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Fixed cast: heavy sword without front

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特邀专栏作者
2019-04-02 07:03
This article is about 8977 words, reading the full article takes about 13 minutes
We choose three typical market states: the starting point of the bull market, the apex of the bull market, and the volatile market. We use BTC historical data backtesting to conduct empirical research and compare fixed investment and one-time investment.
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We choose three typical market states: the starting point of the bull market, the apex of the bull market, and the volatile market. We use BTC historical data backtesting to conduct empirical research and compare fixed investment and one-time investment.



Tongtong Research Institute

Text: Song Shuangjie, CFA; Sun Hanru, analyst; Tang Hao, trainee analyst

guide

Fixed investment is a simple but very effective investment method. Some researchers said, "No matter how the security price fluctuates, this investment method can make investors full of confidence. So far, there is no investment method comparable to the dollar cost averaging method."

Summary

Summary

Fixed investment, that is, regular fixed investment. This investment method includes two aspects of "fixation": regular and fixed. Regular means that the time interval for each investment behavior is fixed, and fixed amount means that the amount used to purchase investment target assets is fixed each time. In addition to regular and fixed quotas, the characteristics of fixed investment also include: investment in batches, and the investment targets are mostly risky assets.

The advantages of fixed investment include: reducing the average cost of purchasing assets; matching with investors' cash flow; reducing the requirement for timing ability. Fixed investment is a more defensive investment strategy, suitable for stable and entry-level investors. There are many entry-level investors in the digital token market, which is suitable for regular investment strategies. The disadvantage of digital token fixed investment is that the price is not uniform due to the large number of exchanges, and the calculation of income is inconvenient.

Risk warning: market volatility risk.

Table of contents

Table of contents

1 Token fixed investment and its characteristics

1.1 What is fixed investment?

1.2 The "smile curve" of regular investment

1.3 Regular investment matches investor cash flow

1.4 Fixed investment lowers the requirement for timing ability

1.5 Are digital tokens suitable for regular investment?

2 Empirical research on fixed investment strategy

2.1 Research on fixed investment at the starting point of the bull market—precise timing, suitable for one-time investment

2.2 Research on fixed investment at the apex of the bull market - long-term expectations are locked in, it is not as good as fixed investment

2.3 Research on regular investment in volatile markets

3 The current market should adopt a fixed investment strategy

3.1 The advantages of regular investment in the current market environment

3.2 Target Token Selection for Regular Voting

3.3 Determination of the frequency of scheduled investment

4 Theoretical basis of regular investment

4.1 Fixed investment cash flow model and income calculation

4.3 Assets with high volatility are suitable for regular investment

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"Most of the happiness of mankind comes not from rare fortunes, but from a little daily gain." -Benjamin Franklin

Token fixed investment and its characteristics

1.1 What is fixed investment?

Fixed investment, that is, regular fixed investment. Graham mentioned the "dollar cost averaging method" in "Smart Investors": The New York Stock Exchange promotes the "monthly purchase plan", that is, investors invest the same amount of money every month to buy one or more stocks. Lucile Tomlinson made a comprehensive study of this kind of stylized investment. She used the Dow Jones Industrial Index constituent stocks as a sample, covering the price data from 1929 to 1952, and came to the following conclusion: "No matter how the security price fluctuates, this Investment law can make investors full of confidence. So far, there is no investment law comparable to the dollar cost averaging method."

More than half a century after the publication of Graham's book, fixed investment (also known as automatic investment plan, AIP) has become an important investment method. The trading strategy it uses is still the investment cost averaging method, and trading orders are generally issued by pre-set rules . This investment method includes two aspects of "fixation": regular and fixed.

Periodic means that the time interval between each investment behavior is fixed. There are usually different cycles such as weekly, biweekly, natural month, and quarterly, depending on the cash flow characteristics or investment preferences of investors.

Quota means that the amount used to purchase investment target assets is fixed each time. Since the price of the target asset will fluctuate each time an investment occurs, the amount of assets purchased in each investment period is also different. When the price of an asset rises, the quantity of assets purchased decreases; on the contrary, when the price falls, the quantity purchased increases, thereby achieving the purpose of reducing the average cost.

In addition to fixed term and fixed amount, the characteristics of regular investment also include:

Invest in batches. Regular investment implies the concept of "fractionation", that is, investors' funds are not all used to purchase target assets at one time. Even when investors do not have stable cash flow income but hold a certain amount of principal, they can adopt a strategy of investing in batches on a regular basis.

