#### How to approach a quantitative exam, quiz or homework problem

- What is the quantity that you are looking for? Write it down.
- What are the different items regarding which you have information? Write them down.
- Write out different formulas that relate the quantity sought to one or more of the quantities which are given.
- Plug in the numbers for the unknowns in each formula; if you don't have any of these unknowns, try to compute them from the other data that is given, iterating between the previous step and this one. Once we can plug in numbers for all the unknowns in the formula, we go on to the next step.
- Once you have all the unknowns in the formula except the quantity that you are looking for, solve the resulting equation.
- If there is more than one unknown, try another formula.
- If none of the formulas lead to an equation in a single variable, look for other formulas.
- Go over your work again.

#### Examples:

**Calculating Portfolio Betas**: Chapter 11, Problem 15, p. 364, Ross, Westerfield and Jordan, 3rd. ed. (used for FIN 301)

You own a portfolio invested 20 percent in Stock Q, 40 percent in Stock R, 25 percent in Stock S, and 15 percent in Stock T. The betas for these four stocks are 1.10, 0.95, 1.40 and 0.70 respectively. What is the portfolio beta?Let us apply the suggested steps in order:

1. The desired quantity (from the last sentence) is the portfolio beta.2. The information given is of two kinds: a) the proportion of money invested in each of the four stocks, and b) the betas of the four stocks.

3. Clearly, we need a formula that involves the portfolio beta. Look in Appendix B, Key Equations. In this case, no formula is given for the portfolio beta. So let's go to the chapter itself. Where might there be such a formula? Look at the section headings. Looking in order, we see that section 11.2 talks about portfolios; however, there is no discussion of betas at that point in the chapter. Going further, we see 11. 5 deals with diversification and portfolio risk; that section, too, has no discussion of betas. Next, in section 11.6, the third subsection is titled 'Portfolio Betas.' Bingo!

Unfortunately, no formulas for portfolio beta. On the other hand, at the bottom of p. 348, an example is worked out. By looking at that example, and at the second line in that subsection [A portfolio beta, however, can be calculated just like a portfolio expected return.], we can get the formula that we want. Comparing the worked-out example with formula 11.2 for portfolio expected returns (p. 337), we see that the required formula is:

Portfolio beta = (percentage of money in asset 1)x(Beta of asset 1) + (percentage of money in asset 2)x(Beta of asset 2) + ... + (percentage of money in asset

*n*)x(Beta of asset*n*).4. When we try to plug in the numbers, we see that all the required information is provided in the question. So we simply plug in the numbers:

Portfolio beta = (0.20)(1.10) + (0.40)(0.95) + (0.25)(1.40) + (0.15)(0.70) = 1.06.

**Using Return Distributions**Problem 24, Chapter 10, Ross, Westerfield and Jordan, 3rd. ed., p. 330 (Used for FIN 301)

Assuming the return from holding small-company stocks is normally distributed, what is the approximate probability that your money will double in value in a single year? What about triple in value?Let us apply our principles again: Step 1 asks us what we seek. In this case, we seek a probability; in particular, the probability that a portfolio invesrted in small-company stocks will double in value.

Step 2 asks us to write down the other pieces of information available. We have, in addition, the information that the return on this portfolio is to be assumed to be distributed normally. Hence we want the probability of a normally distributed variable taking on certain values.

Step 3 asks us to write down formulas. In this particular case, we did not generate formulas as such. However, in class, we considered the problem of how to compute the probability that the return on a given asset would fall within a certain range; but we did not talk about the probability of an investment doubling. Can we rephrase the question about doubling in terms of the random variable falling within a certain range? To do this, we have to think of what it means in terms of returns for an investment to double in value. If it is worth $100 at the beginning, we want it to be worth $200. But this simply means a return of 100%! In other words, what we want is the probability of the return being greater than, or equal to 100%. And tripling, therefore, is equivalent to a return of 200%!.

Now that we have this, we want Prob{Return >= 100%}. From our discussion in class, this is equal to the Prob{ d >= (100-E(R))/std. dev.}. To go further, we need the expected return and standard deviation of returns on our portfolio. These, we see from Table 10.10 are 17.6% and 34.8% respectively. This means that we must compute Prob{ d >= (200-17.6)/34.8} = Prob{ d >=2.37}. This equals 1 - Prob{ d <2.37}. This second probability can be read off from Table A.5 as .991 approximately. Hence the probability of the investment doubling in value is about 0.009%, or slightly less than 1 in a 100.

