A Rational Model for Earnings Benchmarks Luca Ashok Stanford University Lashok1@stanford.edu September, 2013 Abstract I present a rational model of earnings management as an explanation for why the market assigns a premium for firms achieving key earnings benchmarks. I study an infinite period model where in each period firms are probabilistically chosen to have either high or low discretion. Firms with high discretion have the option to either manipulate earnings downward, and have the manipulation reverse next period, or to do nothing at all. I show that, under certain conditions, equilibrium exist where achieving a benchmark earns a market premium and the market assigns the premium rationally. 1 1. Introduction Benchmarks play an important role in managers’ decision making. Managers are typically concerned with meeting analyst forecasts, meeting earnings of prior periods, and avoiding losses (Moehrle, 2002). In many instances managers have been found to manipulate earnings to reach key benchmarks. Brown (2001) finds that over time firms have shifted towards meeting or just beating analyst forecasts, implying an increase in earnings management. Kasznik and McNichols (2002) find that the market assigns a premium to firms that meet analyst’s expectations, giving managers incentives to reach these benchmarks. Burgstahler and Dichev (1997) find a large amount of firms that just beat last period’s earnings or report a small positive gain. They also find unusually low frequencies of firms reporting just below last period’s earnings or small losses, implying that firms manage earnings when close to these benchmarks. Barth et al. (1999) find that the market rewards firms even more for achieving consecutive periods of increased earnings. Managers also have incentives to achieve benchmarks in the form of CEO pay. Matsunaga and Park (2001) find that CEO bonus payments incentivize managers to meet both analyst forecasts and earnings from prior periods. The accounting literature has studied three main ways that managers manipulate earnings to reach these benchmarks: Accrual manipulation, expectations management, and real activities manipulation (Doyle et al. 2013). My model focuses only on accrual manipulation, specifically when a firm lowers earnings in the current period through accounting reserves so that it can report higher than normal earnings in future periods. 2 In my setting each firm is probabilistically assigned a firm value and an amount of discretion in every period. The firm value of each firm is drawn from a finite, discrete distribution and each firm’s discretion can either be low or high. High discretion firms can choose to exercise an option where they misreport their earnings one lower than their true firm value. In exchange, the firm can misreport their earnings one higher in the following period. Firms that choose to exercise this option trade off the difference between their “true” market value (the market value if they report truthfully) and the market value of the report one lower in exchange for a higher report in the future. Therefore firms will choose to exercise if and only if the difference between the market value of a truthful report and the report one lower is sufficiently low. I try to show the existence of equilibrium where the market puts a premium on firms that report above a certain threshold. High discretion firms just above the benchmark will choose to exercise since the difference between their truthful report and the benchmark report is sufficiently low. These high value firms making the benchmark earnings report raise the expected firm value for that specific report maintaining the market premium. The main characteristic of the model that allows for this result is that firms cannot predict their types in the next period. Since they only have the potential option of lowering earnings now so that they can increase them next period, the firms which misreport higher earnings will be spread out evenly throughout all firm types. Meanwhile only firms which find themselves at an opportune point to exercise (firms just above a benchmark) will choose to misreport lower earnings. This pattern of earnings management allows for market premiums on firms which reach a benchmark. 