### Assignment 1: Simulating dollar cost averaging

#### 1 Introduction

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 us further insight in which method our client should 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).

#### 2 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 distribution are 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 times to produce a large number of likely outcomes.

#### 2.1 Lump sum

A lump sum investing strategy entails that you invest your entire sum at one given moment in 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.

Table 1: Monte Carlo Simulation – Lump sum: Descriptive Statistics

Mean | Median | Standard Deviation | CI Lower Bound | CI Upper Bound | |

Jan/21 | 150000.0 | 150000.0 | 0.00 | 150000.0 | 150000.0 |

Feb/21 | 151355.7 | 151339.2 | 7984.38 | 135825.4 | 167109.0 |

Mar/21 | 152858.7 | 152600.5 | 11520.69 | 130695.3 | 176216.8 |

Apr/21 | 154352.8 | 153731.8 | 14241.04 | 128021.6 | 183856.0 |

May/21 | 155699.9 | 154972.9 | 16580.13 | 125316.2 | 190665.8 |

Jun/21 | 157262.8 | 156234.8 | 18770.71 | 123385.8 | 196397.2 |

Jul/21 | 158794.7 | 157777.5 | 20588.86 | 121263.9 | 201807.7 |

Aug/21 | 160362.4 | 159099.5 | 22475.52 | 120083.5 | 208301.2 |

Sep/21 | 161930.4 | 160702.2 | 24152.98 | 118358.4 | 212646.3 |

Oct/21 | 163413.0 | 162097.7 | 25805.90 | 117791.9 | 218719.7 |

Nov/21 | 164944.7 | 163338.3 | 27413.34 | 117672.9 | 224185.6 |

Dec/21 | 166357.0 | 164219.4 | 29084.03 | 116244.0 | 229846.1 |

Jan/22 | 167898.5 | 165217.5 | 30816.61 | 115462.0 | 234504.8 |

*Note: *This table summarizes the descriptive statistics regarding our Monte Carlo Simu- lation where we used the lump sum over a one year horizon.

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.

#### 2.2 Dollar Cost Averaging

Dollar Cost Averaging (DCA) is a strategy where an investor invests a total sum of money 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 based upon 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.

Table 2: Monte Carlo Simulation – Dollar Cost Averaging: Descriptive Statistics

Mean | Median | Standard Deviation | CI Lower Bound | CI Upper Bound | |

Jan/21 | 12500.00 | 12500.00 | 0.00 | 12500.00 | 12500.00 |

Feb/21 | 25235.37 | 25243.21 | 1330.28 | 22555.25 | 23519.61 |

Mar/21 | 38102.59 | 38056.45 | 2415.66 | 33373.90 | 35022.93 |

Apr/21 | 51063.19 | 51015.47 | 3619.57 | 44180.37 | 46441.49 |

May/21 | 64191.25 | 64025.67 | 4947.81 | 54955.12 | 57938.23 |

Jun/21 | 77333.69 | 77062.97 | 6438.72 | 65593.01 | 69275.31 |

Jul/21 | 90594.92 | 90213.19 | 8095.56 | 75778.49 | 80447.96 |

Aug/21 | 104055.71 | 103545.85 | 9979.13 | 85936.00 | 91613.31 |

Sep/21 | 117589.86 | 117026.11 | 11796.63 | 95918.93 | 102770.64 |

Oct/21 | 131224.89 | 130548.96 | 13852.67 | 105608.27 | 114072.82 |

Nov/21 | 145122.68 | 144285.51 | 16015.87 | 116390.84 | 124949.34 |

Dec/21 | 159037.46 | 158004.34 | 18302.89 | 126066.42 | 135908.19 |

Jan/22 | 173109.32 | 171829.68 | 20513.74 | 136145.75 | 147563.22 |

*Note: *This table summarizes the descriptive statistics regarding our Monte Carlo Simula- tion where we used Dollar Cost Averaging over a one year horizon.

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.

#### 2.3 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 has the 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. This is due to the fact that the Dollar Cost Averaging strategy adds 12500 every month. However, in Table 1 and Table 2 we can see that the medians in 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 Cost Averaging strategy. In order to compare our results with the research of Williams and Bacon (1993), we calculate the annualized holding period return (AHPR).

*Notes: *This figure plots Monte Carlo Simulations for the lump sum and Dollar Cost Averaging strategies.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴ℎ

− 1

𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴ℎ

The AHPR for the lump sum strategyis 9.48% and 5.34% for the Dollar Cost Averaging strategy. The AHPR for the lump sum strategy is 4.14% higher. These findings are consistent with the conclusions of Williams and Bacon (1993), who stated that “the sooner the entire endowment is fully invested in the market, the higher the realized return.”

#### 3 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.

#### 3.1 Lump sum

We used the mean and standard deviation of our historic monthly stock returns from 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.

