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We consider the problem of approximating the true time-weighted return when a cash flow occurs at an unknown time during the estimation period, which is usually the case of a traditional portfolio evaluated on a daily basis. We aim to provide the best approximation, in terms of mean square error (MSE), under the following main assumptions: the distribution of the log-returns belongs to a subclass of elliptical distributions; a single flow occurs at a uniformly distributed random time; the amount of the flow and the returns of the period are independent. We derive a closed-form formulation for high evaluation frequencies when the returns satisfy the popular assumption of a Geometric Brownian Motion. Besides, with the further assumption of small flows, the Original Dietz return can be obtained as an approximation of our optimal estimator. This implies that under the above-mentioned conditions the Original Dietz return has a MSE close to the minimum. Although further improvements of the MSE seem to be possible only by increasing the estimation frequency, which in turn is usually infeasible, our model provides a rigorous way to handle large flows, which are especially frequent in applications such as performance attribution.

This paper studies the dynamic relation between trading volume and stock returns from the perspective of out-of-sample stock return predictability. Evidence from the U.S. suggests that higher returns do follow more intensive trading, especially in the pre-2000 period. However, the ex-ante predictability delivers a small economic gain equivalent to an annual return of 0.73% for a risk-averse investor. This weak out-of-sample predictive power of volume is absent in most of the other major markets. Overall, investors are not likely to gain much financially by “riding the volume curve,” at least at the levels of net profits suggested by our findings.

We use the data of 10 thousand accrual of Zenglibao monetary fund of Celestica Fund, and two indicators of the monetary fund market, WIND index of monetary fund and CSI money fund index, to analyze the volatility and compensative rate of return of Yuebao. Based on the time variant CAPM, we quantitatively show that the volatility of return of Yuebao is less than that of the market, and the correlation between the Yuebao and the market is relatively low. These two conditions make the Beta coefficient lower than that in traditional financial products. In this paper, we define the gap between return of Yuebao and the estimated return by CAPM as the externality compensative rate of return, which is the main explanation of the high-return property of Yuebao.

Using the theory of asset securitization, we analyze the feasibility of the securitization of letters of guarantee in theory. In the process of constructing the model of rate of return of securities backed by letters of guarantee, we propose two indices: the risk probability of asset-backed securities and the loss rate of asset-backed securities to analyze the cash flow of securities. On the basis of no arbitrage principle, the expression of rate of return of securities backed by letters of guarantee is put forward. In order to study the relationship between the rate of return of securities and other influential factor in the model, a simulation experiment is designed. The experiment results show that (i) an increasing risk probability of cash flow or a short maturity date also make the return rate of securities increase and (ii) the return rate of securities is higher in economic boom than that in economic recession when other parameters remain unchanged.

IRR , a widely used profitability measure, is the Discount Rate that yields Net Present Value ( NPV ) = 0 for a stream of positive and negative cash flows, at least one of each sign and with no explicit financing payments. A big disadvantage is lack of parameters, such as a project finance rate or the enterprise rate ( ER ), i.e., Return on Investment of the overarching investment group to serve as a measure of opportunity cost. The coupled metrics proposed earlier by the author— NPV project and NPV % –do not suffer these disadvantages, so IRR is analyzed in terms of NPV % . Useful information can be obtained from a projection of IRR values onto the NPV % , ER plane revealing the sensitivity of IRR to risk under meaningful operating conditions.

The evaluation of the efficiency of investments relies on a system of measures based on actuarial techniques that consider the time value of money. One of the common measures used is the Internal Rate of Return (IRR). Commonly, by applying of the efficiency evaluation criteria, result consistent outcomes. In this paper, the author tries to highlight that, based on its theoretical assumptions and practical drawbacks, considering of this measure in evaluation of the investments decisions may lead to erroneous decision. Despite the fact that the Internal Rate of Return (IRR) has never had a favorable academic press, the surveys outline that financial managers seem just to enjoy this measure. The aim of this paper is to summarize the drawbacks of this indicator and to offer a presentation of the Modified Internal Rate of Return (MIRR), as a solution to express a project performance by using of a percentage measure concomitant to discard the unrealistic assumption of reinvestment of cash flow stream just at the value of the IRR, allowing a straightforward calculation.

