Frontiers of Entrepreneurship Research 1995

Frontiers of Entrepreneurship Research
1995 Edition

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    Robert H. Keeley, University of Colorado-Colorado Springs
    Lassaad A. Turki, Purdue University


    This paper presents a method to estimate private company values at any time between market transactions, and constructs a stock price index for venture capital transactions. Using a pilot sample of 1041 transactions from 274 companies an index is presented covering 1980-1993. Using this method a full scale venture capital index could be developed by one of the organizations with a large database of venture capital transactions. Such an index would aid institutional investors, venture capitalists and entrepreneurs in assessing price levels and trends in the venture capital market. However, the errors inherent in such an index are sizable. For example, our pilot study with about 60 stocks has a standard error of about 10 percent of the index value. The error could be reduced to about 2.2 percent if an index of 1000 companies were compiled.


    Stock market indices, such as the Standard and Poor's 500, are the most widely reported economic indicators. They have many functions, such as:
    * summarizing countless daily trades and expressing the net effect of those trades in a single measure of the market's price level and its direction,
    * helping investors and traders to make buy/sell decisions,
    * providing a benchmark against which the performance of a portfolio can be assessed, and
    * serving as a focus of hundreds of research studies (e.g. Fisher (1966), Ohlson & Rosenberg (1982), Schwert (1990)).

    Indices now exist for most public equity markets in the world, but not for private markets such as venture capital. Although an index of venture capital prices would be useful for all of the reasons noted above, to-date none has been created. This paper continues our research on how to establish a price index for venture capital transactions.

    In an earlier paper (Keeley & Turki, 1992) we presented a method for constructing venture capital index, one which could deal with infrequently traded securities. To verify that the proposed method provided reasonable estimates it was tested on a sample of 60 public stocks--stocks for which a true index could be calculated. Then a venture capital index was estimated for the years 1980-1987 using a pilot sample of companies that had received venture capital financing. This study extends that work by estimating a venture capital index for the interval January 1980 through December 1993, and by providing estimates of the likely error in the index.

    In addition to the technical issues discussed in this paper, the private nature of venture capital has been a barrier to the creation of an index. However, several groups, such as "gatekeepers" and trade publications now have sufficient data to create such an index for the United States and for Europe. We believe this study shows that a serviceable, although not perfect, index is technically feasible and hope that one of those organizations will begin providing it on a regular basis.


    Price-weighted, value-weighted and equally-weighted indices

    Most stock market indices are based on a general formula1 :

    = the price level of the index at time 0.
    = a base value of the index at t = 0.

    = an adjustment factor which allows for changes in the membership of the index and capitalization changes in companies comprising the index.

    Wit = the relative weight given to stock i for purposes of computing the index at time t.

    = the value of stock i at time t (for some indices ``value'' may include dividends or other cash equivalent distributions during the interval t-1 to t.

    = the price of stock i at t=0 Various indices can be created depending on the weighting scheme. Three important types of indices are:

    1) Price weighted: the Dow--Jones Industrial Index is a leading example. Weights are set as , which corresponds to buying one share of every company in the index.

    2) Value weighted: the S&P 500 Index, the New York Stock Exchange Index, the CRSP Value Weighted Index (Center for Research in Security Prices, University of Chicago) and the NASDAQ Composite Index are examples. Weights are set as where is the number of shares outstanding at t =0 for company i. A value weighted index corresponds to buying an equal percentage, say one percent, of every company in the index.

    3) Equally weighted: the CRSP Equally Weighted Index is a leading example. Weights are where is the number of stocks in the index. An equally weighted index corresponds to investing an equal amount (say $1) in each company.

    Equally weighted and price weighted indices do not require knowledge of a company's total value; a value weighted index does.

    As discussed in Keeley & Turki (1992), every index has an implicit investment strategy. Price-weighted and value-weighted indices assume a buy-and-hold strategy. Equally-weighted indices assume sale at the end of every period and reallocation of the proceeds in equal amount for every stock in the index.

