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FIN 421 Assignment 1 100% Correct Key
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In this assignment you will use a Monte Carlo simulation to investigate the effect of randomness in returns on an individual’s future retirement wealth. Saving for Retirement Assume an investor begins saving for retirement at age 25 and retires at age 65. Each year, she contributes \$5, 000 to her retirement account and her employer matches this contribution for a total of \$10, 000 in annual savings. To keep things simple, assume that each of the 40 annual payments is added to the account at the end the year.1 Savings are invested as follows: 50% in a broad stock market index and 50% in T-Bills. Your task is to compute the accumulated real retirement savings (at age 65) for different return realizations. As explained below, you will generate returns using a Monte Carlo simulation. On Blackboard, you will find an Excel file containing historical net returns on the S&P 500 and 3-month T-bills, as well as the net inflation rate from 1928 to 2013. The return on the CPI serves as a measure of inflation. STEPS: 1. Compute the annual real return on the 50/50 portfolio for each year in the sample. The resulting set of 86 portfolio returns represents the empirical distribution. These are the returns investors historically realized when investing in a 50/50 mix of stocks and T-bills over this time period. 2. We will use the historical data to assess what may happen in the future via a Monte Carlo simulation. To generate a possible path of future returns, draw 40 times with replacement from the empirical distribution.2 Assuming the historical returns are located in the cell range H11:H96, a random draw can be generated with =INDEX(H11:H96,RANDBETWEEN(1,86)) The set of 40 draws you generated can be viewed as one scenario of what may happen in the next 40 years. 1The first payment is added on the investor’s 25-th birthday and first earns returns during “year 26”. The last payment is added on the investor’s 64-th birthday and first earns returns during “year 65”. The investor retires on her 65-th birthday. Thus, there are 40 total returns and the final retirement wealth is determined on the investor’s 65-th birthday. 2Recall that this procedure is valid under the assumption that returns are independently and identically distributed (i.i.d.). In other words, we assume that each of the return realizations computed in step one represents an equally likely draw from the same distribution of possible returns. 1/2 3. Using the simulated return path, compute the investor’s wealth at age 65. 4. Repeat steps two and three 1, 000 times. The most efficient way of doing so in Excel is to use a data table. An example is contained in the Excel file ”Monte Carlo Simulation Example” on Blackboard. QUESTIONS: A Report the mean and standard deviation of the portfolio returns computed in step one. B Report the mean, standard deviation, 25th and 75th percentiles, minimum, maximum as well as a histogram of the 1, 000 values you generated for the wealth at age 65. Interpret each of these statistics, i.e. explain in words what they tell you in the context of the example. C Assuming a 50/50 mix of both assets, what amount would the investor need to save annually such that her real retirement wealth at at least \$1m with a probability of 75%? Assume that the employer matches the investor’s contribution. [Hint: (1) To find this number, create an input cell for the annual savings (sum of the investor’s and her employers contributions) and use trial-and-error to determine the required amount. (2) The number you find will only be approximate because of simulation noise – that is ok!.] D Assuming annual savings of \$10,000 (including the employer’s contribution), what mix of the two assets ensures that the investor’s savings amount to \$1.5m on average? How do the standard deviation and the minimum savings change in this case relative to the baseline scenario of a 50/50 mix?[Hint: (1) To find the necessary mix, create an input cell for the asset mix and use trial-and-error to determine the required amount. (2) The number you find will only be approximate because of simulation noise – that is ok!.]

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FIN 421 Assignment 1 100% Correct Key
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QUESTIONS: A Report the mean and standard deviation of the portfolio returns computed in step one. Answer: The 50/50 portfolio has a mean of 4.4% and a standard deviation of 10.2%. B Report the mean, standard deviation, 25th and 75th percentiles, minimum, maximum as well as a histogram of the 1, 000 values you generated for the wealth at age 65. Interpret each of these statistics, i.e. explain in words what they tell you in the context of the example. Answer: The distribution of wealth at age 65 has the following (approximate) sample moments: mean = \$1, 080, 585, st. dev. = \$484, 469, 25th percentile = \$755, 066, 75th percentile = \$1, 316, 201, minimum = \$243, 678, maximum = \$3, 700, 469. Because the exact values depend on the random numbers drawn, you will get slightly different numbers. Your results should be close for the mean and standard deviation, but may differ quite a bit for the minimum and maximum. When you repeatedly press F9, Excel will simulate new random numbers, a...
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