**Assignment Task**

**Task **

**Assignment 1: Simulating dollar cost averaging**

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In this report we will compare the lump sum and Dollar Cost Averaging strategies by simulating and boot- strapping the next 12 month period based on the historic monthly stock return data of Australia from MSCI starting in 1969. This will give usfurther insightin which method our clientshould use to invest her 150000 euro. The comparison will be made by calculating the annualized holding period return (AHPR) for both strategies. We will also compare our findings with the research of William and Bacon (1993).

**Monte Carlo Simulation**

A Monte Carlo simulation predicts a set of outcomes based on an estimated range of values versus a set of fixed input values (IBM cloud education, 2020). In our simulation we assume the following: monthly returns are normally distributed and the risk-free rate is 0 percent. The parameters for the distributionare estimated and then finally a random number generator is used to generate data according to the presumed distribution. It recalculates the results over and over, each time using a different set of random numbers between the minimum and maximum values. In our case these are based on the historic MSCI data of Australia. In a Monte Carlo experiment, this exercise can be repeated thousands of timesto produce a large number of likely outcomes.

**Lump sum**

A lump suminvesting strategy entailsthat you invest your entire sum at one givenmomentin time (Muller, 2021). We replicated this by creating a simulation function. This function uses the mean and standard deviation of our historic monthly stock returns from MSCI and simulates this over a twelve month horizon. It uses this input to generate a random number based upon these parameters and calculates the wealth of given period based upon the wealth of the previous period multiplied by this randomly generated number. We replicated this function 10000 times in order to make our simulation more precise.

In Table 1 we show the descriptive statistics of our simulation of the lump sum strategy with 10000 iterations. We calculated the mean, median, standard deviation and the 95 percent confidence interval for the different wealth levels. To determine the end wealth, we look at the December 2021, as this is the twelfth month of the year.

**Dollar Cost Averaging**

Dollar Cost Averaging (DCA) is a strategy where an investor invests a totalsum ofmoney in small increments over time instead of all at once (Muller, 2021). Investing your money at regular intervals such as weekly, monthly, or quarterly allows the investor to mitigate the risk of buying in at an inflated price. This function uses the mean and standard deviation of our historic monthly stock returns from MSCI and simulates this over a 12 month horizon. It uses this input to generate a random number based upon these parameters and calculates the wealth of given period basedupon the starting wealth, which is 12500 euro. Then it adds an additional 12500 and multiplies this sum with the randomly generated number. We replicated this function 10000 times in order to make our simulation more precise.

In Table 2 we show the descriptive statistics of our simulation of the Dollar Cost Averaging strategy with 10000 iterations. We calculated the mean, median, standard deviation and the 95 percent confidence interval for the different wealth levels. To determine the end wealth, we look at the December 2021, as this is the twelfth month

of the year.

**Lump sum vs Dollar Cost Averaging**

We can compare the two strategies based on two different statistics, the mean and the median. We opted for the median, as this is unaffected by extreme outliers and allows us to draw more accurate conclusions. First of we plotted the two functions in order to see which one hasthe higher median in the twelfth month. It is difficult to visually see the difference between the two median levels, as the two graphs follow a different trajectory. Thisis due to the factthatthe Dollar Cost Averaging strategy adds 12500 everymonth. However, in Table 1 and Table 2 we can see that the mediansin the twelfth month are 164219.40 euro and 158004.34 euro respectively.

This shows us that the median of the lump sum strategy is higher than that of the Dollar evidence . In order to compare our results with the research of Williams and Bacon (1993), we calculate the annualized holding period return (AHPR).

**Bootstrap**

We control our findings with the bootstrap method. Instead of generating random observations from a theo- retical distribution, a bootstrap generates observations from the empirical distribution by drawing randomly from the sample (Annaert, 2021). Bootstrap uses the empirical distribution to simulate. It is assumes that one-period returns are identically and independently distributed. This implies that the order in which we observed the actual returns was coincidental. Any other ranking would have been possible. If there are T observations in a sample, each of these T outcomes could have been observed for any given period. This means that there are T H equally likely H – period return paths that could have occurred. In principle, these paths can be used to derive the distribution of all statistics an investor would need to make investment decisions. This is another way to generate the random returns. We take random values from the historical sample rather than assuming a normal distribution,. The

sample() function does this for us.

**Lump sum**

Weused themean and standard deviation of our historicmonthly stock returnsfrom MSCI and bootstrapped this over a twelve month horizon. Instead of generating a random number, the bootstrap method retrieves a random number from our sample. It calculates the wealth of given period based upon the wealth of the previous period multiplied by this randomly selected return. We replicated this function 10000 times in order to make our simulation more precise.

In Table 3 we show the descriptive statistics of our bootstrap of the lump sum strategy with 10000 iterations. We calculated the mean, median, standard deviation and the 95 percent confidence interval for the different wealth levels. To determine the end wealth, we look at the December 2021, as this is the twelfth month of the year.

**Dollar Cost Averaging**

Weused themean and standard deviation of our historicmonthly stock returnsfrom MSCI and bootstrapped this over a 12 month horizon. It uses this sample to randomly select a return and calculates the wealth of given period based upon the starting wealth, which is 12500 euro. Then it adds an additional 12500 and multiplies this sum with the randomly selected number. Wereplicated thisfunction 10000 timesin order to make our bootstrap more precise.

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