    The form chosen for a venture capital index should reflect the investment policy of a typical venture capitalist. This suggests some type of an equally weighted index, because a venture capital fund tends to diversify through relatively equal investments in a number of companies. It also suggests a buy and hold formulation. For example, if a fund invested $1 simultaneously in each of 30 companies, the index would be the total value of the investments at any given time divided by 30.

    Assumptions about stock prices contained in all indices

    All indices assume that the observed stock prices are "market prices" and that such market prices are available for all members of the index. In active public stock markets one can easily accept that the price at any moment represents the decisions of many buyers and sellers, and is therefore a true market price. Private stock transactions are different, because there are a limited group of buyers and only one seller--the company. Thus the price is the result of a negotiation, not a bidding process. Additionally, there is a high transaction cost (incorporating the costs of search, of "due diligence," of price negotiations, and of writing an investment agreement). Thus the price may deviate from a true market price.

    Keeley & Turki (1992) examined 273 investment transactions in detail. They conclude that a transaction, in which a new investor joins the investor group, should be viewed as giving an unbiased estimate of the true market price. Transactions in which only the existing investors participate should be viewed as downwardly biased estimates of market value. They also conclude that such biased will not impact a stock price index because the index utilizes pairs of transactions to find price ratios and then find rates of return. A downwardly biased price may occur in the first or the second of the pair (or both--in which case the ratio will not be biased). The effects lead to upwardly and downwardly biased price ratios in roughly equal numbers. The net effect is that the entire set of price ratios is unbiased, and the price ratios can be treated as ratios of true market prices.

    The second assumption, that a very recent market price can be observed, also creates problems in non-public companies. In young, private companies transactions are very infrequent (about once a year in our sample) and price volatility is high. Thus the true (but unobservable) price on a given date is unlikely to be close to that price at the last trade. Fortunately an unbiased estimate of the true price can be obtained for any point between two transactions. One simply uses the rate of return implied by the two transactions and applies it for the appropriate amount of time.

    Example: Estimate the price of a stock on February 28, 1992, given that the stock sold for $10 on January 1, 1992 and for $13 on May 31, 1992.

    Step 1: Find the rate of return (assuming continuous compounding) between January 1 and May 31--an interval of 0.414 years.

    r=[1/ 0.414] ln(13/10) = 0.6337 ( or 63.37%/year)

    Step 2: Apply the rate of return from step 1 to the initial price of $10 for the period January 1 to February 28--an interval of 0.162 years.

    P Feb. 28 = P Jan. 1 e 0.6337*0.162 = $11.13

    Keeley & Turki (1992) describe the conditions necessary for such an estimate to be unbiased (essentially that percentage changes in a stock's value over a very short interval are normally distributed, and that changes from one such interval to the next are not correlated), and note that empirical studies show such conditions to be reasonably satisfied for public stock markets and for venture capital markets.

    An index, which relies on estimated prices, will always have a lag. One must wait for a sufficient number of transactions before estimating the index. In the United States, venture capital transactions occur at a rate of about 100 companies/month. Thus a reasonably broad index could be estimated with a one month lag, similar lags typify many important economic statistics.

    The estimated price is, of course, subject to error. The true price on February 28 will be described by a lognormal distribution.2The standard deviation of the price can be calculated if one knows the standard deviation of the normal distribution that describes the rate of return over short intervals, and a standard deviation of the index can be estimated from the standard deviations of its constituent companies.

    Allowing for changes in the composition of an index

    The specific companies contained in an stock price index will change over time as new companies qualify to join the index and existing members are acquired, deleted or go out of business. Thus every index must have a process for adding and subtracting companies. The adjustment factor ( in formula 1) provides for changes in the index. As explained in Keeley & Turki (1992) the adjustment process begins by finding the value of the index immediately prior to adding or dropping a company. Then the index is recalculated allowing for the addition or deletion. The adjustment factor is set to a value that maintains equality between the two indices. That is, the addition or deletion of a company will not cause a change in the index.

