# The Harmony Theorem

We know that there is a difference between the present value criterion for selecting investment opportunities and the internal rate of return criterion, and that it is strongly believed by theorists that the present value criterion is the better of the two, provided that account is made for the entire cash flow stream of the investment over all its periods But if you are asked to consider an investment of a fixed amount of dollars {say, in your friend's new venture), you probably would not evaluate this proposition in terms of present value; you would more likely focus on potential return. In fact, if you do make the investment, you are likely to encourage your friend to maximize the return on your investment, not the present value of the firm. Your friend might insist on maximizing present value. Is there a conflict here?

We will try to shed some light on this important issue by working through a hypothetical situation. Suppose your friend has invented a new gismo for which he holds the patent rights. To profit from this invention, he must raise capital and carry out certain operations, The cost for the operations occurs immediately; the reward occurs at the end of a year, In other words, the cash flow stream has just two elements: a negative amount now and a positive amount at the end of a year.

Your friend recognizes that there are many different ways that he can operate his venture, and these entail different costs and different rewards. Hence there are many possible cash flow streams corresponding to different operating plans, He must select one, The possibilities can be described by points on a graph showing the reward (at the end of a year) versus the current cost of operations, as in Figure 5 11(a), Your friend can select any one of the points.

Suppose also that the l~year interest rate is r = 10%, The possibility of depositing money in the bank can be represented on the graph as a straight line with slope 1,10: the current deposit is a cost, and the reward is 1.10 times that amount This slope will be used to evaluate the present value of a cash flow stream.

(a) Maximum present value (b) Maximum return

FIGURE 5„11 Comparison of criteria. U?) Plan A is selected because it has the greatest present value It is the point corresponding to (he highest line of slope, equal to 1 10 (b) Plan B is selected because it is the point on the line from the origin of greatest slope As the text demonstrates, the analysis in (b) is faulty, and when corrected, the maximum return criterion will correspond to the present value criterion

If your friend decides to maximize the present value of his venture, he will draw lines with slopes 1 + / = 1 10 and find the highest one that goes through a possible operating plan The plan that lies on that line is the optimal one This optimal line and plan are shown in Figure 5 11(a); point A is the optimal plan Using a bank, it is possible to move along the line through A In particular, it is possible to move all the way down to the horizontal axis At this point, no money will be received next year, but an amount P of net profit is obtained now

Suppose your friend asks you to invest in his venture, supplying a portion of the operating cost and getting that portion of the reward, You would measure the return on your investment. The operating point that achieves the maximum return is found by swinging a line upward, pivoting around the origin, reaching an operating point of greatest possible slope. The result of this process is shown in Figure 5 1\(h) The optimal point according to this criterion is the point B in the figure. The maximum return is the slope of this maximum-slope line Note that this slope is greater than 110%, So point B achieves a higher rate of return than point A Its present value, however, is just P', which is less than P. There seems to be a conflict

Here is how the conflict is resolved Your friend currently owns the rights to his gismo He has not yet committed any money for operations; but his present value analysis shows that he could go to the bank, take out a loan sufficient to cover the expenses for plan A, and then, at the end of the year, he could pay back the loan and pocket the profit of 1 10P (which is worth P now), He doesn't care about the rate of return, since he is not investing any money; he is just taking out a loan Alternatively, he could borrow the money from you, but he would not pay you any more than the current interest rate.

But you are not being asked to make a loan; you are being asked to invest in the venture—to have ownership in it, As an extreme case, suppose your friend asks you to buy the whole venture You will then have the rights to the gismo He is willing to stay on and operate the venture (if you provide the necessary operating costs), but you will have the power to decide what operating plan to use.

If your friend sells you the venture, he will charge you an amount P because that is what it would be worth to him if he kept ownership So if you decide to buy the venture, the total expense of an operating plan is now P plus the actual operating cost, If you want to maximize your return, you will maximize reward/(cost P) You can find this new best operating plan by swinging a line upward, pivoting around the point ~P, reaching the operating point with the greatest possible slope That point will be point A, the point that maximized the present value ILook again at Figure 5 ..1 !(«)■] Alternatively, once you are the owner, you might consider maximizing the present value That will lead to point <4 as well, Therefore if you decide to buy the venture, and you pay the full value P, you will maximize the return on your investment by operating under plan A\ and your return will be 110% (It does not matter if you decide to borrow some of the operating costs instead of funding them yourself; still you will want to operate at A, and your return will still be 110%,)

We summarize the preceding discussion by a general result that we term the harmony theorem It states that there is harmony between the present value criterion and the rate of return criterion when account is made for ownership.

Harmony theorem Current owners of a venture should want to operate the venture to maximize the present value of its cash flow stream Potential new owners, who must pay the full value of their prospective share of the venture, will want the company to operate in the same way, in order to maximize the return on their investment.

The harmony theorem is justification for operating a venture (such as a company) in the way that maximizes the present value of the cash flow stream it generates Both current owners and potential investors will agree on this policy

The presentation in this section considered only deterministic cash flow streams with two flows The harmony theorem generalizes to multiple periods and to random streams as well—under certain conditions A multiperiod generalization is discussed in Exercise 10