From the theory of effective market, the paper analyzes the relationship between the rate of return and the effective market. Through the theoretical analysis, we can see that the effective market ...

We revisit China's suspected overinvestment problem by examining the rate of return on capital from two perspectives. First, we find that two existing methods of estimating the rate of return generate conflicting results, and we succeed in reconciling them. Our revised estimates show that the rate of return rose sharply from 1998 to 2014, which helps explain the strong investment drive. Second, we explore what caused the rate of return to rise. We find the explanation lies mainly with the long-term factor of rising total factor productivity but that the short-term cyclical factor of low real interest rates has also contributed.

Markowitz Portfolio theory is based on the expected return and risk but investors are more interested in realized return. The considerations, expected return as realized return and variance as investment risk, of Markowitz’s mean – variance model enable the researchers or scholars to further explore on the validity of Markowitz theory. The present study makes an attempt to unfold a new idea in investment scenario where Markowitz theory is empirically tested on realized return and risk as well as on realized return and expected return in the context of India. The findings show that a large variation in Expected Return is explained by the risk (Market Beta) alone and this risk and Expected return are significantly negatively related. However, the risk (Market Beta) and Realized return are insignificantly related. Further, a very low variation in the Realized (Actual) Return is explained by the Expected Return and the Expected Return and the Realised Return are insignificantly positively related. Thus, it is considered that the Markowitz model is not possible to implement in the real world even though the relationship holds good. This study acts as one of the guiding tools for investors in transforming their new age investment philosophy.

Hurley, Rao, and Pardey (2014) argue to replace the internal rate of return with the modified internal rate of return for the evaluation of public research investment. The crux of their argument is that the internal rate of return “assumes intermediate cash flows can be reinvested (or borrowed) at same return as the initial investment, which is generally not correct or reasonable,” (page 1492). This article first demonstrates that reinvestment decisions are embodied in the project specification, and that the internal rate of return makes no inherent reinvestment assumption. The article then clarifies the algebraic properties of the marginal internal rate of return and the reinvestment implications of the internal rate of return and modified internal rate of return within the context of public agricultural research evaluation.

The paper uses various approaches: capital asset pricing, mean-variance, global minimum-variance, Bayes-Stein, Bayesian and multi-prior to develop foreign equity bias measures for Australia’s international equity holdings in 41 countries, over the period 2001 to 2012. Bayesian models allow for various degrees of mis-trust in the ICAPM model. Multi-Prior restricts the expected return for each asset to lie within specified confidence interval around its estimated value. Mean-Variance computes optimal weights by sample estimates of mean and covariance matrix of sample return. Bayes-Stein shrinks each asset’s historical mean return toward the return of the minimum variance portfolio and improves precision associated with estimating the expected return of each asset. The plausible sources of foreign equity bias are trade, GDP per capita, real GDP growth rate, exchange rate volatility, tax credit, stock market development, familiarity and institution variables. The paper finds that economic cost of the observed foreign bias is low. The paper analyses correlation effect on the foreign bias and finds that economic loss decreases with an increase in correlation.

We show that Hau and Rey's (2006) empirical evidence is not sufficient to support their risk-rebalancing theory as an explanation for the negative correlation between the stock market return differential and currency. A simple model combining home-wealth rebalancing and extrapolative expectations on the foreign stock predicts this negative correlation only when the host market is a source of international capital. Panel regressions indicate that the source status of the economy (i.e., whether it is a net receiver or source of international capital) is a main predictor of the stock return differential-currency correlation. Refereed/Peer-reviewed