    A venture capital index will have greater turnover of its constituent companies than an index of public stocks, because its companies are of higher risk and because we drop them from the index when they become public. However, the higher turnover rate is not a conceptual difference, it is simply one of degree. As long as new companies entering the index are good substitutes, on the average, for those leaving, changes in composition will have a minor effect.

    The changing composition of the index suggests that consideration should be given to the "investment strategy" implied by the index. As noted in the examples earlier, a value weighted index implies buying a specified percentage of each company and holding it indefinitely. When a company joins the index, an equal percentage of each company is sold in order to buy that percentage of the new company such that percentage holdings are equal for all companies in the index.

    A venture capital index might logically follow an implied "investment strategy" of putting a specified amount (say $1) in every new investment and then holding the investment until it leaves the index. Such a strategy reflects the illiquidity of such stocks and the tendency of venture capitalists to spread their funds fairly evenly over a number of investments. The adjustment for companies that leave the index assumes that the funds thus liberated will buy a specified (and equal) number of shares in each of the remaining companies, and the adjustment for companies joining the index assumes that equal numbers of shares of every company in the index will be sold in order to obtain the funds to invest $1 in the new company.

    Although a index must have an implicit "investment strategy," there are a number of alternatives worth exploring (although such steps are beyond the scope of this paper). For example, the funds liberated from a liquidated investment might be reinvested in a broad portfolio of public stocks (such as the S&P 500) and the fund for adding new venture capital investments might be obtained from that public portfolio. Such a strategy would "pull" the returns of a venture capital index toward the broad index-it would no longer be purely a venture capital index. But it might represent a more realistic investment strategy.

    Another "investment strategy" for the index would incorporate additional investment in every financing round of a new venture. In this study we assume that a venture capitalist invests only once, and we calculate separate indices for those who invest on the company's start-up financing, and those who invest at a later stage. An index that reflects the realistic strategy of reinvesting at every round, will be a combination of the two indices presented in this study.

    Studies of public stock indices (e.g. Fisher, 1966) find small effects from changing between price-weighting, value-weighting and equal-weighting. The implication is that the price ratios of the constituent companies turn out to be uncorrelated with the weighting scheme (which is determined by the implicit "investment strategy"). However, the effects may be greater in an index of new companies and the exploration of alternative "investment strategies" is an important future extension of this work.


    Having determined the structure of the venture capital price index, we now turn to the three steps needed to calculate it:

    Transaction data must be assembled for a number of companies. Transactions will occur infrequently, e.g. once per year for a given company. Seldom will two companies have a transaction on the same date. This pilot study uses a sample of 274 companies which engaged in 1017 transactions between 1980 and 1994. None of the 274 are represented over the entire time interval. They either started business after 1980 or dropped out of the sample before December 1994 (because they became public, were acquired, went out of business, or did not have a sale of stock in 1994).

    Prices must be estimated for dates between the actual transactions. This provides a set of price estimates for specific times, e.g. the end of each month.

    An index is calculated for each time of interest.

    We will defer discussion of the pilot sample until the next section, and will describe the construction of an index in the remainder of this section.

    A test of the method

    Because the Private Venture Index must be constructed from estimated share prices, one can never know its true value. However, we can test the properties of an index based on estimated prices by constructing one from public stocks (whose market values are continuously observable) and comparing the index using estimated prices with a ``true'' index based on actual market prices. Keeley & Turki (1992) conducted such a test using 60 NASDAQ stocks. To construct the index similar to a venture capital index, they sampled each of the 60 stocks about once a year, and used the price estimation procedure discussed above to fill in all interim prices. A more accurate index was also constructed from the actual prices of each of the 60 stocks at each month end.

