## Net present value (NPV)

* General approach to the evaluation of investment decisions. * As noted above, in order to make the investment solution profitable from a commercial point of view, it is necessary to compensate for cash outflows by cash inflows. However, taking into account the principle of time value of money, i.е. the fact that different cash flows have different meanings, you must first bring them to one point in time.

Therefore, in order to evaluate the investment decision, it is necessary:

o calculate and time all the cash flows generated by the solution to the corresponding time points (intervals)

o bring these flows at a rate corresponding to their risk, to one point in time (the moment of evaluation);

o summarize the resulting cash flows based on their sign (inflows - with a plus sign, and outflows with a minus sign). The resulting value is called * pure reduced value (MRA). MDU - the amount of discounted cash flows related to this investment decision (project). * Thus, this is the amount of cash flows associated with this investment decision, given by the time factor at the time of valuation.

where * РСР ^ - * the cash flow, timed to the 7th time (interval) of time;

*the lifetime of the project.*

**n -** The discount rate * (d) * uses the required level of return, determined taking into account the investment risk. If

*Oh, it means that:*

**YYYY & gt;**o expressed in & quot; today's & quot; evaluation of the effect of the project is a positive value;

o the project has a higher return than the discount rate g, the capital required from the investment capital with this level of risk;

o in an efficient market, the capitalization of the company's shares carrying out the project should increase when this decision is made on a value equal to * MDU. *

* MDU and the purpose of the financial manager. * Last interpretation is most important for us. It follows from the fact that the value of a company is determined by its future cash flows. Therefore,

*the amount that the project adds to the value of the company. And since the goal of financial managers is to maximize the value of the company (its own capital), their main task is to search for projects (strategic decisions) with a positive value of net present value.*

**MRO -**Thus, this criterion is ideal for evaluating individual investment projects, the absolute magnitude of their effect.

** Case **: consider the situation with a mini all-terrain vehicle. We will determine the value of the criterion of the MGU for the project of organizing the production of a mini-ATV at an automobile plant, using the rate of the required yield (the discount rate) of 17% per annum.

The project's cash flow consists of:

o a single payment in the zero period, equal to $ 10,586,000;

o annuity consisting of nine equal cash inflows, but $ 4,842,000 each;

o single monetary inflow, falling on the 10th time interval, equal to 5 428 000 dollars.

This means that:

o The expected absolute effect of this project is $ 12,093,000;

o project profitability is expected to be higher than 17% per annum in US currency;

o In an efficient market, the value of the company (its own capital) should increase by $ 12,093,000.

In order to encapsulate the profitability of the investment decision in percentages per annum, another criterion, called the internal rate of return, is used.

## Internal rate of return (IRR - internal rate of return)

* The concept of IRR. IRR - * is the profitability of the cash flows of the investment project, calculated at a compound interest rate.

Knowing how * NPV, * is interpreted, it's easy to explain why

*is defined as the discount rate at which*

**IRR***0.*

**NPV =** * Calculation of IRR. * Thus, in order to find

*it is necessary to solve the equation*

**IRR y** Then * IRR = x. *

There is no formula to solve such an equation. Therefore, it is solved using linear interpolation:

o at random take two discount rates - rl and r2, rl & lt; g;

o using each of the bids, calculate two values of * NPV NPV1 * and

**NPV2;** o the approximate value of * IRR * is derived from the formula

To get a more accurate value * IRR * the calculations are repeated several times, narrowing the interval between rl and g.

The internal rate of return can also be found using the financial calculator or the standard financial function of the EXCEL application (Figure 3.2).

Fig. 3.2. ** Graphic Interpretation ** * IRR *

* The rule of use and the purpose of the IRR. * If the project is an investment project, it usually has an ordinary cash flow, in which outflows occur at the beginning, and inflows later.

If * IRR & lt; * g, i.e. the profitability of the project is less than that required by investors (barrier) rate of return on invested capital, the project is rejected. The required level of profitability depends on the risk of the project and the state of the financial market. When comparing alternatives, the best one is considered to be a project that has a greater IRR rate when risk equals.

On the other hand, if the project is divisional or financial, i.e. its outflows follow inflows, the best is having a smaller IRR , since in this case * IRR * which & quot; donates & quot; initiator, when he chooses to receive money now instead of receiving additional income in the future. For example, if I attract financing, then I pay a bet for the resources provided to me, and the lower this rate, the better for me. Or if someone sells an asset, he loses the opportunity to receive income from it in the future, and the less lost profitability, the better for the seller of the asset. In both cases, we are dealing with reverse flows (first the tributaries, then the outflows).

* IRR * estimates the yield of

*of the invested capital, unlike*

**per unit***criterion, measuring the absolute value, the mass of the income received. Therefore, when comparing projects*

**NPV-***sometimes & quot; contradicts & quot;*

**IRR***as the project can be more profitable (in terms of the invested ruble), but in absolute terms give less effect due to smaller scale of the investment object.*

**NPV,**Another common cause of the contradiction between

andIRRis that comparing projects byNPVoccursIRRwhile calculatingonly by profitability,through the interest rate at which the discount occurs. Therefore, a more risky project may have a higher yieldNPV takes into account the riskbut when discounting at a high rate corresponding to its risk, it can show a smaller(IRR),NPV.

* Disadvantages of the IRR criterion. * Disadvantages of the

*criterion are as follows:*

**IRR**o difficulty in calculating the multi-step algorithm presented. At present, financial calculators are widely used by specialists for such calculations, which make it easy to overcome this inconvenience;

o the equation of the i-th power, which defines * IRR, * has

*roots, so one project, there are*

**n***values *

**n***True, only one*

**IRR.***If a project with a single stream has a positive return (i.e., the sum of its cash flows without discounting is greater than zero), "true"*

**IRR rate counts for analysis.***is the first positive zero of the equation from zero;*

**IRR** o When calculating * IRR *, it is assumed that the income received is reinvested at a rate equal to

*If the value*

**IRR.***is significantly larger than the discount rate, this assumption introduces significant distortions in the calculation results. However, the conclusions about the profitability of the project remain in effect, so that in the long run*

**IRR***performs its function.*

**IRR** ** Case **: let's go back to the situation with a mini-terrain vehicle. We will determine the internal rate of return of the project for the production of a vehicle with increased terrain.

Take the two discount rates - 17 and 50% per annum. * NPV * for 17% per annum is $ 12,093. At 50% per annum

*$ 1,060,000 From which the approximate value*

**NPV = -***can be defined as follows:*

**IRR** After several more iterations, successively narrowing the interval between rates, we can significantly clarify the value of * IRR, * which for the conditions of this task will amount to 44.66% per annum.

Since this value is somewhat larger than the required yield of 17% per annum, we can conclude: the project will not be rejected by the criterion * IRR. * It remains to find out where the figure for the required yield of 17% per annum appeared, which we used as a discount rate when calculating

*as the barrier level for the bet < i>*

**NPV vi****IRR.**To answer this question, you need to get acquainted with the

**model of the value of capital (long-term) assets (SARM-**capital asset prising model). However, before doing small additions to the technique of calculating the criteria for evaluating investment decisions.

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