Future Value

The future value of a sum of money invested at interest rate i for one year is given by:

FV = PV ( 1 + i )

where

FV = future value
PV = present value
i = annual interest rate

If the resulting principal and interest are re-invested a second year at the same interest rate, the future value is given by:

FV = PV ( 1 + i ) ( 1 + i )

In general, the future value of a sum of money invested for t years with the interest credited and re-invested at the end of each year is:

FV = PV ( 1 + i ) ^t

Solving for Required Interest Rate or Time

Given a present sum of money and a desired future value, one can determine either the interest rate required to attain the future value given the time span, or the time required to reach the future value at a given interest rate. Because solving for the interest rate or time is slightly more difficult than solving for future value, there are a few methods for arriving at a solution:

1. Iteration - by calculating the future value for different values of interest rate or time, one gradually can converge on the solution.

2. Financial calculator or spreadsheet - use built-in functions to instantly calculate the solution.

3. Interest rate table - by using a table such as the one at the end of this page, one quickly can find a value of interest rate or time that is close to the solution.

4. Algebraic solution - mathematically calculating the exact solution.

Algebraic Solution

Beginning with the future value equation and given a fixed time period, one can solve for the required interest rate as follows.

FV = PV ( 1 + i ) ^t

Dividing each side by PV and raising each side to the power of 1/t:

( FV / PV ) ^1/t = 1 + i

The required interest rate then is given by:

i = ( FV / PV ) ^1/t - 1

To solve for the required time to reach a future value at a specified interest rate, again start with the equation for future value:

FV = PV ( 1 + i ) ^t

Taking the logarithm (natural log or common log) of each side:

log FV = log [ PV ( 1 + i ) ^t ]

Relying on the properties of logarithms, the expression can be rearranged as follows:

log FV = log PV + t log ( 1 + i )

Solving for t:

t =	log ( FV / PV ) log ( 1 + i )

Interest Factor Table

The term ( 1 + i ) ^t is the future value interest factor and may be calculated for an array of time periods and interest rates to construct a table as shown below:

Table of Future Value Interest Factors

t \ i	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
1	1.010	1.020	1.030	1.040	1.050	1.060	1.070	1.080	1.090	1.100
2	1.020	1.040	1.061	1.082	1.103	1.124	1.145	1.166	1.188	1.210
3	1.030	1.061	1.093	1.125	1.158	1.191	1.225	1.260	1.295	1.331
4	1.041	1.082	1.126	1.170	1.216	1.262	1.311	1.360	1.412	1.464
5	1.051	1.104	1.159	1.217	1.276	1.338	1.403	1.469	1.539	1.611
6	1.062	1.126	1.194	1.265	1.340	1.419	1.501	1.587	1.677	1.772
7	1.072	1.149	1.230	1.316	1.407	1.504	1.606	1.714	1.828	1.949
8	1.083	1.172	1.267	1.369	1.477	1.594	1.718	1.851	1.993	2.144
9	1.094	1.195	1.305	1.423	1.551	1.689	1.838	1.999	2.172	2.358
10	1.105	1.219	1.344	1.480	1.629	1.791	1.967	2.159	2.367	2.594
11	1.116	1.243	1.384	1.539	1.710	1.898	2.105	2.332	2.580	2.853
12	1.127	1.268	1.426	1.601	1.796	2.012	2.252	2.518	2.813	3.138
13	1.138	1.294	1.469	1.665	1.886	2.133	2.410	2.720	3.066	3.452
14	1.149	1.319	1.513	1.732	1.980	2.261	2.579	2.937	3.342	3.797
15	1.161	1.346	1.558	1.801	2.079	2.397	2.759	3.172	3.642	4.177

Present Value

The present value of a sum of money to be received at a future date is determined by discounting the future value at the interest rate that the money could earn over the period.

Starting with the future value equation:

FV = PV ( 1 + i ) ^t

where

FV = future value
PV = present value
i = annual interest rate

we see that the present value is given by:

PV =	FV ( 1 + i ) t

The term 1 / ( 1 + i ) ^t is known as the discount factor.

