Problem 1 (simulation of a replacement problem)
The lighting director of a theater is worried about the maintenance and replacement of five floodlights. They fail according to the number of weeks they have been installed and used. The probability that a bulb still functions after it has been used for t
weeks (i.e., it is presently in its (t
+ 1)-st week of use) is denoted by P
). The numerical values are shown in Table 14.9
. In addition, the single-digit random numbers associated with the survival events are shown in the last row of the table.
Conditional survival probabilities and associated random digits
Whenever a bulb fails, it has to be replaced immediately, as “the show must go on.” Changing a bulb individually is expensive, as scaffolding must be put up. The entire process costs $350. On the other hand, changing all five bulbs once regardless if they still work or not costs $800.
The director ponders two replacement policies: either replace the bulbs only when they actually fail, or, alternatively, in addition to failures during the week which have to be attended to immediately, change all bulbs every 3 weeks regardless if they still work or not. It should be pointed out that even if multiple bulbs fail during the same week, they may do so at different times, so that this case has to be treated and paid for as individual failures.
Use the following random numbers to generate specific instances of survival and failure:
83638 51597 70322 35984 03933 30948 36142 72865 63348 28024
then display for each light its age in a given week, the random number, and an indication, if a bulb works during any given week (W
), or if it fails (F
). Consider, for instance, Light 4. In week 1, its age is 0 as the bulb is new. According to Table 14.9
, the random digits associated with not failing are 1–9, and since the random digit is 3, it will symbolize proper functioning. This means that in the beginning of week 2, Light 4 is of age 2. The next random digit is 9, which, for a bulb in week 2, means failure. This means that during week 2, bulb 4 will be replaced and its age in week 3 will again be 0. The process continues in this fashion for all bulbs. The result of this replacement policy is that 16 bulb replacements are necessary for a total cost of $5,600.
Simulation given only individual replacements
Simulation given individual replacements in addition to triweekly group replacements
The simulation for the second policy is shown in Table 14.11. Here, we use the same random digits as before. As an explanation of the numbers in the table, consider Light 1. It works during weeks 1 and 2, but fails during week 3, when it has to be replaced during the week. At the beginning of week 4 (indicated by an “a” in the leftmost column), a group replacement is made, at what time the bulb of Light 1 is replaced again, even though it was just replaced individually during the previous week.
It turns out that this policy requires only 12 individual replacements for a total of $4,200, plus three total replacements for $2,400 for a grand total of $6,600. This is more than the costs of individual replacement alone, making the former strategy preferable.
Problem 2 (evaluation of investment strategies via simulation of stock prices)
The manager of an investment company manages a specific fund. There are $1,000,000 available for investment and three stocks, currently priced at $17, $59, and $103 per share, respectively, are considered for that fund. Planning is made on a weekly basis and any amounts that are not invested will be kept in a short-term money-market account that pays 0.01 % per week.
The manager considers three investment strategies. The first strategy would be to keep all of the money in the short-term account. This is the benchmark strategy. The second strategy will invest 50 % of the available money at the end of any week that has seen at least a 2 % increase in value, and will sell at the end of any week that has seen a decline in stock price, provided that a gain can be realized. Otherwise, the stock is held until a gain can be made. The third strategy is to invest 50 % of the available money in a stock whenever its price declines, and it will be sold as soon as a gain can be realized. The manager will not purchase new shares of a stock that is still held.
Stock prices are thought to follow two overlapping trends. On the one hand, their value will be determined by the “state of the economy,” (measured by the relative value of the currency, unemployment figures, manufacturers’ receipts, and similar factors), and, on the other hand, by stock-specific factors. The change of the state of the economy is denoted by Δ, while the changes of the standings of the three industries are Δ1
, and Δ3
, respectively. The stock prices are then thought to be the sum of Δ and Δ j
for stock j
= 1, 2, 3. The probabilities of these changes are denoted by P
), and P
). These probabilities have been observed and are displayed in Table 14.12
for changes in the overall economy, Table 14.13
for the specific case of the first industry, and Table 14.14
for the second and third industry. In each table, double-digit random numbers # are listed that are associated with the individual changes Δ and Δ j
Probabilities P(Δ) and random numbers #
Probabilities P(Δ1) and random numbers #
Probabilities P(Δ2) and P(Δ3) and random numbers #
In addition, we will use the random numbers shown in Table 14.15
. They are read row by row from left to right.
The task is to evaluate the different investment strategies of the investment manager.
We first use the random numbers to simulate the states of the economy Δ, followed by the simulation of the states of the three different industries Δ1
, and Δ3
. This allows us to compute the stock prices. All of these computations are shown in Table 14.16
Simulation of stock prices
Strategy 1: Leaving the entire amount of $1,000,000 in the short-term account for 10 weeks will net us $1,001,000.45, or a 0.1 % gain.
Strategy 2: In week 1, none of the stocks has increased by at least 2 %, so that we keep our entire amount in the short-term account. By the end of week 2, we have $1,000,200.01. Since Stock 2 increases by 3 % in week 2, we purchase it with half of the available money, i.e., $500,100. At the price of $61.38 a share, we obtain 8,147.6051 shares. The first time we can realize a gain is at the end of week 8, at which point we sell the shares at $61.99 a share for a total of $505,070.04. This money is kept in the short-term account for 2 weeks, resulting in a payoff at the end of week 10 in $505,171.06. The remaining $500,100 that were not invested in week 2 will remain in the short-term account, resulting in $504,114.83 for a total of $1,009,285.89 or an increase of 0.929 %.
Strategy 3: By the end of week 1, Stock 3 has declined in value, which leads the investor to invest half of the available money in that stock. The $1,000,000 has appreciated due to its investment in the short-term account for 1 week, so that 1,000,100 are available, half of which ($500,050) are invested in Stock 3. Each share costs $101.97, so that 4,903.8933 shares are purchased. Since the shares will never exceed that value again during the 10 weeks, they will not be sold.
The remaining $500,050 are left in the short-term account for 2 weeks until the end of Week 3, when they have appreciated to $500,150.02. As Stocks 1 and 2 decreased in value in Week 3, half of the available amount is invested in each. (Note, by the way, that Stock 3 decreased in Week 2, but since we still hold shares of that stock, we do not invest in it again). The sum of $250,075.01 is invested in Stock 1, which costs $16.66 per share, so that we obtain 15,010.5048 shares. We hold them until the end of Week 8, when their price increases to $16.83, which gives us $252,626.80.
Back to the end of Week 3, we invested $250,075.01 in Stock 2 at $60.77 a share, so that we obtain 4,115.1063 shares. We sell these shares at the end of week 5 for $61.38 each, resulting in $252,585.22. We hold this money in the short-term account until the end of Week 8, when the investment in Stock 1 is liquidated. By that time, we have $505,212.02. Since none of the stocks declined in Week 8, we hold the amount in the short-term account for a week, resulting in $505,262.54. We are now at the end of Week 9. During Week 9, we observed all stocks declining. As we still hold Stock 3, we cannot invest in it, so that we invest the entire remaining money in Stocks 1 and 2 in equal parts. For the 252,631.27 invested in Stock 1, we obtain 15,163.9418 shares, while for the same amount invested in Stock 2, we obtain 4,116.5271 shares of Stock 2.
Since none of the stock prices increases during Week 10, the account by the end of the planning period consists of 15,163.9418 shares of Stock 1, 4,116.5271 shares of Stock 2, and 4,903.8933 shares of Stock 3. The total value of the portfolio is thus $982,989.70, for a loss of 1.7 %.
Comparing the three strategies, it appears that the second investment strategy is best.