    The results showed that the index of estimated prices was much smoother than the true index, as would be expected, but that it tracked the overall trend of the true index very well. The overall annual rate of return from the estimated index was 19.6 percent compared to a rate of return of 19.9 percent for the true index. On the whole, the algorithm for calculated an index based on estimated prices worked well.


    Two venture capital funds, which requested anonymity, allowed the authors to collect data on their transactions during the period from 1975 through 1994. Three other funds provided information about transactions for shorter periods of time.

    All five venture capital firms are similar in many respects, which made us optimistic that the sets of transactions would not differ greatly. The five are of similar size, age and standing--being among the leading firms in the industry. They prefer to invest in technology based, start-up companies. The internal records of the funds were different so our collection procedure varied. Fund A did not keep its transactions on a database, but provided access to the documents related to the transactions (e.g. investment agreements, closing statements). Such documents provided information about stock splits, liquidation preferences, anti-dilution provisions, changes in management options, sweeteners and other elements which could influence the company's value, as well as identifying the other venture capital funds that invested in the financing. The fund's annual audited statements and investments logs served to verify that we had recorded all of its transactions.

    From fund A we collected 442 transactions in 107 companies. Allowing for the fact that a typical transaction had several investors, 231 different venture capital funds were identified as participants in one or more of these financings. In Keeley & Turki (1992) we investigated how often financing would include a new investor. In 14 percent of the transactions the records were insufficient to determine whether any new investor were involved. In 21 percent of the financings, no new outside investor was involved, and 65 percent of the transactions had at least one new investor. This information was used in Keeley & Turki (1992) to determine whether the presence of a new investor influenced the price, and they found that it did. However, the ratios of prices in successive financings of a company did not appear to vary on the average between those that had new investors in both transactions and all other cases (a new investor in the first only, a new investor in the second only, or missing information regarding new investors). Thus we concluded that other databases of prices could be used, whether or not they identify the investor group, and that all prices could be used to construct an index.

    Fund B provided a summary of its transactions from a database. It had 386 transactions in 119 companies. The information included the share price (adjusted for stock splits), the size of the transaction, the overall value of the company, and a code designating the company's maturity. The other three funds provided data on 213 transactions in an additional 48 companies for a total of 1041 transactions in 274 companies--providing 767 price ratios from which an index could be prepared.

    The earliest transaction was in 1975, but the number of companies with data did not reach 20 until January 1980. Thus the index begins in 1980. In the fifteen years from 1980 though 1994 the number of transactions average 67 per year. The number of companies in the index reaches a maximum of 91 in mid-1984, and never drops below 45 until late 1993. Thus the data in this pilot study are sufficient to calculate a fairly broad-based index--although one formed from the portfolios of venture capitalists who emphasize technology-based start-ups.


    Figure 1 shows two indices created from the pilot sample.3 The index with the higher value, Index1, includes all transactions in the sample. Because it reflects an equally weighted, buy and hold strategy, it is equivalent to investing only in the first outside financing by a company--which is usually at start-up. Further financing rounds by a company set prices for the stock purchased at the initial investment but no further purchase is implied. A further purchase would change the weights.

    (figure 1)

    Index2 excludes all initial investments by venture capitalists. It reflects the return to an investor who invests in the second external financing of a company. It is substantially lower than Index1, which implies that initial investments in this sample (largely start-ups) earn higher returns than follow-on investments on the average.

    The performance of the two venture capital indices is dramatic, when compared to the public indices for the same period. Index1 increases by a multiple of 374 over a 14 year period, Index2 by 69.0. During the same interval, the Standard & Poor's 500 Index, representing major public companies in the United States, rose by a multiple of 4.4.

    Much of the performance comes from investing in the initial external financings as indicated by the difference between Index1 and Index2. The implied investment strategy of the Index clearly influences its performance. Index1 is a ``buy and hold'' strategy of investing $1 in the first external financing of each company with no further investment in subsequent financings. Index2 is the same except the investment occurs only in the second financing.