Table 3: Bootstrapping – Lump sum: Descriptive Statistics

Mean | Median | Standard Deviation | CI Lower Bound | CI Upper Bound | |

Jan/21 | 150000.0 | 150000.0 | 0.00 | 150000.0 | 150000.0 |

Feb/21 | 150159.9 | 148924.0 | 10508.14 | 133287.0 | 178976.7 |

Mar/21 | 151652.4 | 150768.2 | 15204.35 | 124042.1 | 185588.7 |

Apr/21 | 153710.2 | 152840.5 | 18510.60 | 119834.4 | 192183.8 |

May/21 | 155888.0 | 155091.3 | 21127.30 | 116307.4 | 199651.3 |

Jun/21 | 157949.6 | 157325.9 | 23323.88 | 113217.9 | 205362.1 |

Jul/21 | 159825.1 | 159155.1 | 24978.71 | 110417.1 | 211005.6 |

Aug/21 | 161443.9 | 160667.0 | 26288.54 | 109359.1 | 215722.4 |

Sep/21 | 162938.5 | 162233.4 | 27338.55 | 108350.6 | 219884.9 |

Oct/21 | 164130.8 | 163276.7 | 28230.27 | 108542.5 | 222802.2 |

Nov/21 | 165257.7 | 164304.2 | 29082.30 | 108712.5 | 226078.4 |

Dec/21 | 166450.7 | 165207.5 | 30020.30 | 108539.1 | 229534.0 |

Jan/22 | 167887.4 | 166786.9 | 31025.75 | 108549.4 | 232455.6 |

*Note: *This table summarizes the descriptive statistics regarding our bootstrap where we used the lump sum over a one year horizon.

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.

#### 3.2 Dollar Cost Averaging

We used the mean and standard deviation of our historic monthly stock returns from 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. We replicated this function 10000 times in order to make our bootstrap more precise.

Table 4: Monte Carlo Simulation – Dollar Cost Averaging: Descriptive Statistics

Mean | Median | Standard Deviation | CI Lower Bound | CI Upper Bound | |

Jan/21 | 12500.00 | 12500.00 | 0.00 | 12500.00 | 12500.00 |

Feb/21 | 25005.73 | 24820.66 | 1773.32 | 21400.86 | 29829.45 |

Mar/21 | 37843.48 | 37697.62 | 3237.35 | 31668.88 | 44815.56 |

Apr/21 | 51083.40 | 51011.89 | 4788.96 | 42006.47 | 60793.33 |

May/21 | 64535.86 | 64570.63 | 6312.02 | 52220.57 | 76919.39 |

Jun/21 | 78122.77 | 78199.74 | 7781.20 | 62931.37 | 93156.47 |

Jul/21 | 91795.48 | 91851.22 | 9054.89 | 73806.87 | 109298.41 |

Aug/21 | 105294.45 | 105463.85 | 10210.97 | 85118.67 | 125115.71 |

Sep/21 | 118889.31 | 118846.64 | 11346.12 | 96616.37 | 141192.44 |

Oct/21 | 132447.22 | 132498.48 | 12521.77 | 108012.90 | 157215.37 |

Nov/21 | 146051.33 | 146049.65 | 13867.54 | 118760.82 | 173587.49 |

Dec/21 | 159614.88 | 159668.25 | 15287.57 | 129097.67 | 189496.78 |

Jan/22 | 173469.93 | 173694.67 | 16962.74 | 138618.27 | 206750.79 |

*Note: *This table summarizes the descriptive statistics regarding our bootstrap where we used Dollar Cost Averaging over a one year horizon.

In Table 4 we show the descriptive statistics of our bootstrap 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.

#### 3.3 Lump sum vs Dollar Cost Averaging

As previously mentioned, we compare the two strategies based upon their median. We plotted the bootstrap of both strategies to visually see the difference in medians.

However, in Table 3 and Table 4 we can see that the medians in the twelfth month are 165207.55 euro and 159668.25 euro respectively. This shows us that the median of the lump sum strategy is higher than that of the Dollar Cost Averaging strategy. In order to compare our results with the research of Williams and Bacon (1993), we calculate the annualized holding period return (AHPR).

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴ℎ

− 1

𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴ℎ

2021.0 2021.2 2021.4 2021.6 2021.8 2022.0 2021.0 2021.2 2021.4 2021.6 2021.8 2022.0

Time Time

Figure 2: Plot Monte Carlo Simulations

*Notes: *This figure plots the bootstraps for the lump sum and Dollar Cost Averaging strategies.

The AHPR for the lump sum strategy is 10.14% and 6.45% for the Dollar Cost Averaging strategy. The AHPR for the lump sum strategy is 3.69% higher. These findings align with our previous results of the Monte Carlo simulations. Furthermore, these results are once more consistent with the conclusions of the research of Williams and Bacon (1993).

#### 4 Conclusion

Our results show that the annualized holding period returns for both the Monte Carlo simulation as the bootstrap method are higher for the lump sum method. These results were computed based upon a the sample of historic monthly stock return data of Australia from MSCI starting in 1969. This inclines us to state that the lump sum method outperforms the Dollar Cost Averaging method. Furthermore, these results are consistent with the findings of Williams and Bacon (1993). Therefore we would advice our client to invest her capital using the lump sum strategy.

#### 5 References

Annaert, J. (2021) Quantitative Methods in Finance

IBM cloud education (2020). Monte Carlo Simulation Retrieved from https://www.ibm.com/cloud/learn/ monte- carlo-simulation

Muller (2021). Dollar Cost Averaging vs Lum Sum which is better from https://www.moneyunder30.com/ dollar- cost-averaging-vs-lump-sum-investing

Williams, R., Bacon, P. (1993). Citation Lum Sum Beats Dollar Cost Averaging. Journal of Financial Planning

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