For investors, the time-weighted return only provides a partial picture considering their performance. The money-weighted return or internal rate of return calculation delivers a more precise measurement of how they performed for the time of being invested. The difference between those two types of return calculations is commonly referred to as the performance gap.Since the money-weighted return and therefore the performance gap is driven by the timing and magnitude of flows into and out of the asset, the behavior of investors plays an important role. Considering different asset classes, that behavior tends to change. Understanding what drives real estate fund flows is of importance for the size of the performance gap of these funds. It is said, that the return chasing behavior of investors is a source of bad timing skill. The convexity of the flow-performance relationship for open-end real estate funds suggests that the best-performing funds tend to be chased less as their risk of becoming illiquid increases. Furthermore, investors are more sensitive to poor performance when fund liquidity is low. Being more sensitive to fund-level underperformance, when there is increased liquidity risk, leads to investors being more likely to sell. Apart from commonly known drivers of the performance gap like volatility, all these vehicle-specific aspects also influence fund flows and therefore the gap between money-weighted and time-weighted returns.For a sample of German open-end real estate funds over the 1990-2016 period, we analyze the determinants of the performance gap, considering the special conditions of that investment vehicle. In contrast to the academic literature, mainly telling about the performance gap of mutual equity funds, the difference in the two return calculation methods for open-end real estate funds will deliver new implications for investors.

Public interest about stocks and bonds increasingly growing, it's because the wider public curiosity about the conditions in the capital market. Indicator in profitability ratio is Return on Equity and Return on Investment. The reason why the ROE and ROI selected as measure is due in ROE and ROI using net income as a benchmark in measuring profitability. ROI and ROE affect the volume of trade, even though it gives weak results. Trading volume was influenced by other factors beyond profitability ratio. Investors do not just look at the profitability factor in this case is represented by the ROE and ROI.

Fundamental and technical analysis is used by analysis to predict the trend ofstock price and trading volume. Studies conducted aimed to determine the effect of fundamental analysis to technical analysis. Combining two forms of analysis can produce a more accurate prediction of the stock price movement of listed cement companies in Indonesia Stock Exchange. Research experts indicate that the fundamental and technical analysis can be used independently with the ability to predict stock price movements. This study combines both analysis in a model that can provide a more robust predictive capability in the Company's share price movements of cement. Fundamental analysis is the economy wide scope, one of the predictions of financial performance. In this study the total asset turnover, return on assets and return on equityto determine which stocks are pretty good. While technical analysis is usedaccumulation distribution line that has a better ability to predict future stock prices because the data contained technical stock price and trading volume to determine when to buy and sell momentum. These results indicate that the total asset turnover, return on assets and return on equity significantly influence the accumulation distribution line. While the individual that the return on equity has no significant effect. The results of this study are expected to improve knowledge for the readers, especially investors in order to obtain optimal benefits.

Existing studies on capital asset pricing model (CAPM) have basically focused on point data which may not concern about the variability and uncertainty in the data. Hence, this paper suggests the approach that gains more efficiency, that is, the interval data in CAPM analysis. The interval data is applied to the copula-based stochastic frontier model to obtain the return efficiency. This approach has proved its efficiency through application in three stock prices: Apple, Facebook and Google.

This study investigates whether non-performing loans effect the bank’s profitability in Turkey. The study applies a panel regression method to the quarterly data set including 1809 observation belongs to 55 Banks in Turkey during the period from 1st quarter of 2005 to 3rd quarter of 2016. It is found that there is a significant, negative relationship between non-performing loans and bank profitability which is measured by return on equity and return on asset. The higher non-performing loans, the lower asset quality, leads to the lower return on equity and return on asset, and the lower non-performing loans, the higher asset quality, leads to the higher return on equity and return on asset.

I share some thoughts regarding realized, required, and expected returns and risk-free rate. I provide simple mathematics to divide return to change in real equity value and change in demand-supply curve of an asset. I also consider risk-free rate in terms of inflation and additional interest rate that compensates for liability risk. Finally, I show mathematically how this risk-free rate affects implications about realized and expected returns of assets with certain payoff. This is a perspective paper.

The banking industry still uses the return on equity (RoE) as a measure of performance and as an instrument of internal capital allocation. Some bankers even claim they have to aim at high RoE targets in order to cover the return on capital (RoC) required by shareholders.