    Although equal investments across companies may roughly characterize venture capitalists, investing in only one financing round does not. Thus we could (but do not) calculate an index based on a strategy of investing $1 (approximately) in every financing round. It would lie between Index1 and Index2.

    A second type of index would focus on a single financing stage. A security would drop out of the index when the financing of the next stage occurred. Although this does not follow a realistic strategy, it would provide useful guidance to investors.

    The growth by a factor of 374 in Index1 does not mean that a typical investment in the initial financings of our sample companies grew by a factor of 374 over the 1980--1993 period. It means that one could realize a multiple of 374 by following a ``buy and hold''4 policy in which one invests remaining securities in proportion to their values. The actual portfolio changes substantially over time. Our sample of 274 companies produces a set of about 60 estimated prices at most times, which suggests that about 4.5 companies are needed to produce 14 years of prices. Thus the portfolio turns over in about 3.1 years, and during that time a typical investment grows in approximately $1 in any new opportunity and reinvests any realized returns in the value by a multiple of 3.72. That typical multiple of 3.72 in one portfolio cycle grows to 374 when carried through 4.5 cycles.

    Public indices have much lower turnover in membership. Thus their changes in value should be viewed them as reflecting the change in an average stock within a slowly changing population. The venture indices represent the growth in value of very young companies, and the membership of the indices are constantly changing in order to eliminate older companies and replace them with start-ups.

    Although the private firms in our index exceed the public indices by a wide margin, Table 1 shows that the largest advantage occurred between 1980 and 1983 when Index1 exceeded 100 percent per year and Index2 was close to 65 percent per year. In the subsequent 10 years, covering January 1984-December 1993, Index1 had an average annual rate of return of 35.0 percent, Index2 had 26.7 percent, and the S&P 500 had 11.2 percent. The advantage of Index1 over Index2 also narrowed considerably after 1983.

    (Table 1)


    (1) VC Index1 is the Venture Capital Index with all the financing rounds included.
    (2) VC Index2 is the Venture Capital Index with the first financing excluded.
    (3) S&P 500 is the Standard & Poor's 500 Stock Index, a value weighted index. Rates of return are calculated from annual average values of the index. Others are Jan.1 to Dec. 31.
    (4) Venture Capital Indices are equally weighted with a "buy and hold"approach. Month end prices are estimated.

    The venture capital indices rely on estimates of prices for the months falling between two observed transactions. Although the estimates are unbiased, they may have a considerable error, and the estimation errors will in turn cause errors in the index. Estimation of an error in the index involves several steps:

    1. Use the variation in rates of return to estimate the drift rate and standard deviation of the stochastic process producing changes in stock prices. Although the values are slightly different for the two indices, the standard deviations for the rates of return are about 105 percent per year.

    2. Convert the standard deviation of the normally distributed rate of return to the standard deviation of the lognormally distributed price. For any individual stock the standard deviation varies depending on the length of time from the last transaction and the length of time to the next. At the time of a transaction, the standard deviation is zero. That is, the exact price is known. The error of estimation reaches a maximum half way between transactions, when it is about 59 percent per year. We assumed that the individual stocks are uniformly positioned along the interval between transactions, which reduces the standard deviation of an average stock to about 23 percent per year (from 105 percent per year).

    3. The index is a weighted sum of the estimated prices of the individual stocks. That is, it is a portfolio. Thus the formula for calculating the standard deviation of a portfolio can be used to find the standard deviation of the index. Although the portfolio follows an investment strategy of investing $1 in every stock, it is not an equally weighted portfolio, because subsequent growth gives the high performing stocks a greater weight than low performers. As is well known, the standard deviation of a portfolio depends heavily on the correlation of price movements of the constituent stocks. That correlation is very difficult to estimate for our data set, but it appears to be near zero. If we assume that it is zero, the standard deviation of each index is about 10 percent of the value of the index at any moment.