3 The rest of the paper is organized as follows. Section 2 discusses related theoretical literature. Section 3 presents the model and solves for specific equilibrium. Section 4 discusses a numerical example. Section 5 discusses limitations of the model. Section 6 discusses empirical implications of the model and Section 7 concludes. Appendix A contains proofs. 2. Related theoretical literature Guttman, Kandel, and Kadan (2006) offer an alternative explanation for a discontinuity in the distribution of earnings around a key benchmark. In their model they have a more standard earnings management setting similar to Fischer and Verrecchia (2000). They find that a discontinuity in earnings may be due to a special form of equilibrium where there is a pooling interval. A key difference in their model is that managers face a personal cost for misreporting that increases in the amount of earnings management taking place. Their results are driven by investor’s beliefs that all firms in the pooling interval will make the same report. Managers trade off the higher stock price from increasing their report against the personal cost of earnings management. In my setting the result is driven from a different process. Here only high discretion firms can manipulate, and all firms that manipulate do so by the same amount and without personal cost to the manager. Managers that choose to exercise trade off a loss in current period stock price in exchange for a future gain in stock price. 4 3.1 The Model Each period firms are assigned randomly to a firm value ∈ {0,1, … , 𝑁 − 2, 𝑁 − 1} according to a probability distribution 𝑑. The market is risk neutral and cannot distinguish between two firms making the same report. Therefore it assigns a value 𝑀(𝑥) = 𝐸[𝑓𝑖𝑟𝑚 𝑣𝑎𝑙𝑢𝑒|𝑟𝑒𝑝𝑜𝑟𝑡 𝑥] 𝑓𝑜𝑟 𝑥 ∈ {0,1,2 … 𝑁}. Firms are assigned high or low discretion according to the following function: 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐻𝑖𝑔ℎ 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 = { 𝑝 0 𝑖𝑓 𝑓𝑖𝑟𝑚 𝑑𝑖𝑑 𝑛𝑜𝑡 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑖𝑓 𝑓𝑖𝑟𝑚 𝑑𝑖𝑑 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 Firms with high discretion can choose to report one lower than their true value in exchange for an automatic reversal the next period (they must report one higher than their true value the next period). 𝑁 is greater than four and there is a discount rate 𝑟. Each high discretion firm faces a choice to either report truly or report lower in exchange for a higher report the next period. The expected value of exercising the option minus the expected value of not exercising the option is the following (for a firm of type x): 𝑁−1 − 𝑀(𝑥) + 𝑀(𝑥 − 1) − 𝜀 𝑁(1 + 𝑟) Where 𝜀 is the expected gain from having the potential of high discretion next period. If this value is greater than zero, the firm will exercise its option, otherwise it will report truthfully. Another way to state this is: 𝑁−1 A firm x will exercise if and only if 𝑀(𝑥) − 𝑀(𝑥 − 1) < 𝑁(1+𝑟) − 𝜀 5 𝑁−1 Notice that 𝑁(1+𝑟) − 𝜀 is a constant. Therefore this is the constant threshold value of 𝑀(𝑥) − 𝑀(𝑥 − 1) that determines whether or not firm x will exercise. 3.2 Equilibrium Conditions 1) 𝑀(𝑥) = 𝐸[𝑓𝑖𝑟𝑚 𝑣𝑎𝑙𝑢𝑒|𝑟𝑒𝑝𝑜𝑟𝑡 𝑥] 𝑓𝑜𝑟 𝑥 ∈ {0,1,2 … 𝑁}. 