This paper proposes excess stock market return as a way to measure the impact of marketing strategy on firm value. First, it provides an overview of event study method. An event study examines the excess return to a firm's stock price after the release of information that is relevant to the firm's financial success. Second, it shows how excess return captures a marketing strategy's impact on firm value. It presents a model that illustrates how a marketing strategy impacts consumers, future cash flows, firm value, investor's expectations, and excess return. Third, a comparison shows that excess return stacks up well against standard marketing metrics. Excess return yields unbiased estimates, allows direct causal inference, is future oriented, includes all cash flows, accounts for opportunity costs, factors in risk, and takes into account the time value of money.

In the Fama–French three-factor model, the market return is not the return to market beta. By including a separate beta factor, the market portfolio without a coefficient can be described as only “half” a factor. Documenting the returns to a pure beta factor in the US equity market, the authors show that the distinction between the market return and the return to the cross-sectional variation in security betas also applies to portfolio performance measurement. The realized alphas of low-beta (high-beta) portfolios are reduced (increased) when a separate beta factor is included.

Diversification return is the amount by which the geometric mean return (i.e., average compounded return) of a portfolio exceeds the weighted average of the geometric means of the portfolio’s constituent assets. Diversification return has been touted as a source of added return, even if markets are informationally efficient. Portfolio rebalancing has been advocated as a valuable source of diversification return. The authors demonstrate that diversification return is not a source of increased expected value. However, portfolio rebalancing can be an effective mean-reverting strategy. Any enhanced expected value from rebalancing emanates from mean-reversion, rather than from diversification or variance reduction.

This article shows that a given series of single-period economic rates of return can be aggregated into either an economic average internal rate of return or a modified internal rate of return, depending upon the aggregation process (weighted average or geometric mean) and on the use of traditional economic depreciation to determine capital values for each time period. The results are shown analytically and numerically and evaluated under different levels of discount rates. The modified internal rate of return is also a form of the average internal rate of return based on the geometric mean, rather than the arithmetic mean.

In this paper we consider the problem of selecting an absolute return portfolio. This is a portfolio of assets that is designed to deliver a good return irrespective of how the underlying market (typically as represented by a market index) performs. We present a three-stage mixed-integer zero-one program for the problem that explicitly considers transaction costs associated with trading. The first two stages relate to a regression of portfolio return against time, whilst the third stage relates to minimising transaction cost. We extend our approach to the problem of designing portfolios with differing characteristics. In particular we present models for enhanced indexation (relative return) portfolios and for portfolios that are a mix of absolute and relative return. Computational results are given for portfolios derived from universes defined by S&P international equity indices.

This paper introduces new money-weighted metrics for investment performance analysis, based on arithmetic means of holding period rates weighted by the investment’s market values. This approach generates rates of return which measure a fund’s or portfolio’s performance and a fund manager’s performance. It also enables to show that the Internal Rate of Return (IRR) is a weighted mean of holding period rates associated with interim values which differ from market values, so that value additivity is violated. The manager’s Arithmetic Internal Rate of Return (AIRR) is shown to be the true period equivalent of the cumulative Time Weighted Rate of Return (TWRR), whereas the period TWRR (a geometric return) provides a different ranking. The method is easily generalized for coping with varying benchmark rates. We also cope with the practical problem of estimating interim values whenever they are not available.

At odds with a vast body of economic evidence reporting exceptionally high rates of return to investments in agricultural research and development (R&D), growth in public R&D spending for food and agriculture has slowed in numerous, especially rich, countries worldwide. The observed R&D spending behavior is consistent with a determination that the reported rates of return are perceived as implausible by policy makers. We examine this notion by scrutinizing 2,242 investment evaluations reported in 372 separate studies from 1958 to 2011. We find that the internal rate of return (IRR) is the predominant summary measure of investment performance used in the literature despite methodological criticisms dating back more than a half century. The reported IRRs imply rates of return that are implausibly high. We investigate the reasons for these implausibly high estimates by analytically comparing the IRR to the modified internal rate of return (MIRR). The MIRR addresses several methodological concerns with using the IRR, has the intuitive interpretation as the annual compounding interest rate paid by an investment, and is directly related to the benefit-cost ratio. To obtain more credible rate of return estimates, we then develop a novel method for recalibrating previously reported IRR estimates using the MIRR when there is limited information on an investment's stream of benefits and costs. Our recalibrated estimates of the rate of return are more modest (median of 9.8% versus 39% per year); however, they are still substantial enough to question the current scaling back of public agricultural R&D spending in many countries.