    With a larger number of stocks on which to base the index, the standard error could be reduced. For example, a 400 stock index would have a standard deviation of 3.5 percent of the value of the index, and a 1000 stock index would have a 2.2 percent standard deviation. An index of 400 venture capital-backed companies is probably possible with existing databases, but 1000 may not be. The implication is that annual rates of return will be subject to considerable error, because they are the ratio of two values which each have a considerable error. Over longer periods, the rates of return become more accurate, since the errors in the index do not grow with time.


    How useful is the venture capital index? Clearly several problems exist, beyond the fact that this is simply a pilot study and lacks a sufficient number of transactions to be representative. In the first place, an investor could not realistically achieve a performance matching the index. The growth of 374X in the index over a 14 year period exceeds the growth in the venture capital pool by a factor of 10. That is, under the investment strategy implied by the index, in which an investor reinvests any distributions into the index, an investment of about $100 million in 1980 would have owned the entire venture capital pool by 1993. In reality investors as a group have not reinvested their entire proceeds from venture capital. An alternative index might "distribute" the value of any stock leaving the index to a public portfolio (such as the S&P 500), and then transfer $1 from that public portfolio to the venture index every time a new venture joins the index. Such an implied strategy would be the overall yield to an investor on the pool of funds that the investor had earmarked for venture capital. It would be a blend of the return on public stocks and on venture capital.

    The large standard error (about 10 percent of the index's value) of the index makes an annual rate of return impossible to calculate with any degree of accuracy. The problem is stems from the high risk of the individual ventures in the index, and indicates that most conventional approaches to measuring the performance of a portfolio manager (i.e. how much risk did the manager assume and did the manager beat the market after due allowance for that risk.) will not work well for venture capital portfolios. Any venture capital portfolio will have many investments that have not traded for some time. Given the high variability of their prices, any estimate of the current value will contain a large error. Index1, with 60 or more stocks, has a standard error of about 10 percent. A typical venture capital fund with, say, 25 investments will have a higher standard error--probably 20 percent (the calculation is different from that of an index).

    Bygrave, (1989) and Brophy & Guthner (1988) have examined the performance of venture capital portfolios during the 1980's, so we might briefly compare results and methods. Our annual returns are generally above theirs although patterns are similar. 5} Brophy & Guthner (1988) study the twelve venture capital funds which were publicly traded during 1981-1985. Their average annual return (using market values of the funds) is 18.5 percent. Our sample shows annual average returns of 89.8 percent and 57.4 percent for Index1 and Index2 respectively. We believe that the venture capital funds that provided data for this pilot study were among the industry's top performers. The inherent error in our index indicates the difficulty that investors in a public fund would encounter in trying to evaluate it. If the risks of individual ventures are comparable to those in our indices, the standard error of a public portfolio will be 20 percent or more--assuming values are set at the most recent prices of the individual ventures.

    Bygrave. et. al. (1987) appear to find returns which are close to zero during 1984 and 1985, following a period of high returns. The pattern is the same as shown by our indices, although returns from our pilot sample are higher than theirs. Bygrave, et. al. measure compound rates of return from the inception of individual funds. This is a geometric average of performance over individual years. Thus it will change more slowly than our index. It also reflects each fund's continuing investment activity.

    An index such as ours has advantages over a geometric average. First, it may be calculated with a shorter lag, because it does not require overall data about fund performance, which is usually available only once a year. Second, its construction is very similar to that of other stock market indices, allowing relatively easy comparison. Third, it responds more quickly to market changes. It is an efficient use of the information contained in individual prices. Thus it can reduce the `recognition lag'' of venture capitalists or investors as to changes of price levels in the market. Fourth, it has information about the likely degree of error. Understanding that error, which is imbedded but not measured in the geometric averages, assists one in assessing whether a given performance is the result of ability or simply a chance outcome. In the case of venture capital investing, the estimation errors appear to be so large that systematic differences between the performance of funds are difficult to assess with any confidence.