Note that the N(d) value for 2.37 cannot be found explicitly in Table A.5 and must be interpolated.

The probability of the investment tripling in value will be almost non-existent, since the relevant d-value becomes (200-17.6)/34.8 or 5.24, which is off the charts!

**Computing the Value of a Levered Firm Under M&M I with corporate taxes**

Chapter 15, Problem 20, Ross, Westerfield and Jordan, Chapter 15. (Used for FIN 301)

Clines Manufacturing has an expected EBIT of $2880 in perpetuity, a tax rate of 35%, and a debt/equity ratio of 0.5. Th efirm has $8,000 in outstanding debt at an interest rate of 7%, and its WACC is 10.6%. What is the value of the firm according to M&M Proposition I with taxes?Applying our principles (point 1), we note that the quantity that we are looking for is the value of the levered firm.

We then write down the other information that we have (point 2):

- EBIT in perpetuity, per year = $2880
- corporate tax rate (T
_{C}) = 35% - outstanding debt (D) = $8,000.
- debt/equity ratio (D/E) = 0.5.
- Interest rate on debt (r
_{D}) = 7% - WACC = 10.6%.

Going to point 3, we look for a formula that involves the quantity that we are looking for. Equation (15.3) tell us:

V_{L}= V_{U}+ T_{C}DFollowing the dictates of point 4, we try to plug in numbers, but we soon realize that, although we have T

_{C}, and D, we don't have V_{U}. Hence, we have to go back to point 3, and try to figure out V_{U}.Point 3 asks us now to find a formula involving V

_{U}. This is provided in the lower half of page 485, as a simple application of the present value of a perpetuity, recognizing that the after-tax cash flows to an unlevered firm are equal to EBIT(1-T_{C}):

V_{U}= EBIT(1-T_{C})/R_{U}Again, applying point 4, we come to a halt, realizing that even though we have information on EBIT and T

_{C}, we don't have information on R_{U}. Hence, we go back to point 3.We don't have an explicit formula in the text for R

_{U}. However, we do have formula (15.4) on page 486 that involves R_{U}:

R_{E}= R_{U}+ (R_{U}- R_{D})(D/E)(1-T_{C}).Applying the requirements of point 4, we find that we cannot identify R

_{U}from this formula, because we do not have R_{E}. So, it's back to point 3 again.Following point 3, we look for another formula for R

_{E}, and we see that we have such a formula in (14.6) on page 453 from Chapter 14:

WACC = (E/V)R_{E}+ (D/V)R_{D}(1-T_{C}). At this point, we find that we can plug in all the other pieces of information demanded by this formula, except for E/V and D/V. Fortunately, though, we can compute D/V and E/V from the information provided, viz. the debt/equity ratio. We know that V = D+E. Hence, D/V is simply D/(D+E). From the information that D/E = 0.5, we infer that D = 0.5E. Substituting for D everywhere, we discover that D/V = 1/3 = 0.33. E/V is, then, 2/3 or 0.67. Hence,

.106 = (0.67)R_{E}+ (0.33)(0.07)(1-0.35)Following point 5, we solve this equation to find that R

_{E}= 0.13625. We now go back and plug it into the last formula relating R_{E}and R_{U}:

0.13625 = R_{U}+(R_{U}-0.07)(0.5)(1-0.35). Solving the equation, we find that R_{U}= 0.12.This can then be used to find the value of the unlevered firm: V

_{U}= 2880(1-0.35)/0.12 = 15600.Going back to the formula for the levered firm, we finally discover that V

_{L}= 15600 + (0.35)8000 = $18400.#### How to solve an equation

An equation is simply a relationship between two or more quantities. Solving an equation means isolating one of these quantities on the left hand side. If the desired quantity that you want to solve for, is on the left hand side already, move all other quantities to the right hand side. This is done by undoing the operation that keeps those undesired quantities on the left hand side. Undoing an operation is carried out by performing the reverse operation on both sides of the equation.If the desired quantity is on the right hand side, bring it to the left hand side by undoing the operation that keeps it on the right hand side. Once you have the desired quantity on the left hand side and all other quantities on the right hand side, you have solved the equation.