Most of the investment targets are risky assets such as stocks, stock funds, and indices. If investors regularly purchase bonds, currency funds and other wealth management products with low volatility and low risk, they do not need to use the strategy of fixed investment. Because the advantage of fixed investment is to buy assets with large fluctuations at a lower average price.

1.2 The "smile curve" of regular investment

Compared with other long-term investment strategies, the biggest advantage of fixed investment is that it can reduce the average cost of purchasing assets. The "smiling curve" of fixed investment is widely known. When the market shows a trend of first falling and then rising, the yield curve of the fixed investment strategy will form a shape similar to a smile. The longer the decline process lasts, the more time there is to buy assets at a lower price and dilute the cost. When the market starts to pick up, as long as the price exceeds the average cost, it can be profitable.






Taking the 32-month historical price of BTC from November 2013 to July 2016 as an example, the following compares the average cost and return rate of the weekly fixed investment and the initial one-time investment within the selected time period.

1.3 Regular investment matches investor cash flow

Fixed investment is suitable for investors with stable cash flow income. For example, investors can use part of their monthly income for regular investment. Its advantages are as follows: the amount of investment matches your own income and budget, even if the investment suffers a loss, the investor will not fall into bankruptcy crisis; using spare money to invest can reduce the psychological burden of the investor when the loss occurs; the investment cycle is long and can Through bull and bear, effectively reduce investment costs and improve capital utilization efficiency in the bear market stage.

Even investors who have a certain amount of capital and no stable cash flow can adopt the investment method of fixed investment. Use the principal to buy short-term wealth management with stable yields such as monetary funds, and redeem a fixed amount according to the planned time and use it for fixed investment.

1.4 Fixed investment lowers the requirement for timing ability

"Time selection" refers to the selection of trading points through fundamental or technical analysis. For ordinary investors, without long-term investment experience, it is difficult to grasp the judgment of timing. The fixed investment can effectively reduce the risk of choosing "entry timing" by dispersing a single investment over a longer period of time.

If the investor's entry timing is not good, the price of the asset will start to fall when the first purchase is made, then the subsequent fixed investment will allow the investor to buy more assets at a lower price, thereby reducing the average cost .





Compared with investors who have no timing ability, assuming that their buying timing is a random variable distributed within a period of time, then their expected cost is the arithmetic mean of the prices during this period. Comparing the average cost of fixed investment with the expected cost of one-time investment of investors without timing ability, it can be concluded that the cost of fixed investment is always better than the expected cost of one-time investment.




When the regular investment is carried out for a long time, the average cost will be less affected by the impact of new additional investment in each period. Therefore, the long-term profitability of fixed investment depends more on the price of assets at the time of exit, that is, more attention is paid to the timing of selling and whether the assets are of high quality.

However, in a unilateral rising market, the profitability of fixed investment may not be as good as the one-time investment at the beginning of the period. In a market that rises first and then falls, fixed investment will continuously increase the average cost. When the asset price is lower than the average cost of fixed investment, it is the most efficient period for investors to use funds.

All in all, fixed investment is a more defensive investment strategy, suitable for stable and entry-level investors.

1.5 Are digital tokens suitable for regular investment?

Most of the investment targets of regular investment are risky assets with high volatility. The specific reasons will be analyzed later. At present, the mainstream fixed investment wealth management products in the market include fund fixed investment and stock fixed investment. With the vigorous development of the blockchain industry and the digital token secondary market, digital tokens such as BTC and ETH have gradually entered the field of vision of traditional investors.

However, the professional ability of current digital token investors is not as good as that of traditional investors. The threshold for investing in digital tokens is low, and entry-level investors also have shortcomings in terms of trading concepts, strategies, and execution capabilities. As a programmed and defensive investment strategy, fixed investment can effectively prevent investors from investing all their principal at the wrong time The resulting losses are suitable for investors who do not grasp the timing of entry accurately and are preparing for long-term investment. There is a high probability of making profits and reducing risks. The target of investment can choose a certain token or a fund that tracks the blockchain index.

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At present, the disadvantages of regular token investment include that there is a price difference between different exchanges, and there is no industry index and corresponding tracking fund recognized by the mainstream in the current market. There will be some inconvenience.

Empirical Research on Fixed Investment Strategy

In the token market, BTC has the longest history, the highest proportion of market value, and the strongest consensus, and has experienced two complete cycles. Therefore, in the empirical study, we used BTC for analysis.

