Wednesday, October 10, 2012

Time Value of Money (TVM)

Time Value of Money (TVM) is the most important concept in finance. It is the underlying concept for majority of finance related calculations, such as project analysis (capital budgeting or investment appraisal), equity valuation, bond valuation and indeed any asset valuation.

Time value of money can be summarized in the statement, "£1 received today is more valuable than £1 received in the future." The reasons for this include the ability to invest £1 today to receive more than £1 in the future, and inflation, which is generally positive reduces the value of money in the future.

When doing most financial analysis, cash flows at different periods is involved. Some cash flow is typical involved now (year 0) and some cash flows in future periods. Any of these cash flows could be positive (cash inflows) or negative (cash outflows). Any of these cash flows can be converted to an equivalent value today, which is termed the present value of the future cash flow (PV).  If the sum of the present values of a series of future cash flows is sought, the approach is to find the present value of each cash flow, prior to adding them together to find the Net Present Value (NPV) of the series of cash flows. The concept of Net Present Value (NPV) is central to the study and understanding of finance. This is because the value of any asset can be evaluated by finding the NPV of the net cash flows generated by the asset.

In finding the present value of any cash flow, what is required is the amount of the cash flow or the future value of the cash flow (e.g. £1,000),  the timing of the cash flow (e.g. 3 years), and an appropriate discount rate to be applied to the cash flow (e.g. 10%). The formula for finding the present value, PV of a future value (or future cash flow), FV is given by:

PV = FV / (1+r)^t

where PV is the present value, FV is the future value, r is the discount rate in decimals and t is the time of the cash flow in years.
As an example, the PV of £1,000 to be received in 3 years time at a 10% discount rate is:

PV = £1000 / (1+0.1)^3 = £751.31    (to 2 decimal places).

Thus, at 10% discount rate, £1000 received in 3 years is equivalent to £751.31. We can look at this discount rate in another way. Let us consider the 10% as another investment opportunity available to us. Thus, if we invested £751.31 today, it would grow at 10% compounded annually to £1000. The 10% discount rate is now effectively a compound rate, or the opportunity cost of capital for the investment. When doing time value of money analysis of this type, the discount rate takes on a few names, such as opportunity cost of capital, capitalization rate, expected return on capital and hurdle rate.

The time value of money concept can be applied to more than just a single cash flow as mentioned earlier, such as when calculating a Net Present Value. A set of stylised payments are central to understanding financial products that involve multiple payments or cash flows such as mortgages, loans, bonds and stocks. These are annuities, growing annuities, perpetuities and growing perpetuities.  The formulas for the Net present value of these types of cash flow arrangements are given below:

NPV of an Annuity:   PV(A) = (C/r)*[1- {1/(1+r)^n}]
NPV of a growing Annuity: PV(GA) = [C/(r-g)]*[1- {(1+g)^n/(1+r)^n}]
NPV of a perpetuity: PV(P) =  C/r
NPV of a growing perpetuity: PV(GP) = C/(r-g)

These values assume that constant payments of C are made from period 1 (or year 1) or that the payments grow at a constant rate g after the first payment of A in period 1 (or year 1). As a reminder, Net Present Values refer to values now (period 0 or year 0). The formulas assume discrete compounding at an interest or discount rate of r. If the period of compounding is different from a year, the number of periods, n, needs to be adjusted as well as the interest rate, r in these formulas. The derivations of these formulas and equivalent formulas for continuous compounding are addressed in another post.

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