    This pilot study suggests that a price index for venture capital investments can be constructed, but that it may not provide the benchmark for venture capital performance that the S&P 500 provides for managers of portfolio of public stocks. Infrequent trading in new ventures, combined with their high inherent risk, leads to an index with a sizeable standard error. If the index were very inclusive with perhaps 1000 ventures, its accuracy would become adequate. But any specific comparison between the benchmark and a particular venture capital fund would still be of limited value, because the effects of chance are so large in venture capital investing--even the funds in this pilot study, who appear to be superior performers, made investments whose standard errors exceeded 100 percent per year. When the inherent risks of individual investments are so high, the random variation of a portfolio will also be considerable, and estimates of the portfolio's value, based on the most recent transactions in each investment, will be subject to wide errors.


    Brophy, D.J. and Guthner, M.W. 1988 "Publicly Traded Venture Capital Funds: Implications for Institutional 'Funds of Funds' Investors" Journal of Business Venturing 3(3): 187-206.

    Bygrave, W., Fast, N., Khoylian, R., Vincent, L. and Yue, W. 1989 "Early Rates of Return of 131 Venture Capital Funds Started 1978--1984" Journal of Business Venturing 4(2): 93-105.

    Fisher, L. 1966 "Some New Stock Market Indexes" Journal of Business 29(1): 191-225.

    Keeley, R. and Turki, L.A. 1992 "An Equity Price Index for Venture Capital Investments" in S. Birley and I. MacMillan, eds. International Perspectives on Entrepreneurship, Advanced Series in Management, vol. 18, North Holland, New York:173-198.

    Ohlson, J. and Rosenberg, B. 1982 "Systematic Risk of the CRSP Equal-Weighted Common Stock Index: A History Estimated by Stochastic-Parameter Regression" Journal of Business 55(1): 121-145.

    Ruhnka, J.C. and Young, J.E. 1987 "A Venture Capital Model of the Development Process for New Ventures" Journal of Business Venturing 2(2): 167-184.

    Schwert, G.W. 1990 "Indexes of U.S. Stock Prices from 1802 to 1987" Journal of Business 63(3): 399-426.

    Venture Capital Journal annual "Survey of Venture Capital Fund Disbursements" e.g. May, 1987.

    1 The indices discussed in the paper use arithemetic averages. "Geometric" indices, such as the Value Line Index, exist but are not common. 2 In our example the return is normally distributed with a mean of 63.37 percent. The calculation to find the price involves using the (normally distributed) rate of return as an exponent. Thus the price is not normally distributed, it is lognormally distributed. The index will be a weighted sum of many prices, each lognormally distributed. The central limit theorem implies that estimation errors for the index will be approximately normally distributed.

    3 Index1 has only 20 companies on January 1, 1980 but grows to 45 by mid 1981 and stays above that level until mid 1993. By January 31, 1993 it drops to 39 companies. That is, only 20 companies in this database had transactions before January 1, 1980 and only 39 had transactions after January 1, 1993. Index 2 begins with 13 companies on January 1, 1981 and reaches 40 by mid 1982. It stays above 20 companies until early 1994.

    4 "Hold" in this case means holding until an initial public offering, a sale of the company, or a write-off. If none of these events have happened by 1994, a security is removed from the index at the date of its latest private transaction, because extrapolating beyond the last price would be inappropriate. The movement of prices described in footnote 2 means that any extrapolation would reflect only the average drift in prices, not the deviation from the average. Since the appropriate estimate of the drift at any specific time is the movement of the index itself, extrapolation would provide no additional information.

    5 This is a pilot study of a method and uses data from only five funds--with only two of those funds providing complete sets of transactions. We do not suggest it is representative. However, very little has been published on the performance of venture capital investments; so our pilot sample of 1041 transactions in 274 companies (in which over 200 venture capital firms invested) is a significant database.

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