Similar to most other assets, for the regular investment of tokens, the choice of entry time has a greater impact on the rate of return, and investors have great autonomy in deciding when to enter the market. Therefore, we choose three typical entry points to analyze the impact of one-time investment and fixed investment strategies on the rate of return.

Due to doubts about the reliability of BTC's early transaction data, we took the peak of the second round of bull market in November 2013 as the starting point of the study, and set the end point as January 24, 2019.

2.1 Research on fixed investment at the starting point of the bull market—precise timing, suitable for one-time investment





If a trader makes a one-time investment at the starting point of the third round of bull market and holds it until now, the rate of return will be greater than that of a fixed investment strategy. The third round of the BTC price cycle has not yet been fully completed, and is currently in the price downward stage of this round of price cycle. The price upward cycle of this round started in August 2015 and lasted for a total of 845 days until December 2017. The starting price of BTC in this price upward cycle was 199.57 US dollars per piece, and the highest price was close to 20,000 US dollars per piece. up to 99 times. Beginning in December 2017, the price of this round of cycle began to decline. So far, the lowest drop has reached $3,200, a drop of 84%.

If a trader buys BTC at 00:00 on August 25, 2015 and holds it until now, the cost of BTC holdings will be 221.60 USD per piece. Since then, BTC has never fallen below this price. The function of low average cost is not only impossible to take effect, but will increase the cost of holding positions. As of February 22, 2019, the price of BTC was $4005 per piece, and its yield was about 2047%.

However, if a trader chooses to start a fixed investment plan with an interval of 30 days on August 25, 2015, then after 42 fixed investment periods, his BTC holding cost will be US$869/coin, which is much higher than the one-time investment plan of US$221.60/coin Sexual investment cost line. After multiple scheduled investments, due to the significant increase in the cost of holding positions, as of February 22, 2019, the rate of return of this scheduled investment plan was only 343%.

2.2 Research on fixed investment at the apex of the bull market - long-term expectations are locked in, it is not as good as fixed investment

If investors enter the market at the peak of the second round of bull market, the one-time investment will make them in a state of floating loss for a long time, and the fixed investment will reduce the time of floating loss and the average holding cost, and the rate of return of fixed investment will also be higher than that of one-time investment. From November 2011 to November 2013, it was the price upward cycle stage of the second round of price cycle, which lasted for 743 days. In this price upward cycle, the starting price of BTC was 1.99 US dollars per piece, and the highest price was 1242 US dollars / piece, an increase of 623 times. From November 2013 to August 2015, it is the price downward cycle stage of the second round of price cycle, which lasts for 634 days. In this price downward cycle, the starting price of BTC is 1242 USD/piece, and the lowest price is 199.57 USD / piece, a drop of 84%.

If an investor buys BTC at 00:00 on November 30, 2013, the investor will have to endure a floating loss period of more than three years at a holding cost of $1129 per coin , During these three years, BTC fell to a minimum of $199.57 per piece, that is, the maximum retracement of this investment reached 84%. As of January 24, 2019, the investment income was 218%. During the investment period, he experienced a three-year floating loss, two retracements greater than 80%, and multiple doublings. Regardless of the rate of return, in the process of investing in BTC, if the account cannot be operated due to objective reasons such as forgetting one's own private key, the value of spiritual experience obtained must be greater than the rate of return of 218%.





On the contrary, if the investor starts a regular investment plan with an interval of 30 days at 00:00 on November 30, 2013, without interruption, the situation will be much better. As the price of BTC fell, the implementation of the fixed investment plan led to a rapid decline in the cost of holding positions. After 38 fixed investment periods, the cost of holding positions dropped to the lowest point of $347.63 per coin on November 21, 2015. In November 2015, the rate of return of the regular investment plan was positive for the first time. Although this investor still needs to endure up to two years of floating losses, his maximum drawdown is far lower than that of a one-time investor, which is 65%, and the duration of the large drawdown is shorter. Considering the discipline and rules of implementing the fixed investment plan, the psychological pressure of the investor who chooses the fixed investment in the four years after starting the fixed investment will be much less than that of the investor who has been on the peak for three years. As of February 22, 2019, the rate of return of this regular investment plan was 548%, significantly higher than the rate of return of one-time investment of 254%.

