Life of Luxury or Life of Crime

Yan Li1 and Sean Wilkoff2

1Wells College2California University at Berkley

Introduction

In the past few decades retirement options have become much more complex. Major investment firms have started to offer a wide variety of investment tools for retiring. People now have the choices of investing in different asset classes and multiple asset classes. Depending on how much risk one wants to take with their retirement fund, they can allocate their money accordingly. Today the Monte Carlo method helps in predicting how one should invest the money. The algebraic model is a riskless method of calculating the future value of money. A Monte Carlo model is a method for including risk which gives a distribution of the future value of money.

MODELS

In the algebraic model there were two models. One was Hyde’s single asset model. The next one was an extension of the first into a multiple asset model. Give models were considered for the Monte Carlo model. The first uses Forsyth’s. The second extends that model to multiple assets. After Forsyth’s model was extended to account for multiple assets, we were able to add new extensions to answer more questions. The first addition to the model was periodic investment. The second addition was to rebalance the multiple asset model every year. The last model is for inflation.

Market Historical Interest Rate Historical Average Standard Deviation

S&P500 0.125 0.198

Corp. Bond 0.063 0.063

TBill (Cash) 0.031 0.008

Inflation 0.027 0.018

The Interest Rates and Standard Deviations

Table 1 Harvey: Historical Perspective January 1926- December 2004

Allocation Rates for Different Assets

Market Allocation Rate

S&P500 0.7

Corp. Bond 0.2

TBill (Cash) 0.1

Table 2 Our allocation for testing results unless otherwise stated

RESULTS

The results we got included Algebraic model and Monte Carlo model. At the last section, we compared the result Monte Carlo model with Algebraic Model.

Monte Carlo Model

The Graph below shows the portfolio value at each time step of a single asset with the same starting value but different Standard Deviations. This is just one possible outcome of this asset. If the model was run again the graph would look different.

Figure 1 Possible value for single asset with different Standard deviations

The Distribution of Portfolio Values for a single asset with a standard Deviation of .2.

Figure 2 shows the distribution is not Gaussian

Figure 3 has a graph of the distribution of final portfolio values for the multiple asset model after 10 years.

Figure 3 Possible portfolio value after ten years

Figure 4 Possible portfolio values after 30 years

Figure 4 has a graph of the distribution of final portfolio values for the multiple asset model after 30 years. If one looks at the final portfolio value one can see the increase in expected outcomes from 10 to 30 years.

Algebraic vs. Monte Carlo

We wanted to solve the problem: What percentage of one’s initial portfolio can be withdrawn yearly? We both ran the problem and created a table to compare our results. The Monte Carlo model says if you take out 13% of your one’s initial portfolio every year then there is an 80% chance that one will run out of money in 18 years. The Algebra model says on can only withdraw 10% of one’s initial portfolio each year in order for the money to last 18 years.

Mean 50% less than 80% are less than Algebraic

4% 2891.15 1850 4050 2599.09

6% 2168.06 1150 3150 1853.78

8% 1481.91 450 2250 1108.48

10% 933.20 50 1150 363.19

12% 592.54 50 350 -382.11

13% 475.94 50 50 -754.76

Table 3 Comparison of how much money one will have left

The Value of Money

This table is to show the time value of money. The Table compares starting with 1000 dollars and letting the money sit for 10-40 years to investing 50 dollars every year.

Time (Years) $1000 $50

10 $2127.71 $820.33

20 $4838.97 $2532.05

30 $11453.16 $6529.08

40 $27725.54 $16132.64

Table 4 Example of the time value of money

This table is to show the time value of money, which is similar to above. But the allocation is different, 30% in cash, and 70% in bonds.

Time (Years) $1000 $50

10 1299.82 $628.35

20 $1718.55 $1387.63

30 $2304.92 $2400.0

40 $3127.70 $3767.22

Table 5 Example of the benefits of risk

CONCLUSION

Both the Monte Carlo and algebraic models are used by financial planners. The Monte Carlo model does not by any means say for certain the outcome of any investments but can give a probability of reaching a goal. The method also only accounts for normal deviations and random walks. Market shocks are not taken into account by this model.The future work will focus on developing the Monte Carlo model, which involves risk, and the result will be more accurate by comparing to the realistic. Furthermore, we will include the calculation for tax and transaction fees, so that people can use for calculating their investments.