This note demonstrates that when there is a discount on uniform cash flow, the rate of return would not increase to the extent of the discount. The extent to which the rate of return would increase depends on the investment horizon.

Teoreticky nejvhodnějsim kriteriem pro hodnoceni investicnich projektů je cista soucasna hodnota. Vnitřni výnosove procento neumi řesit situace s nekonvencnim peněžnim tokem a se vzajemně se vylucujicimi projekty. Modifikovane vnitřni výnosove procento ma univerzalnějsi charakter a nemusi se při něm použivat iterace. Předpoklada reinvestici ve výsi požadovane miry výnosu. Je tak zavisle na požadovane výnosnosti jako cista soucasna hodnota.

The expected return to equity--typically measured as a historical average--is a key variable in the decision making of investors. A recent literature based on analysts' forecasts and practitioner surveys finds estimates of expected returns that are sometimes much lower than historical averages. This study presents a novel method that estimates the expected return to equity using only observable data. The method builds on a present value relationship that links dividends, earnings, and investment to market values via expected returns. Given a model that captures this relationship, one can infer the expected return. Using this method, the estimated expected real return to equity ranges from 4 to 5.5 percent. Furthermore, the analysis indicates that expected returns have declined by about 2 percentage points over the past forty years. These results indicate that future returns to equity may be lower than past realized returns.

One of the key factors in investment analysis is the risk free rate of return. The interest rate on a government bond is considered a risk free rate of return, but the bond should be free from the risk of non- payment and reinvestment risk. However, the practical determination of the risk free rate can often be followed by a number of difficulties. Therefore, in this paper we discuss the difficulties faced by an analyst in determining the risk free rate of return on investment and portfolio analysis such as: - Lack of long-term government bonds on the capital market - Exposure to the risk of government default - Change the risk free rate of return over time. I. INTRODUCTION The numerous model od assessment of risk and rate of return start with a financial asset that is defined as the risk-free asset and therefore the expected return of that asset is used as a risk-free rate of return. Then, the expected returns of risky investments are determined in relation to the risk free rate, by adding expected risk premium. To understand what makes an asset as a risk free asset, it is necessary to consider the way of the risk is measured in investment decisions. Investors who buy assets (financial or real) expect to achieve a yield in the period in which they plan to hold the asset. Realized yields that investors will achieve in a given period may vary and differ from expected returns as a result of the risk that carries the appropriate investment. To some an investment to be characterized as risk-free it needs the realized yields always to be equal to the expected yield. Such investment is risk free because the realized returns do not vary around the expected yield, i.e. the standard deviation of realized yields in terms of the expected return is equal to zero. (1)

The cost of capital is a key variable in the financial analysis because it provides adequate assessment of the value of capital, and the value of individual securities. Hence, the important question is how to determine the cost of capital. Initial basis for determining the cost of capital is a determining risk free rate of return that is subject of research in the paper. Risk free rate of return exists when the expected rate of return is known with certainty, that means that realized proceeds of an asset is equal to the expected yields of the same asset. Rate of return on that asset is characterized as risk-free rate of return. To be able to adequately determine the risk free rate of return is necessary to determine the factors affecting its value, such as the currency in which it is expressed risk free rate of return and inflation. Therefore, the paper explored the reasons that may lead to inadequate assessment of the risk free rate of return due to inadequate treatment of the impact of currency and inflation in the analysis.

The objective of this paper is to determine which point of the business cycle offers investors the best reward to risk ratio in the stock market. Expected reward is defined as expected return in excess of the risk free rate, whereas risk is defined as the standard deviation of return. Thus, the expected reward to risk ratio is measured by expected return in excess of risk free rate relative to the standard deviation of return. Expected return in excess of the risk free rate and standard deviation of return are generated on a continuum of time periods and the GDP growth rate. The point where the expected reward to risk ratio peaks, would signify the best time for investment. Being able to identify this point could help investors in deciding the best time to invest as well as help firms in choosing a favorable time for raising equity capital. While most people think that the best time to invest is near the bottom, it is not clear whether the best time for investing is before, at, or after the economic trough. The interesting finding in our model is that the best time is after the point of the economic trough.