Comparing the profit and loss time of the two investment methods, it can be found that when the investment plan is started at the same time, the number of days when the fixed investment loses money is far lower than the one-time investment. This is due to the sharply lowering effect of regular investment on the average position price at the apex of the bull market.




2.3 Research on regular investment in volatile markets

In most cases, it is difficult for investors to accurately buy the bottom, and it is also difficult to accurately buy at the apex of the bull market. They usually enter the market in a volatile market. Therefore, we analyzed the historical price of BTC from May 23, 2017 to February 22, 2019, and calculated the cumulative income of two strategies of fixed investment and one-time investment every day during the period until February 22, 2019, and then Some conclusions were drawn.

First of all, it is suitable to adopt a fixed investment strategy in a market that has just begun to fluctuate and go down. From March 2018 to November 2018, the price of BTC was generally in a state of shock and downward. If you start a regular investment plan with an interval of 30 days during this period, the investment results are better than simultaneous one-time investment. During this period of time, the price of BTC showed a fluctuating downward trend. Starting fixed investment at this moment will give full play to one of the biggest advantages of fixed investment - lowering the average holding cost.

As shown in the figure, if you start the fixed investment plan within this range and persist until February 22, 2019, the cost of holding positions for investors will be lower than that of the one-time buy-and-hold strategy on the start date of the fixed investment in most cases. In terms of yield, it is also advisable to choose a fixed investment strategy when starting an investment plan during this period. Due to the decrease in the cost of holding positions, the maximum retracement range and duration of fixed investment are much less than those of one-time investors.




Secondly, in the later stage of the volatile downward phase, the decline begins to ease, and after the BTC price has experienced a large decline, it is advisable to adopt a one-time investment strategy. After December 2018, BTC has lost most of its market value from the peak of the bull market. Although it is in a volatile downward phase, the fixed investment strategy during this period cannot achieve very good results. There are two reasons for this. First, the core of fixed investment - buying in batches with fixed quotas is a double-edged sword. In the current market, it will instead increase the average holding cost; second, there is not much room for the current price to continue to fall, which makes the advantage of fixed investment-reducing costs play a limited role.

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Finally, in a volatile upward market, it is suitable for a one-time investment strategy. In the early stage of the bull market, the price of BTC fluctuated and went up. Starting a fixed investment on any day during this stage will continue to increase the cost of holding positions. Conversely, if you make a one-time investment on any day during this stage, as long as the two start at the same time, then the fixed investment has no worth mentioning advantage over the one-time investment.

The current market should adopt a fixed investment strategy

3.1 The advantages of regular investment in the current market environment

First of all, fixed investment can match cash flow, which is suitable for young investors.

Regular investment strategies can match cash flow. Generally speaking, young investors are more inclined to invest in risky assets like BTC. Although they don't have much savings, they have a gradual increase in income expectations. Regularly investing part of their income in BTC will not affect the quality of life, but will add hope to their future life. In addition, if you examine this investment plan on a yearly basis, long-term fixed investment in a new type of asset that is developing rapidly is likely to allow young investors to achieve financial freedom as soon as possible.

Secondly, fixed investment does not require high timing ability, but the effect is even better than subjective timing.

The fixed investment strategy does not need to choose a time, and is suitable for investors who do not have the energy to keep an eye on the market every day. The token market is open for trading 24 hours a day. Due to limited energy and funds, it is difficult for individual investors to artificially choose the right time to achieve good investment results. Moreover, the token market is currently in the late stage of the bear market, the market value has shrunk sharply, the liquidity has decreased, and the trend in the short term is even more difficult to predict. If the token price continues to fall after the one-time investment, investors will likely face great pressure. On the contrary, if a fixed investment strategy is adopted, investors can buy more tokens at a lower unit price when the market outlook continues to fall. It is pleasing to admit that buying an asset at a lower price that is likely to rise in the future, while falling will bring negative emotions such as anxiety and regret. Then when making a fixed investment, buying assets that will rise in the future at a cheaper price and the current temporary decline in asset prices will form a hedging relationship in terms of investor sentiment, and vice versa. In this way, investors will be able to win without pride and lose, avoid being passive in the volatile market, and thus invest in tokens with a healthier attitude.

Most investors are non-professional traders. Due to their busy work, investors often cannot devote enough energy to analyze the short-term trend of BTC and make precise timing. From the avenue to simplicity, from evolution to complexity. Instead of struggling to predict the short-term trend and worrying about the ups and downs of a few points, it is better to choose a low-risk fixed investment strategy, which can not only enjoy the dividends brought by the rapid development of the token economy, but also allocate energy to enjoy life. After all, in investment, the energy invested is not directly proportional to the income.

