MBA Math Monday: Normal Distribution

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The normal distribution is essentially the familiar symmetrical bell curve that characterizes many phenomena.  Even when distributions of interest are asymmetrical, the normal distribution is central to sampling and confidence intervals that help guide efforts to make sense of massive data sets by working with representative data samples.  Working with the normal distribution highlights the importance of thinking about intervals measured by units of standard deviations away from the mean.

Hal Varian, Chief Economist at Google, was recently quoted in the New York Times from a McKinsey Quarterly interview as saying “that the sexy job in the next 10 years will be statisticians.”  Perhaps more on point for MBAs, Tom Davenport and Jeanne Harris are releasing “Analytics at Work: Smarter Decisions, Better Results” this month as a follow up to their influential 2007 book Competing on Analytics.  Understanding normal distributions is a core skill in developing the statistical numeracy required to apply analytics to strategy.

Exercise:

Suppose the daily customer volume at a call center has a normal distribution with mean 4,600 and standard deviation 950. What is the probability that the call center will get fewer than 3,400 calls in a day?

Click here to view a standard normal (z-score) table, if you know how to use it.

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Supply and Demand

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The market interaction of supply and demand is one of the classic business concepts that people encounter from school snack swaps to shopping at the mall, from start ups to too-big-too-fail banks, and from subprime real estate to U.S. Treasury bond auctions.

The concepts can be used qualitatively and quantitatively.   Like much in economics, supply and demand can be introduced equivalently in pictures or with equations. 

Here, we use equations for a situation involving a perishable agricultural commodity market.

Exercise:

Assume that the demand curve D(p) given below is the market demand for apples:

Q = D(p) = 270 – 15p, p > 0

Let the market supply of apples be given by:

Q = S(p) = 42 + 15p, p > 0

where p is the price (in dollars) and Q is the quantity. The functions D(p) and S(p) give the number of bushels (in thousands) demanded and supplied.

What is the consumer surplus at the equilibrium price and quantity?

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Journal and T-Accounts

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The first MBA Math accounting exercise explained that balance sheets provide a snapshot of a firm’s financial condition at a moment in time, with “balance” referring to the equality between the left side’s assets and the right side’s combination of liabilities and equity.  The second accounting exercise introduced transactions as the means by which the balance sheet changes over time.  As long as the balance sheet equation (assets = liabilities + equity) is maintained for each transaction then the new balance sheet that results from a large sequence of transactions will remain in balance. 

This exercise introduces journals and t-accounts as recording systems to handle the volume of transactions that companies generate.  Modern accounting recording systems are automated, of course, but putting pencil to paper with journals and t-accounts helps beginning students to internalize the logic of accounting.

Spending time in the accounting trenches working with transactions is critical to developing an informed understanding of the financial statements that MBAs will analyze in their classes and careers. 

Exercise:

Ruston Company
Balance Sheet
As of January 4, 2009
(amounts in thousands)
 
Cash 9,300 Accounts Payable 2,500
Accounts Receivable 5,000 Debt 2,300
Inventory 5,500 Other Liabilities 6,500
Property Plant & Equipment 15,900 Total Liabilities 11,300
Other Assets 1,400 Paid-In Capital 5,700
    Retained Earnings 20,100
    Total Equity 25,800
Total Assets 37,100 Total Liabilities & Equity 37,100

 

Transfer the journal entries to T-accounts for the transactions below, compute closing amounts for the T-accounts, and construct a final balance sheet to answer the question.

Journal amounts in thousands

Date Account and Explanation Debit Credit
Jan 4 Accounts Payable 8  
     Cash   8
  Paid money owed to supplier    
Jan 5 Property, Plant & Equipment 49  
     Cash   49
  Paid cash for machine    
Jan 6 Cash 70  
     Paid-In Capital   70
  Issued stock    
Jan 7 Cash 20  
     Inventory   16
     Retained Earnings   4
  Sold and delivered product to customer    
Jan 8 Cash 51  
     Debt   51
  Borrowed money from bank    
Jan 9 Inventory 14  
     Accounts Payable   14
  Bought manufacturing supplies on credit    
Jan 10 Cash 10  
     Accounts Receivable   10
  Received customer payment    

 

What is the final amount in Total Assets?