𝑁−1 2) 𝑀(𝑥) − 𝑀(𝑥 − 1) < 𝑁(1+𝑟) − 𝜀 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ𝑖𝑔ℎ 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 𝑓𝑖𝑟𝑚𝑠 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑖𝑛𝑔 𝑁−1 3) 𝑀(𝑥) − 𝑀(𝑥 − 1) > 𝑁(1+𝑟) − 𝜀 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ𝑖𝑔ℎ 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 𝑓𝑖𝑟𝑚𝑠 𝑛𝑜𝑡 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑖𝑛𝑔 3.3 Equilibrium with a single benchmark Suppose we have an equilibrium with a single benchmark Y ∈ {0,1 … 𝑁} such that the market assigns a premium for that benchmark and for no others. (We will also assume 𝑌 > 1.) I will define a premium as the market valuing the report for more than the actual report made. I will solve for the conditions (𝑝, 𝑁, 𝑟, 𝑑) such that this type of equilibrium can be sustained. This can only happen in equilibrium when firms of type 𝑌+1 with high discretion choose to report down. It will suffice to show that the equilibrium price 𝑀(𝑌) > 𝑌, and that the only high discretion firm that will exercise the option to report lower is the 𝑌+1 firm. I will assume that only high discretion firms of type 𝑌+1 will exercise and then verify that this is in fact an equilibrium. 6 Given the behavior of each type of firm, we can deduce the market value for each type of report. There are five different types of reports to calculate. The highest report (𝑁+1), the lowest report (0), the benchmark report (𝑌), the report just above the benchmark (𝑌+1), and all the rest. 𝑀(𝑁 + 1) = 𝑁, since only firms of type 𝑁 can report 𝑁+1 𝑀(0) = 0, since only firms of type 0 will report 0 Let 𝑃 be the proportion of firms that will be reporting higher this period because they exercised last period. 𝑃= 𝑀(𝑌) = 𝑝 ∗ 𝑑(𝑌 + 1) 1 + 𝑝 ∗ 𝑑(𝑌 + 1) (1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1) (1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1) 𝑀(𝑌 + 1) = 𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) 𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1) 𝑀(𝑥) = 𝑃𝑑(𝑥 − 1)(𝑥 − 1) + (1 − 𝑃)𝑑(𝑥)𝑥 𝑃𝑑(𝑥 − 1) + (1 − 𝑃)𝑑(𝑥) Next we check all the possible values of 𝑀(𝑥) − 𝑀(𝑥 − 1): There is a general form and then four different forms when x is close to Y or to 0. The general form is: 𝑀(𝑥) − 𝑀(𝑥 − 1) = 𝑑(𝑥)(1 − 𝑃)𝑥 + 𝑑(𝑥 − 1)𝑃(𝑥 − 1) 𝑑(𝑥)(1 − 𝑃) + 𝑑(𝑥 − 1)𝑃 − 𝑑(𝑥 − 1)(1 − 𝑃)(𝑥 − 1) + 𝑑(𝑥 − 2)𝑃(𝑥 − 2) 𝑑(𝑥 − 1)(1 − 𝑃) + 𝑑(𝑥 − 2)𝑃 7 The four corner cases are: 𝑀(1) − 𝑀(0) = 𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃 ∗ 0 𝑑(1)(1 − 𝑃) −0 = 𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃 𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃 𝑀(𝑌 + 2) − 𝑀(𝑌 + 1) = 𝑑(𝑌 + 2)(1 − 𝑃)(𝑌 + 2) + 𝑑(𝑌 + 1)𝑃(𝑌 + 1) 𝑑(𝑌 + 2)(1 − 𝑃) + 𝑑(𝑌 + 1)𝑃 − 𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) 𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1) 𝑀(𝑌 + 1) − 𝑀(𝑌) = 𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) 𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1) − (1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1) (1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1) 𝑀(𝑌) − 𝑀(𝑌 − 1) = (1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1) (1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1) − 𝑑(𝑌 − 1)(1 − 𝑃)(𝑌 − 1) + 𝑑(𝑌 − 2)𝑃(𝑌 − 2) 𝑑(𝑌 − 1)(1 − 𝑃) + 𝑑(𝑌 − 2)𝑃 In order to have such an equilibrium we need the following conditions to hold: 1) 𝑀(𝑌) > 𝑌 𝑁−1 2) 𝑀(𝑌 + 1) − 𝑀(𝑌) < 𝑁(1+𝑟) 𝑁−1 3) 𝑀(𝑥) − 𝑀(𝑥 − 1) > 𝑁(1+𝑟) − 𝜀 ∀𝑥 ≠𝑌+1 8 PROPOSITION 1: There exists an equilibrium with a single benchmark for uniform 𝑑 and for any 𝑝. A sufficient condition for such an equilibrium is: 1 − 𝑝(1 − 𝑃)(1 + 2𝑃) 𝑁−1 < < 1−𝑃 2 2 1 − 𝑝 (1 − 𝑃) 𝑁(1 + 𝑟) 4. Numerical Example 1) There are 5 firm types: -2, -1, 0, 1, and 2. 2) There are 6 possible reports: -2, -1, 0, 1, 2, and 3. 3) The probability of having high discretion is .5, and there is no rate of return. The market value of -2 will be -2 since only firms of type -2 will report -2. 10 1 12 The market value of -1 will be: −1 ∗ 11 + −2 ∗ 11 = − 11 1 10 5 11 The market value of 0 will be: (−1 ∗ 11 + 0 ∗ 11 + 1 ∗ 11 ) 16 = 5 1 The market value of 1 will be: (1 ∗ 11 + 0 ∗ 11) 1 11 6 10 The market value of 2 will be: 1 ∗ 11 + 2 ∗ 11 = = 1 4 5 6 21 11 The market value of 3 will be 2 since only firms of type 2 will report 3. Next we check if all the choices made are in fact rational: 4 The expected gain from reporting one more than true value in the next period is 5. The 1 expected gain from doing nothing is 10 times the gain from reporting down when you are of type 7 1 and of high discretion. The loss from reporting down when you are of type 1 is 12. 