When the price of BTC continues to fall, investors who choose to make a one-time investment will not only have doubts about their own decisions, but also have the idea of ​​giving up and admitting defeat due to a large number of external doubts and multiple pressures. The most regrettable thing is that some investors may even fall before dawn. In fact, there are two stories with very different endings circulating in the token circle:

At the beginning of 2014, an investor known as "Brother 480,000" live-broadcasted the story of himself buying 480,000 RMB worth of BTC at one time on Baidu Tieba. Finally, under the persecution of his own doubts and relatives, he bought the BTC before June 2016. Lost and left the market with a loss of 180,000. On the contrary, the registered user "Old Captain" of the Babbitt Forum started the regular investment plan on March 11, 2015, and terminated it in February 2017. As of today, the "Old Captain" has invested a total of 110,000 yuan. If the forked coins generated by the multiple forks experienced by BTC are not counted, its value currently exceeds 1 million.

3.2 Target Token Selection for Regular Voting

BTC has the value of regular investment. The longer the history of the token and the more active the transaction, the stronger its consensus, and the establishment of consensus has a strong Matthew effect. As difficult as it is to establish a consensus, it is as difficult as it is to be eliminated. Therefore, BTC is the least likely of all tokens to go to zero. In addition, even in a bear market, BTC has strong liquidity, and most individual investors do not need to consider the problem of insufficient BTC capital capacity. Although from the perspective of maximum income, regular investment in BTC may not be comparable to regular investment in some counterfeit tokens. However, the above two points jointly determine that from the perspective of risk, fixed investment in BTC is far superior to fixed investment in other counterfeit tokens. Since the birth of the first BTC in 2009, its price has gone through three cycles. The time span of the first BTC price cycle was from March 2010 to November 2011, which lasted 610 days, which was the largest increase among the three BTC price cycles in history. The time span of the second round of BTC price cycle was from November 2011 to August 2015, which lasted 1377 days. During this time, the unit price of BTC surpassed that of gold for the first time. The third round of the BTC price cycle has not yet been fully completed, and is currently in the price downward stage of this round of price cycle. The price upward cycle of this round started in August 2015 and lasted for a total of 845 days until December 2017. The starting price of BTC in this price upward cycle was 199.57 US dollars per piece, and the highest price was close to 20,000 US dollars per piece. up to 99 times. Since December 2017, the price of this cycle has started to go down. So far, the lowest drop has reached $3,200, a drop of 84%, and it has recently rebounded to around $4,000. Although BTC has experienced a large decline since the peak, it may still hit a new low in the future. Adopting a fixed investment strategy at this time can reduce the possibility of full position buying at a staged high point, and reduce the risk of excessive cost when building a position. BTC is also the least likely of all tokens to go to zero. Therefore, regular investment in BTC is a strategy with a good risk-benefit ratio worth adopting at present.

3.3 Determination of the frequency of scheduled investment

Our empirical analysis shows (if you need detailed backtest data, please contact Tokentong Research Institute), there is no significant difference in income between weekly and monthly fixed investment. Investors can decide the frequency of fixed investment according to the frequency of obtaining stable cash flow.

We selected the BTC price data from April 8, 2013 to January 24, 2019, and adopted regular investment strategies at the frequency of 30 days and 7 days respectively. The final income of the 30-day frequency fixed investment is 499.8%, and the final income of the 7-day frequency fixed investment is 497.1%. The t-test under the assumption of equal variances was performed on the daily net value of the two, and it was found that the p value was 0.46, which was much greater than 0.05, showing that there was no statistically significant difference between the two.

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In addition to the 30th and 7th, we also compared a large number of frequency combinations, and we found that when the total number of fixed investment periods exceeds 20, the impact of the fixed investment frequency on the final return is minimal. If investors decide to adopt a fixed investment strategy of more than 20 periods, they do not need to consider the impact of fixed investment frequency on the rate of return, they only need to match it with their own cash flow.

Theoretical basis of fixed investment

4.1 Fixed investment cash flow model and income calculation

Annuity refers to the receipt or payment of corresponding payments at regular intervals. By definition, regular investment payouts can also be seen as a type of annuity. Since the investment behavior occurs at the beginning of each period, it is an initial annuity.

