. Present value asks: what is a future sum worth right now? Accumulated value asks the reverse: what will a sum today be worth at some later date? These two ideas are the practical payoff of the time value of money, and almost every financial calculation is really one of them in disguise.
To find the accumulated value, you carry money forward by multiplying by the accumulation factor: a payment of C grows to C times (1 + i) to the power n after n periods. To find the present value, you bring money back by multiplying by the discount factor: a payment of C due in n periods is worth C times v to the power n today, where v equals 1 divided by (1 + i). The two are inverse operations — discounting then accumulating over the same period returns the original amount. For a stream of payments, you simply present-value (or accumulate) each one and add the results.
Present value is arguably the single most-used concept in all of actuarial science. The price of a bond is the present value of its coupons and redemption; the liability for a pension is the present value of future benefits; an insurance reserve is the present value of expected future outgo less future income. When an actuary says a policy is worth a certain amount today, they almost always mean its present value at an assumed interest rate. Choosing that rate is consequential: a higher discount rate makes future obligations look smaller today.
You need 10,000 dollars in 5 years for a goal. At 4 percent, the present value is 10,000 times v to the fifth, about 8,219 — that is how much to set aside today to reach it.
Present value discounts the future to now; accumulated value pushes the present into the future.
A present value is only as trustworthy as the interest rate assumed; quoting a present value without stating the rate hides the most important assumption in the calculation.