9 So this will be in fact the rational choice if 4 7 1 4 7 − 12 > 10 ∗ (5 − 12) or if 5 4 7 − 12 > 0. 5 Which is in fact the case. 4 Finally we must check that no other type will lose less than 5 from reporting down one. This is in fact the case. A possible way to interpret this example is a firm that has either negative, slightly negative, zero, slightly positive, or positive earnings. The gap in market price is smallest between a report of zero and slightly positive gain. Therefore firms with a slightly positive gain and high discretion will choose to exercise their option and report zero earnings for a chance to report higher in the next period. Because so many firms with a true value of slightly positive gain are actually reporting zero earnings, the market price for firms which report zero earnings is above zero and sufficiently high to maintain the equilibrium. 5. Limitations of the Model The model leaves out many details both for simplicity and tractability. For example the distribution of firm values is assumed to be uniform and the market is assumed to be risk neutral. Some of the other details left out are more worrisome, however. The model assumes that the market cannot distinguish between firms making the same report. This is assumed for tractability so that the market can simply price the firm based on its report without having to condition on the past. This does not line up perfectly with reality since in practice the market is generally able to connect past reports with current ones for any firm. 10 If the market were able to condition on the past, for example if it could look back and see every firm’s report last period, it would be able to pick out the firms which were more likely to have exercised in the last period thus reducing the benefit of exercising the option. This could either weaken or completely reverse the results. Another worrisome limitation is that firms are forced to immediately reverse the manipulation. In practice firms can choose to reverse their manipulations at a time which they most need it. This may not qualitatively affect the results of the model. For example if 𝑁 is large and there is no reason to wait a period since it will be too unlikely to be in a better position next period. Alternatively, if the discount rate is high then firms may almost never want to wait for the next period. Finally, firms with high discretion only have the option to first misreport down and then misreport up the next period. In reality, we might expect firms with high discretion to also have the capability of doing the opposite. We might expect them to be able to first misreport up and then misreport down. If it were equally easy for a firm to do so (misreport up then down), then the market premium would vanish. One possible justification of the model is that if it were significantly easier for firms to first misreport down and then up, the results of the model would not change qualitatively. 6. Empirical Implications of the Model The model predicts that if there is a discontinuity around a single earnings benchmark, it is the product of firms just above the benchmark creating an accounting reserve and reporting low. This would correspond to the following empirical predictions: 11 1) A larger frequency of reports at or just above an earnings benchmark (relative to a scenario with no earnings management). 2) A drop in frequency of reports somewhere above and close to the earnings benchmark (relative to a scenario with no earnings management). 