# Net present value

*Investment appraisal.*

(NPV).

Net present value is the total present value of all of the cash flows of a proposal - both positive and negative - netting off negative present values against positive ones.

For example, the expected future cash inflows from an investment project LESS the initial capital investment outflow at Time 0.

Each present value (PV) is calculated as:

PV = Future value x Discount factor (DF)

Where:

DF = (1 + r)^{-n}

- r = cost of capital per period;
*and* - n = number of periods into the future that the cash flow is expected

**Example 1: cost of capital 10%**

A project requires an investment today of $100m, with $120m being receivable one year from now.

The cost of capital (r) is 10% per annum.

The NPV of the project is calculated as follows:

PV of Time 0 outflow $100m

= $(100m) negative

PV of Time 1 inflow $120m

= $120m x 1.1^{-1}

= $109.09m

NPV = -$100m + $109.09m

= **+$9.09m** (positive)

**Decision rule**

In very simple *Net Present Value analysis* the decision rule would be that:

(1) All positive NPV opportunities should be accepted.

(2) All negative NPV opportunities should be rejected.

So the project in the example above would be accepted (on the basis of this simple form of the NPV decision rule) because its NPV is positive, namely +$9.09m.

However this assumes the unlimited availability of further capital with no increase in the cost of capital.

A more refined decision rule is that:

- All negative NPV opportunities should still be rejected; while
- All positive NPV opportunities remain eligible for further consideration (rather than automatically being accepted).

**NPV drivers**

NPV is driven by the amounts of expected cash flows, their timing, and the cost of capital.

**Example 2: cost of capital rises to 20%**

Taking the same example of a project requiring an investment today of $100m, with $120m being receivable one year from now.

The cost of capital (r) rises to 20% per annum.

The NPV of the project is now calculated as follows:

PV of Time 0 outflow $100m

= $(100m)

PV of Time 1 inflow $120m

= $120m x 1.2^{-1}

= $100m

NPV = -$100m + $100m

= **$NIL**

*Now the project decision is marginal, following the change in the cost of capital assessment.*

**Example 3: cost of capital rises further to 30%**

Continuing with the same example of a project requiring an investment today of $100m, with $120m receivable one year from now.

The cost of capital (r) rises further to 30% per annum.

The NPV of the project would now be calculated as follows:

PV of Time 0 outflow $100m

= $(100m)

PV of Time 1 inflow $120m

= $120m x 1.3^{-1}

= $92.31m

NPV = -$100m + $92.31m

= **-$7.69m** (negative)

*Now the project would be rejected, following the further rise in the cost of capital evaluation.*