The time value of money refers to the idea that there is generally a greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later developed concept of time preference. The time value of money refers to the observation that it is better to receive money sooner than later. Money you have today can be invested to earn a positive rate of return, producing more money tomorrow. Therefore, a dollar today is worth more than a dollar in the future. The time value of money is among the factors considered when weighing the opportunity costs of spending rather than saving or investing money. As such, it is among the reasons why interest is paid or earned: interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the loss of their use of their money. Investors are willing to forgo spending their money now only if they expect a favorable net return on their investment in the future, such that the increased value to be available later is sufficiently high to offset both the preference to spending money now and inflation (if present); see required rate of return.
Overview The time value of money compares cash flows that occur at different dates by converting them to a single valuation date (often called “time 0”). In a simple discrete-time model, time is measured in equal-length periods t = 0 , 1 , 2 , … {\displaystyle t=0,1,2,\dots } and a constant effective interest rate i {\displaystyle i} is applied once per period. If an amount P V {\displaystyle \mathrm {PV} } is invested at time 0, its value after n {\displaystyle n} periods (the future value) is
F V = P V ( 1 + i ) n . {\displaystyle \mathrm {FV} =\mathrm {PV} (1+i)^{n}.}
Discounting reverses this relationship, such that the present value of a sure amount F V {\displaystyle \mathrm {FV} } due at time n {\displaystyle n} is
P V = F V ( 1 + i ) − n = F V ( 1 + i ) n . {\displaystyle \mathrm {PV} =\mathrm {FV} (1+i)^{-n}={\frac {\mathrm {FV} }{(1+i)^{n}}}.}
The factor ( 1 + i ) n {\displaystyle (1+i)^{n}} is the accumulation factor over n {\displaystyle n} periods, and ( 1 + i ) − n {\displaystyle (1+i)^{-n}} is the corresponding discount factor. For a sequence of dated cash flows C F t {\displaystyle \mathrm {CF} _{t}} (positive for receipts and negative for payments), present value is the discounted sum of each cash flow:
P V = ∑ t = 0 n C F t ( 1 + i ) t . {\displaystyle \mathrm {PV} =\sum _{t=0}^{n}{\frac {\mathrm {CF} _{t}}{(1+i)^{t}}}.}
This discounted-sum form underlies net present value calculations used in valuation and capital budgeting. Nominal and real analyses must be kept consistent. Nominal cash flows are discounted at nominal rates, while real cash flows (with general inflation removed) are discounted at real rates; mixing conventions changes results mechanically even when the underlying economics is unchanged.
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