The quiz is closed-book. You don't need a calculator; you can write the answer in terms of unevaluated formulas, such as (F/P,i,N).

Assume the inflation rate is zero unless the question says it isn't. Assume all interest rates are rates per year, compounded annually on the year-end balance.

1. What is an isoquant?

   It's a graph of one variable versus another, showing the pairs of values that give the same present worth. It can be used in sensitivity analysis.

2. The weekly costs of a company's salaried staff, rent, and equipment depreciation is $55,000. The company produces floppy disks for PC's. Each disk costs $0.50 to produce, and sells for $1.60. At what weekly production rate does the company break even?

   Breakeven when

   Fixed costs = Number-of-Units * Contribution/Unit

   So number of units = 55,000/1.10 = 50,000.

3. What is a leverage? If you own a company, would you rather have a small or a large amount of leverage?

   It's the ratio of debt to assets. In general you'd prefer low leverage, but if your company is growing quickly, high leverage will concentrate the growth in your share of the equity. Of course, this works in reverse if the company is shrinking.

   Questions 4-6 are about inflation. The inflation rate is f%, and on January 1, 1998, a real dollar and an actual dollar are worth exactly the same amount. (You'll need to know the difference between real and actual dollars in each question; since the two words are easily confused, it may help to remember that real dollars are also known as `constant dollars' or `now dollars', while actual dollars are also known as `then-current dollars' or `then dollars'.)

4. On January 1, 1998, a loonie falls out of your pocket and rolls under the couch. You find it a year later. When you find it, it is worth:

   1. A real dollar
   2. An actual dollar
   3. Both (a) and (b).
   4. Neither (a) nor (b).

   An actual dollar

5. On January 1, 1998, you deposit $100 in a bank that pays interest at f%. A year later, you have in the bank:
   1. 100 real dollars
   2. 100 actual dollars
   3. Both (a) and (b).
   4. Neither (a) nor (b).

   100 real dollars

6. If you deposit a dollar on Jan 1, 1998, in a bank that pays interest at i%, the value of your investment ten years in the future, in actual dollars, will be:
   1. (1+f)¹⁰
   2. (1+i)¹⁰/(1+f)¹⁰
   3. (1+f)¹⁰/(1+i)¹⁰
   4. (1+i)¹⁰
   5. (1+i)¹⁰(1+f)¹⁰

   d

7. If you deposit a dollar on Jan 1, 1998, in a bank that pays interest at i%, the value of your investment ten years in the future, in real dollars, will be:
   1. (1+i)¹⁰(1+f)¹⁰
   2. (1+i)¹⁰/(1+f)¹⁰
   3. (1+f)¹⁰/(1+i)¹⁰
   4. (1+if)¹⁰

   b

8. In 1981, a change occurred in the way Revenue Canada calculates depreciation on assets. What was this change? Did the change increase or reduce the tax burden on a typical company?

   In 1981, the rules changed so that only 50% of the cost of an item bought in any given year could be included in the depreciation calculation for that year, the other 50% being included in the depreciation calculation for the subsequent year. This increased the tax burden on all companies affected.

9. Distinguish between a slack variable, an artificial variable, and a surplus variable. Which of these must be zero in the final solution to a well-posed linear programming problem? How is this ensured?

   A slack variable symbolises the unused fraction of a resource. An artificial variable doesn't symbolise anything, but is introduced into a simplex problem to provide a suitable starting point. A surplus variable symbolises the extent to which a decision variable exceeeds its lower bound in a `greater-than' constraint. Artificial variables must be zero at the end of a problem, and this is ensured by the Big M method: the artificial variable is put into the objective function with a large penalty attached, so that optimization forces it to zero.

10. In the simplex method, what is a basic variable?

    At each stage of the simplex method, we set n variables to zero, and solve for the remaining m variables. The variables we chose to set to zero we call non-basic, the others we call basic. At the first step of the simplex method, all the slack variables are basic, all the decision variables are non-basic; however, as the solution proceeds, any variable can become basic.

11. Why is the Monte Carlo method called ``the Monte Carlo method''?

    Monte Carlo is a famous gambling resort in Europe; both the resort and the method employ random number generation as a key activity. [This knowledge is not part of the syllabus, and won't be examined in the final.]

12. Give two advantages and two disadvantages of being a corporation rather than a single proprietorship.

    See notes, ``The Company".

13. If I invest $2,000 at 12% interest, compounded annually, about how long will it be before I have $8,000?

    This can be solved using the (P/F,i,N) formula, but it's quicker and easier to use the Rule of 72: an investment doubles in 72/i years, where i is the interest rate.