Higher returns are not compensation for a higher uncertainty but rather for a higher probability of losing a part or the whole investment. With uncertainty I refer to a situation where an investor does not know what will happen in the future and, at the same time, he cannot expect a specific outcome to happen more probably than another. If risk is nothing but uncertainty, higher risk is not rewarded with higher returns. Otherwise there would be an anomaly in the required return curve. I, however, leave space for a debate how to define risk.I also conclude that expected return is actually a risk-free rate of return. Because expected return covers all the possible returns of a probability space, by gaining all these returns an investor gets for certain the expected return. This is a case when probabilities are dependent on each other. If probabilities are independent, expected return is gained only when repeats approach to infinite. This is a perspective paper.

NCREIF/PREA Reporting Standards for computing gross and net time weighted rates of returns (TWRs) are flawed in two respects, thereby producing distorted results for both real estate managers, as well as for the fund-level industry indices against which the managers’ performance may be compared. The far less frequent, yet potentially far more serious flaw impacts only investment entities, as well as the industry indices, which have accrued incentive fees or carried interest.The introduction of the flaws seems to have emanated from a loss of industry perspective, where real estate TWRs came to be regarded as indicators of investor performance. However, as detailed herein, TWRs were created for the express purpose of comparing the performance of managers. Before fee TWRs help to assess a manager’s ‘pure skill’. After fee TWRs help to assess the manager’s ‘cost-adjusted skill’, meaning the manager’s pure skill adjusted for the cost it took to avail oneself of that skill. Both types of TWR can be important, depending upon what insight an investor wants to glean. The flaws identified herein are subtle and, to be fair, remain somewhat controversial to those who have not fully embraced the fact that the primary role of TWRs is one of measuring comparative manager skill. For those reasons, drafts of this article have been circulated among TWR experts and aficionados, both inside and outside of real estate, seeking critical feedback, with the vast majority of experts consulted agreeing that the flaws are both real and potentially serious. The flaws, in principle, are extremely easy to fix. However, in practice, fixes can be time consuming and costly and, as a result, resistance to change from various parties should be anticipated. The alternative to never changing is to knowingly continue to perpetuate these flaws both in existing managed investments and index products, something that unfortunately will, for understandable reasons of consistency, surely cause the flaws to be introduced into all subsequently created investments and fund level indices as well. No doubt, some investors don’t take producing accurate TWRs that seriously, believing that they are little more than mere numbers on a piece of paper, numbers that don‘t impact or reflect ‘true economics’. However, even if, indeed, they do not use TWRs to make decisions on hiring or firing managers, it should be noted that TWRs may impact their incentive fee payments. Specifically, as more real estate indices have been created, more real estate incentive fee structures seem to be incorporating a TWR-based performance test of some form or another. If the TWR test fails, meaning that the manager has underperformed relative to some pertinent index, then some contracts then dictate that the incentive fee payment is to be reduced, if not eliminated entirely. As such, this TWR problem is not limited to mere “performance reporting” distortion, but can also result in sizable errors in the amount of incentive fees actually paid to managers by their investors.

The equity premium, namely the expected return on the aggregate stock market less the government bill rate, is of central importance to the portfolio allocation of individuals, to the investment decisions of firms, and to model calibration and testing. This quantity is usually estimated from the sample average excess return. We propose an alternative estimator, based on maximum likelihood, that takes into account information contained in dividends and prices. Applied to the postwar sample, our method leads to an economically significant reduction from 6.4% to 5.1%. Simulation results show that our method produces more reliable estimates under a wide range of specifications.

The long-term performance of any portfolio can be decomposed as the sum of the weighted average long-term return of its assets plus the volatility return of the portfolio. Hence, maximizing the volatility return of portfolios of assets with similar characteristics, such as factor portfolios, yields an important increase in performance and risk-adjusted return relative to market-cap weighted factor portfolios. Partitioning a universe of assets into homogeneous groups and applying a block-wise maximum volatility return strategy (MVR) also yields a more efficient index than standard market-cap and equal-weighted indices.