3) Low discretionary accruals for firms at or just above an earnings benchmark. Suppose the benchmark is report 𝑌. The first two points come from the fact that a relatively large amount of firms of type 𝑌+1 will be exercising and reporting 𝑌 instead. Of course this is the result of a discrete distribution and real earnings are more easily described in terms of continuous distributions. Another way to summarize the first two points is to say that the model predicts, relative to a scenario with no earnings management, for an interval just above the earnings benchmark, a shift in earnings towards the benchmark. The third point is perhaps the most counterintuitive. The common story is that firms just below an earnings benchmark manage earnings through increased discretionary accruals to reach the benchmark. You would then expect relatively high discretionary accruals for firms just at or above an earnings benchmark. Burgstahler and Dichev (1997) were looking for this effect but instead found the opposite. They found that “other accruals” were lower for the small profit group than for the small loss group. This finding is consistent with my model. 7. Conclusion This paper presents a theoretical explanation for why in equilibrium both firms and investors may care whether or not a benchmark is reached. This result comes from firms just above a benchmark that manage earnings and report lower so that they can report higher in future 12 periods. I show that if we assume the distribution of firm types is uniform, for any given probability of high discretion there exist equilibrium where there is a single benchmark and the market assigns a premium to firms which reach that benchmark. Intuitively, one might expect that firms that end just above a certain benchmark via earnings management would have reached that benchmark by boosting earnings. In this paper, however, I present a model where the opposite takes place: firms find themselves just above a benchmark as a result of managing earnings down. This provides a counterintuitive empirical prediction that firms that just meet or beat a key earnings benchmark may have particularly low discretionary accruals. In future research I hope to relax many of the strong assumptions in the model to see if the results still hold and such equilibrium are still possible. 13 REFERENCES BARTH M., J. ELLIOTT, AND M. 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PARK “The Effect of Missing a Quarterly Earnings Benchmark on the CEO's Annual Bonus” The Accounting Review 76 (2001): 313-332 MOEHRLE, S. R. “Do Firms Use Restructuring Charge Reversals to Meet Earnings Targets?” The Accounting Review 77 (2002): 397-413 14 Appendix A PROPOSITION 1: With 𝑑 uniform, we have: 𝑀(𝑥) − 𝑀(𝑥 − 1) = 1 𝑀(1) − 𝑀(0) = 1 − 𝑃 𝑀(𝑌 + 2) − 𝑀(𝑌 + 1) = 1 − 𝑃 + 𝑃 1 − 𝑝(1 − 𝑃) 𝑀(𝑌 + 1) − 𝑀(𝑌) = 1 − 𝑝(1 − 𝑃)(1 + 2𝑃) 1 − 𝑝2 (1 − 𝑃)2 𝑀(𝑌) − 𝑀(𝑌 − 1) = 𝑝(1 − 𝑃) − 𝑃 +1+𝑃 1 + 𝑝(1 − 𝑃) 𝑀(1) − 𝑀(0) < 𝑀(𝑥) − 𝑀(𝑥 − 1) 𝑀(1) − 𝑀(0) < 𝑀(𝑌 + 2) − 𝑀(𝑌 + 1) We can write 𝑃 in terms of 𝑝 and 𝑁 𝑃= 𝑝/𝑁 1 + 𝑝/𝑁 𝑀(𝑌) − 𝑀(𝑌 − 1) = 𝑝(1 − 𝑃) − 𝑃 +1+𝑃 1 + 𝑝(1 − 𝑃) 𝑝 𝑁 𝑝(1 − 𝑃) − 𝑃 = 𝑝− 𝑝 >0 1+𝑁 1+𝑁 𝑝 This implies: 𝑀(𝑌) − 𝑀(𝑌 − 1) > 1 + 𝑃 > 𝑀(1) − 𝑀(0) Need to show: 𝑀(𝑌 + 1) − 𝑀(𝑌) < 𝑀(1) − 𝑀(0) 1 − 𝑝(1 − 𝑃)(1 + 2𝑃) − (1 − 𝑃) < 0 1 − 𝑝2 (1 − 𝑃)2 15 1− 𝑝 1 + 3𝑝/𝑁 𝑝 ( 1 + 𝑝/𝑁 ) 1+𝑁 1 − 𝑝 2 𝑝 <0 1−( ) 1 + 𝑝 𝑁 1+𝑁 2 3𝑝 1+ 𝑁 𝑝 𝑝 (1 − 𝑝( 𝑝 )) (1 + 𝑁) − 1 + ( 𝑝) < 0 1+𝑁 1+𝑁 1+𝑁 𝑝 2 3𝑝 1+ 𝑁 𝑝 𝑝 (1 + ) − 𝑝 ( 𝑝 )−1+( 𝑝) < 0 𝑁 1+𝑁 1+𝑁 𝑝 3 𝑝 2 𝑝 3𝑝 (1 + ) − (1 + ) − 𝑝 (1 + ) (1 + ) + 𝑝2 < 0 𝑁 𝑁 𝑁 𝑁 𝑝 𝑝 2 𝑝 3𝑝 (1 + ) − 𝑝 (1 + ) (1 + ) + 𝑝2 < 0 𝑁 𝑁 𝑁 𝑁 1 𝑝 2 𝑝 3𝑝 (1 + ) − (1 + ) (1 + ) + 𝑝 < 0 𝑁 𝑁 𝑁 𝑁 1 2𝑝 𝑝2 4𝑝 3𝑝2 + 2+ 3−1− − 2 +𝑝 <0 𝑁 𝑁 𝑁 𝑁 𝑁 Want to show this is true for all 𝑝, 𝑁 𝑓(𝑝, 𝑁) = 1 2𝑝 𝑝2 4𝑝 3𝑝2 + 2+ 3−1− − 2 +𝑝 𝑁 𝑁 𝑁 𝑁 𝑁 𝜕𝑓 2 2𝑝 4 6𝑝 = 2+ 3− − 2+1 𝜕𝑝 𝑁 𝑁 𝑁 𝑁 𝜕𝑓 > 0 ∀𝑝, 𝑁 > 4 𝜕𝑝 So suffices to show that: 𝑓(1, 𝑁) < 0 ∀𝑁 𝑓(1, 𝑁) = 1 2 1 4 3 + 2+ 3−1− − 2+1 𝑁 𝑁 𝑁 𝑁 𝑁 𝑓(1, 𝑁) = − 3 1 1 − 2 + 3 < 0 ∀𝑁 𝑁 𝑁 𝑁 So equilibrium is always possible. Just need 𝑟, 𝑁 such that: 16 𝑀(𝑌 + 1) − 𝑀(𝑌) < 𝑁−1 < 𝑀(1) − 𝑀(0) 𝑁(1 + 𝑟) 1 − 𝑝(1 − 𝑃)(1 + 2𝑃) 𝑁−1 < < 1−𝑃 2 2 1 − 𝑝 (1 − 𝑃) 𝑁(1 + 𝑟) Note that: 𝑀(𝑌) > 𝑌 Since 𝑃 < 𝑝 ∀𝑁 17