14. My MARR is 25%. I am considering two proposals: Proposal X involves an immediate expenditure of $10,000, which will yield a large single influx of cash in ten years time. Proposal Y involves the same initial expenditure, and an annual influx of cash every year for the next eight years. Both proposals have the same present worth. If my MARR is reduced to 20%, which proposal will I favour?

    This is the most important question on the quiz, and you should make sure you have a firm grasp of the principle involved. The principle is that as the interest rate grows, future cashflows become less and less visible. (You could envisage the interest rate as a kind of fog that obscures your view of the future.) So if at the poor visibility of 25% interest, the ten-years-distant payback from X looks as big as the next-eight-years payback from Y, then when the visibility improves to 20% interest, the payback from X will loom larger still.

15. Three men are having breakfast: the CEO of an insurance company, the CEO of a mortgage company, and a retired CEO. The morning paper has a headline: ``Inflation rates expected to rise sharply!''. For whom is this good news, and for whom is it bad news?

    It's definitely bad news for the retired CEO, unless his pension is inflation-linked.

    I had been thinking of the insurance company as a life insurance company, in which case inflation is good news, since life insurance companies usually promise a fixed sum on death, and that sum can now be paid out in inflated dollars. But the question leaves open the possibility that the company insures against fire or theft; in which case, if the amount to be paid out is set at replacement cost, the news is nether good nor bad. (Since the replacement cost in real dollars will be constant.)

    For the mortgage company, it's bad news: they've lent a sum of money in the past, and they're now going to get paid back in deflated dollars. (If the mortgages are variable-rate, they may be able to put the rates up to cushion the effect.)

16. Company T is using linear programming to solve a problem in resources allocation. Company T makes fruit pies, of two kinds: apple and apricot. Their production is constrained by limitations in the supplies of apples, apricots, pastry, and the capacity of their oven. They make a dollar profit on each apple pie, two dollars on each apricot pie. This is the final form of the linear programming tableau, where x₁ stands for the number of apple pies produced, x₂ stands for the number of apricot pies, and x₃ to x₆ are the slack variables corresponding to the contraints imposed by the named limitations, in the order listed. The objective function, Z, is in dollars. From the final form of the tableau, answer the following questions:

    	Coefficient of							Right Side
    Basic Variable	Z	x1	x2	x3	x4	x5	x6
    Z	1	0	0	0	0	1	4	100
    x1	0	1	0	0	0	4	3	20
    x2	0	0	1	0	0	3	0	40
    x3	0	0	0	1	0	2	3	100
    x4	0	0	0	0	1	5	6	200

    1. How many apple and apricot pies does the company make at the optimum point?

       This part of the question (and the other parts) make use of the features of the `Standard Form' for representing a simplex problem as a tableau: each row of the tableau contains exactly one basic variable, with a coefficient of 1, so that variable's value can be read directly from the right-hand side of the tableau; thus, x1 (apple pies) is 20 and x2 (apricot pies) is 40.

    2. What profit do they realise as a result?

       You can either answer this by substituting x1 and x2 in the formula for profit, or, more quickly, note that Z is the basic variable in the first line of the tableau, and is therefore equal to $100.

    3. What resources is it worth increasing, and what is it worth paying for an increase of one unit in each resource?

       From the bottom two lines of the tableau, we can see that x3 and x4, the slack variables associated with the supply of apples and apricots respectively, are both non-zero. I.e., when we're at optimum production, there are still crates of unused apples and apricots lying around the factory, so there's certainly no point in buying more of them.

       Looking at the top line, we see that the coefficients of x5 and x6 are 1 and 4 respectively. This indicates (from the discussion on `shadow prices') that it's worth paying $1 and $4 for an extra unit of the resources associated with these two variables (pastry supplies and oven capacity respectively.)

17. A project is analysed using the PERT method, and found to have a completion time of 15 weeks. A more careful look at the data shows that the time for one of the activities on the critical path will actually be a week longer than estimated, but one of the activities not on the critical path can be sped up by a week. Repeating the analysis on the basis of the new data will give a completion time of
    1. Exactly 16 weeks
    2. Between 15 and 16 weeks
    3. Exactly 15 weeks
    4. Impossible to come to any conclusion

    Exactly 16 weeks

18. What is a simplex?

    A closed, convex polyhedron in n dimensions. (The region bounded by the constraint inequalities in the graphical solution of a linear programming problem is a two-dimensional simplex; the simplex method relies on the mathematical properties of an n-dimensional simplex.)