If both the future value and present value are known, one can solve for the time or the interest rate using one of the techniques discussed in future value calculations.

Present Value of Multiple Future Cash Payments

When there is more than a single cash payment at a future date, the present value is calculated by taking the present values of the individual cash payments and summing them. It is helpful to draw a time line depicting the timing of the cash payments:

Time Line

0		1		2		3
PV		C1		C2		C3

In this model, the cash payment at each date may be either an inflow or an outflow; the direction is designated by the sign. The present value of the above cash flow is:

PV = C₁ / ( 1 + i ) + C₂ / ( 1 + i )² + C₃ / ( 1 + i )³

Discount Factor Table

The discount factor 1 / ( 1 + i ) ^t may be calculated for a range of time periods and interest rates and tabulated for quick reference.

Table of Discount Factors

t \ i	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
1	0.990	0.980	0.971	0.962	0.952	0.943	0.935	0.926	0.917	0.909
2	0.980	0.961	0.943	0.925	0.907	0.890	0.873	0.857	0.842	0.826
3	0.971	0.942	0.915	0.889	0.864	0.840	0.816	0.794	0.772	0.751
4	0.961	0.924	0.888	0.855	0.823	0.792	0.763	0.735	0.708	0.683
5	0.951	0.906	0.863	0.822	0.784	0.747	0.713	0.681	0.650	0.621
6	0.942	0.888	0.837	0.790	0.746	0.705	0.666	0.630	0.596	0.564
7	0.933	0.871	0.813	0.760	0.711	0.665	0.623	0.583	0.547	0.513
8	0.923	0.853	0.789	0.731	0.677	0.627	0.582	0.540	0.502	0.467
9	0.914	0.837	0.766	0.703	0.645	0.592	0.544	0.500	0.460	0.424
10	0.905	0.820	0.744	0.676	0.614	0.558	0.508	0.463	0.422	0.386
11	0.896	0.804	0.722	0.650	0.585	0.527	0.475	0.429	0.388	0.350
12	0.887	0.788	0.701	0.625	0.557	0.497	0.444	0.397	0.356	0.319
13	0.879	0.773	0.681	0.601	0.530	0.469	0.415	0.368	0.326	0.290
14	0.870	0.758	0.661	0.577	0.505	0.442	0.388	0.340	0.299	0.263
15	0.861	0.743	0.642	0.555	0.481	0.417	0.362	0.315	0.275	0.239

Capital Budgeting

A capital expenditure is an outlay of cash for a project that is expected to produce a cash inflow over a period of time exceeding one year. Examples of projects include investments in property, plant, and equipment, research and development projects, large advertising campaigns, or any other project that requires a capital expenditure and generates a future cash flow.

Because capital expenditures can be very large and have a significant impact on the financial performance of the firm, great importance is placed on project selection. This process is called capital budgeting.

Criteria for Capital Budgeting Decisions

Potentially, there is a wide array of criteria for selecting projects. Some shareholders may want the firm to select projects that will show immediate surges in cash inflow, others may want to emphasize long-term growth with little importance on short-term performance. Viewed in this way, it would be quite difficult to satisfy the differing interests of all the shareholders. Fortunately, there is a solution.

The goal of the firm is to maximize present shareholder value. This goal implies that projects should be undertaken that result in a positive net present value, that is, the present value of the expected cash inflow less the present value of the required capital expenditures. Using net present value (NPV) as a measure, capital budgeting involves selecting those projects that increase the value of the firm because they have a positive NPV. The timing and growth rate of the incoming cash flow is important only to the extent of its impact on NPV.

Using NPV as the criterion by which to select projects assumes efficient capital markets so that the firm has access to whatever capital is needed to pursue the positive NPV projects. In situations where this is not the case, there may be capital rationing and the capital budgeting process becomes more complex.

Note that it is not the responsibility of the firm to decide whether to please particular groups of shareholders who prefer longer or shorter term results. Once the firm has selected the projects to maximize its net present value, it is up to the individual shareholders to use the capital markets to borrow or lend in order to move the exact timing of their own cash inflows forward or backward. This idea is crucial in the principal-agent relationship that exists between shareholders and corporate managers. Even though each may have their own individual preferences, the common goal is that of maximizing the present value of the corporation.