For convenience, assume that the number of scheduled investments per year is M times, that is, the interval between each period is 360/M days, the amount of investment in each period is 1, the number of continuous periods is M, and the price of assets at the beginning of the t-th period is P_t. The quantity of assets is Q_t=1/P_t.




After T-period regular investment, the number of assets held is Sigma(1/P_t), and its average cost is




At the end of period T, the price of the asset is P_T+1. Then the end-of-period (cumulative) rate of return of the fixed investment plan after the scheduled investment period




The annualized rate of return converted in the form of simple interest is




If the time value of money is considered, it is necessary to discount the investment in each period and the asset price at the end of T period. Suppose the risk-free annual interest rate is r. Then the discount factor for each period is




The discounted value of the asset price at the end of period T is




rate of return




rate of return




In addition to simply calculating the cumulative rate of return and annualized rate of return, taking into account the time cost of funds, we can also use the modified internal rate of return (MIRR) to calculate the return of the regular investment plan. This method uses a predetermined discount rate to discount the investment amount of each period to the beginning of the first period and the return to the end of the last period. And assume an unknown rate of return, discount the final value of the total return to the beginning of the period according to this rate of return, and make it equal to the discounted value of the total investment, then the rate of return is the modified internal rate of return.

Here is the annualized modified internal rate of return. The advantage of this method is that, considering the time cost, the time value of the investment amount and the reinvestment income of the received return are calculated at a predetermined discount rate. This method is usually suitable for cash inflows such as redemption in the middle of a fixed investment, but this method is more complicated, and the time factor is generally not considered when comparing the income of a fixed investment, and a simple cumulative rate of return is used.

4.2 The principle of fixed investment to reduce average cost




Assume that an investor wants to invest in a certain asset N times within a certain period of time, and further assume that the investor can only invest at certain fixed time points, and the price of the asset at these time points is P1, P2, P3,..., the quantity of purchased assets is Q1, Q2, Q3,..., then the average cost for investors to obtain this asset is




If the investor chooses to invest all of his funds at a certain point in time, then




where t is a random variable, and




Then the investor's expectation of obtaining the cost of assets is

is the arithmetic mean of asset prices at these time points.




If investors buy the same amount of assets every time, that is, Q1=Q2=...=Qn, then

Show that the cost of investing in this strategy is the same as the expected cost of buying at random.




If the method of investing equal funds each time is adopted, that is, Qi=1/Pi, the average cost




is the harmonic mean of asset prices at these time points. Since the harmonic mean of any number of positive real numbers is always not greater than the arithmetic mean, that is

Therefore, it is more effective to reduce the average cost by adopting the fixed amount investment method than the fixed amount and one-time investment.

4.3 Assets with high volatility are suitable for regular investment

In the financial field, people usually use geometric Brownian motion to describe the random fluctuation of stock price. Because geometric Brownian motion has the following characteristics:

It satisfies the normal distribution and can be used to describe the continuous compound interest rate of return of stocks.

Brownian motion is a Markov process, that is, the future price distribution is only related to the current price and has nothing to do with the historical price. That is, the current price has fully reflected the information needed to predict future prices, which is consistent with the weak efficient market hypothesis.

Brownian motion is continuous everywhere but not differentiable everywhere, and its quadratic variation is greater than zero, which is in line with people's perception of stock price changes.




We assume that the rate of return of investment assets in a very short period of time conforms to the geometric Brownian motion:




Among them, u is the expected rate of return of the asset, and o is the volatility of the asset, that is, the standard deviation of the rate of return of the asset in a short period of time. W(t) is a standard Brownian motion with expectation o, and the expression for variance TP(t) over an interval of length T is:




If an investor makes N times of fixed investment within time T, the average cost of fixed investment is:




Assuming that investors invest "continuously" within time, P_D can be written in integral form:




If the investor makes a one-time investment at time 0, the cost is P(0), compare P_D with P(0):




Due to some reasons, some nouns in this article are not very accurate, mainly such as: general certificate, digital certificate, digital currency, currency, token, crowdsale, etc. If readers have any questions, they can call or write to discuss together.

Note:

Due to some reasons, some nouns in this article are not very accurate, mainly such as: general certificate, digital certificate, digital currency, currency, token, crowdsale, etc. If readers have any questions, they can call or write to discuss together.

This article was originally created by TokenRoll Research Institute (ID: TokenRoll). Unauthorized reprinting is prohibited.


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