Time value of 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.
In Tractate Makkos page 3a the Talmud discusses a case where witnesses falsely claimed that the term of a loan was 30 days when it was actually 10 years.
The false witnesses must pay the difference of the value of the loan "in a situation where he would be required to give the money back (within) thirty days..., and that same sum in a situation where he would be required to give the money back (within) 10 years...The difference is the sum that the testimony of the (false) witnesses sought to have the borrower lose; therefore, it is the sum that they must pay.
In a typical case, the variables might be: a balance (the real or nominal value of a debt or a financial asset in terms of monetary units), a periodic rate of interest, the number of periods, and a series of cash flows.
That is, £100 invested for one year at 5% interest has a future value of £105 under the assumption that inflation would be zero percent.
[3] This principle allows for the valuation of a likely stream of income in the future, in such a way that annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream; all of the standard calculations for time value of money derive from the most basic algebraic expression for the present value of a future sum, "discounted" to the present by an amount equal to the time value of money.
: Some standard calculations based on the time value of money are: There are several basic equations that represent the equalities listed above.
The solutions may be found using (in most cases) the formulas, a financial calculator, or a spreadsheet.
The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).
In the case of the standard annuity formula, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
For an annuity that makes one payment per year, i will be the annual interest rate.
For example, the annuity formula is the sum of a series of present value calculations.
A perpetuity is payments of a set amount of money that occur on a routine basis and continue forever.
When the perpetual annuity payment grows at a fixed rate (g, with g < i) the value is determined according to the following formula, obtained by setting n to infinity in the earlier formula for a growing perpetuity: In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications.
Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.
The future value (after n periods) of a growing annuity (FVA) formula has five variables, each of which can be solved for by numerical methods: Where i ≠ g : Where i = g : The following table summarizes the different formulas commonly used in calculating the time value of money.
Applying the formula for geometric series, we get: The present value of the annuity (PVA) is obtained by simply dividing by
The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount: Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula: Initially, before any payments, the present value of the system is just the endowment principal,
In that case, the discount factor, and thus the present value, of a cash flow at time T is given by the integral of the continuously compounded rate r(t): Indeed, a key reason for using continuous compounding is to simplify the analysis of varying discount rates and to allow one to use the tools of calculus.
Using continuous compounding yields the following formulas for various instruments: These formulas assume that payment A is made in the first payment period and annuity ends at time t.[10] Ordinary and partial differential equations (ODEs and PDEs)—equations involving derivatives and one (respectively, multiple) variables—are ubiquitous in more advanced treatments of financial mathematics.
While time value of money can be understood without using the framework of differential equations, the added sophistication sheds additional light on time value, and provides a simple introduction before considering more complicated and less familiar situations.
The fundamental change that the differential equation perspective brings is that, rather than computing a number (the present value now), one computes a function (the present value now or at any point in future).
Formally, the statement that "value decreases over time" is given by defining the linear differential operator
Applied to a function, it yields: For an instrument whose payment stream is described by f(t), the value V(t) satisfies the inhomogeneous first-order ODE
The standard technique tool in the analysis of ODEs is Green's functions, from which other solutions can be built.
In mathematical terms, this instantaneous cash flow is modeled as a Dirac delta function
In other words, future cash flows are exponentially discounted (exp) by the sum (integral,
for future, r(v) for discount rates), while past cash flows are worth 0 (
Note that the value at the moment of a cash flow is not well-defined—there is a discontinuity at that point, and one can use a convention (assume cash flows have already occurred, or not already occurred), or simply not define the value at that point.
is given by combining the values of these individual cash flows: This formalizes time value of money to future values of cash flows with varying discount rates, and is the basis of many formulas in financial mathematics, such as the Black–Scholes formula with varying interest rates.
