PROF. DAVID SCHREINDORFER FIN 421 – SPRING 2016 Assignment 2 Due: Wednesday, 01.27.16 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.

PROF. DAVID SCHREINDORFER FIN 421 – SPRING 2016 Assignment 2 Due: Wednesday, 01.27.16 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 calendar 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 generatedreturns using two alternative Monte Carlo simulation techniques. On Blackboard, you will find an Excel file containing historical net returns on the S&P 500, 3-month T-bills and the CPI 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 access 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. 3. Using the simulated return path, compute the investor’s wealth at age 65. 1The first payment is added at the end of ”year 24” and first earns returns in ”year 25”. The last payment is added at the end of ”year 64” and first earns returns in ”year 65” 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 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. What do these numbers mean in the context of the example? C Next, you will repeat the analysis using an alternative Monte Carlo simulation technique. Instead of drawing returns from the empirical distribution (step 2), assume that log returns follow a normal distribution and simulate from this distribution.3 Specifically, replace steps one and two above with the following: 1. Compute the annual real log return of the 50/50 portfolio for each year in the sample. Next compute the sample mean and standard deviation of these returns. 2. Generate a path of returns by drawing 40 times from the normal distribution with mean and standard deviation equal to the sample moments computed in the previous step. Assuming the sample mean and sample variance are located in the cells I3 and I4 respectively, a random draw can be generated with = NORMINV(RAND(),I3,I4) Convert the log returns to net returns before continuing the analysis with step three. i 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. Do these statistics differ substantially from the results of the first approach? [Hint: They should not.] ii Assuming that the investor chooses a 50/50 mix of the two assets, what amount would she 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 a 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!.] iii Think about advantages and disadvantages of the two simulation approaches (resampling vs. log-normal draws). For example, are there situations where one approach may work better than the other? [Hint: No need to hand part iii in!] 3We will simulate log returns rather than gross returns because normally distributed variables take on values on the whole real line. Gross returns are bounded below by zero, i.e. they cannot take on numbers lower than this threshold. If we simulate gross returns from a normal distribution, we may get some returns that don’t make sense because they are negative. 2/2

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