Common knowledge is that the IRR is some sort of an average rate of return but, in fact, it is a constant rate of return. The distinction may sound immaterial, but the IRR’s fixation on a constant rate of return is a major shortcoming. The fact that, in certain circumstances, it can result in multiple solutions or no solutions is well-known, but is deemed unlikely enough as to be inconsequential to most. However, what is not well known at all is that, in forcing the rate of return to be constant for each cash flow period, in essence, IRR is concocting interim valuations for the investment that contradict the true valuations of the investment at the end of each and every cash flow period that have likely either been estimated (e.g., real estate appraisals) or may even be precisely known (e.g., stock prices). Less obvious, but no less valid, is that these concocted values render the constant IRR result artificial which, in this case, is another way of saying that the IRR is simply the wrong rate of return answer.Where IRR’s more subtle failings are more evident is when one attempts any sort of attribution, even if only informally, meaning that one wants to reconcile how the IRR result has changed when either another asset has been added to the portfolio or when another time period has been added to the investment horizon. In short, the requirement of IRR to always produce a constant answer for each cash flow period forces it to retroactively ‘change its mind’ about the values of the assets in all prior time periods. This chameleon-like property of IRR is a serious flaw that, in effect, makes IRR unworkable for any sort of incremental analysis.Fortunately, in 2011, a more flexible money weighted rate of return (“MWRR”) metric was created by a scholar named Carlo Alberto Magni [see Magni, 2011], a metric which avoids all of the IRR’s flaws by computing a simple 'weighted-average' of the periodic rates of return which, as a result of the aforementioned information on periodic valuations of the investment, are inherently variable. Not only is this metric simpler to calculate in that it does not require a computer iterating in search of a solution; but, in fact, it can be calculated on the back of an envelope. Yet, it has none of the problems that the IRR has, the most prominent one being that, unlike the IRR, it always produces a single, unique rate of return solution.

The rate of return on corporate investments need not be the same as the equity discount rate. Indeed, rational, agency, and behavioral theories all suggest it will be lower when averaged across firms and over time. We introduce a flow-based approach to capture the return on corporate investments, with long time series and aggregate data to reduce the business-cycle and idiosyncratic noise. Using a Kalman filter with ARMA(1,1) transitory noise and a minimum of 58 years, we find the return on corporate investments to be about 55% -- 70% of the equity discount rate, a highly significant difference. In the cross-section, firms with a lower cost of capital (proxied by size or book-to-market ratio) achieve a lower return on corporate investments. Industries prone to the conservatism bias of financial accounting achieve a high apparent return on corporate investments, consistent with their quantity of investment being understated (e.g., with research and advertising being expensed rather than capitalized).

Do “cost of capital” and “required rate of return” equal opportunity cost? The Hypothetical Investor’s opportunity cost is not the subject private company’s (owner’s/management’s) borrowing cost (or internal rate of return), nor the typical real-world buyer’s (or even the Hypothetical Buyer’s) required rate of return. It is the reasonably-expected market rate of return on the best alternative asset that he forgoes when (hypothetically) investing in the subject company under appraisal. In the competitive Hypothetical Market, the set of alternative assets is not limited to similar private companies with similar risk; it includes marketable publicly-traded assets. The rational Hypothetical Investor considers his work-leisure options when investing and, upon a thorough investigation of the Fair Market Value Standard’s logical implications, we are led to the internally-consistent conclusion that it is rational for him to select a long-term U.S. Treasury bond as his best alternative asset, which would earn him investment return without having to work. For both partial and control private company interests, his opportunity cost is not higher than the long-term risk-free rate.

By using the binomial option pricing model, this paper proves that with no arbitrage and no transaction costs, (i) under riskless debt, increasing the debt-equity ratio increases the variance of the rate of return on equity; and (ii) under risky debt, increasing the debt-equity ratio increases the variance of the rate of return on debt but does not affect the probability density function of the rate of return on equity. This finding refutes the Modigliani-Miller second proposition that the expected rate of return on the equity of the levered firm increases in proportion to the debt-equity ratio.