Alternative Rules for Capital Budgeting

While net present value is the rule that always maximizes shareholder value, some firms use other criteria for their capital budgeting decisions, such as:

• Internal Rate of Return (IRR)
• Profitability Index
• Payback Period
• Return on Book Value

In some cases, the investment decisions resulting from the IRR and profitability index methods agree with those of NPV. Decisions made using the payback period and return on book value methods usually are suboptimal from the standpoint of maximizing shareholder value.

Annuities

An annuity is a series of equal payments over a specified time frame. For example, a cash payment of C made at the end of each year for four years at annual interest rate i is shown in the following time line:

4-Year Annuity Time Line

0		1		2		3		4
PV		C		C		C		C

This time line is for an ordinary annuity, in which the cash payments are made at the end of each year. For example, the first payment is made exactly one year from the present. The present value of this cash flow is calculated by:

PV = C / ( 1 + i ) + C / ( 1 + i )² + C / ( 1 + i )³ + C / ( 1 + i )⁴

In general, for a t year annuity:

PV = C / ( 1 + i ) + C / ( 1 + i )² + ... + C / ( 1 + i )^t

From this potentially long series, a present value formula can be derived. First, multiply each side by 1 / ( 1 + i ).

PV / ( 1 + i ) = C / ( 1 + i )² + C / ( 1 + i )³ + ... + C / ( 1 + i )^t+1

In order to eliminate most of the terms in the series, subtract the second equation from the first equation:

PV - PV / ( 1 + i ) = C / ( 1 + i ) - C / ( 1 + i )^t+1

Solving for PV, the present value of an ordinary annuity is given by:

PV =

C i

1 -

1 ( 1 + i ) t

This equation assumes that the first payment of the annuity is made at the end of the first time period. If instead the payments are made at the beginning of each time period, then the present value calculation would be similar to the above, except that all payments would be shifted forward by one year. This shift can be accomplished by multiplying the entire present value expression by ( 1 + i ). Such an annuity with the payments occurring at the beginning of each time period is called an annuity due.

Annuity Factor Table

The factor for calculating the present value of an ordinary annuity may be calculated for a range of time periods and interest rates and tabulated for quick reference. The annuity factor is the value of the following expression:

1 i

1 -

1 ( 1 + i ) t

The following table shows the value of this factor for various interest rates and time periods.

Table of Present Value Annuity Factors

t \ i	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
1	0.990	0.980	0.971	0.962	0.952	0.943	0.935	0.926	0.917	0.909
2	1.970	1.942	1.913	1.886	1.859	1.833	1.808	1.783	1.759	1.736
3	2.941	2.884	2.829	2.775	2.723	2.673	2.624	2.577	2.531	2.487
4	3.902	3.808	3.717	3.630	3.546	3.465	3.387	3.312	3.240	3.170
5	4.853	4.713	4.580	4.452	4.329	4.212	4.100	3.993	3.890	3.791
6	5.795	5.601	5.417	5.242	5.076	4.917	4.767	4.623	4.486	4.355
7	6.728	6.472	6.230	6.002	5.786	5.582	5.389	5.206	5.033	4.868
8	7.652	7.325	7.020	6.733	6.463	6.210	5.971	5.747	5.535	5.335
9	8.566	8.162	7.786	7.435	7.108	6.802	6.515	6.247	5.995	5.759
10	9.471	8.983	8.530	8.111	7.722	7.360	7.024	6.710	6.418	6.145
11	10.368	9.787	9.253	8.760	8.306	7.887	7.499	7.139	6.805	6.495
12	11.255	10.575	9.954	9.385	8.863	8.384	7.943	7.536	7.161	6.814
13	12.134	11.348	10.635	9.986	9.394	8.853	8.358	7.904	7.487	7.103
14	13.004	12.106	11.296	10.563	9.899	9.295	8.745	8.244	7.786	7.367
15	13.865	12.849	11.938	11.118	10.380	9.712	9.108	8.559	8.061	7.606