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Bonds

The MBA Math Monday series returns after a break since mid-July for classroom teaching.  The series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first-year curriculum.

As described in the first three MBA Math Monday finance exercises, quantitative finance builds incrementally.  The first and second finance exercises deal with converting a single amount of money at one point in time into a different single amount at a different time.  The third exercise, covering constant annuities, deals with converting multiple future cash flows into a single current cash flow.   

Bonds, which we cover in this lesson, combine a steady stream of small, periodic coupon payments with a single large payment in the last period.  The coupon payments, typically paid every six months, are a constant annuity and the single large payment is a future value. 

Companies and governments (local, state, and national) raise money by issuing bonds, typically for long-term projects.  The key to valuing bonds is determining the appropriate discount rate to reflect the chance that the borrower will default and fail to pay the borrowed money.

The bond market can turn quickly on companies or governments that appear vulnerable financially.  President Bill Clinton’s advisor James Carville famously captured the immense power of the bond market when he quipped that if he could be reincarnated he wanted to come back as the bond market.  Understanding bonds begins with the mechanics underlying this exercise.

Exercise:

What is the current value of a $1,000 bond with a 10% annual coupon rate (paid semi-annually) that matures in 5 years if the appropriate stated annual discount rate is 12%? 

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Linear Regression

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

Linear regression helps to identify the best line characterizing two sets of data.  Regression generally is used where one factor managers control, such as advertising, is believed to influence another factor of interest, such as sales.  The notion of causality can be wrong, however, so like much of statistics it is important to understand the tool and use it wisely.  Linear regression also quantifies the degree of linearity in a relationship, which you can see in a scatterplot by the extent to which the data create a line versus a scattered set of dots. 

Various nonlinear regression options are available at the click of a button in Excel, all with risks of incorrect assumptions and unwise overprecision. 

One need only look at the monumental financial consequences of failure to anticipate the end of rising house prices to understand that extrapolating the past blindly into the future can be disastrous.  Understanding linear regression is an important step in absorbing the value and limits of statistics.

Exercise:

Consider the following sample data for the relationship between advertising budget and sales for Product A:

 

Observation 1 2 3 4 5 6 7 8 9 10
Advertising ($K) 60 70 70 80 80 90 100 100 100 110
Sales ($K) 362 416 417 499 485 536 602 623 616 663


What is the slope of the “least-squares” best-fit regression line?

 

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Marginal Analysis by Calculus

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The first two economics exercises examined marginal analysis. which discovers a firm’s optimal production quantity and profit.  Marginal analysis problems can be posed equivalently in terms of tables, formulas, or charts.  The first exercise used data in tables.  The second exercise used formulas but without invoking calculus.  This exercise uses formula and calculus.

The calculus bit is the only aspect of this exercise that is different from the second one.  The key idea, which is explained in the full MBA Math course, is that the derivative of the total cost function is the marginal cost function.  The derivative examines the incremental change in a function (e.g., y = f(x)) given a small change in the underlying value (e.g., x).  That’s just what we want when we are considering the marginal cost, which shows how much total cost (e.g., TC = f(q)) has changed for a small change in quantity (e.g., q). 

Similarly, the derivative of the total profit function is the marginal profit function and the derivative of the total revenue function is the marginal revenue.  At least it’s consistent.

Scary as calculus may be to many incoming b-school students, the main idea is quite straightforward if you can keep your brain from locking up in fear.  And the mechanics of taking the derivative of a polynomial, which is a fancy name for the functions determining cost, are refreshingly simple, taking no more than 15 minutes to recover from the memory vault of lost ideas from high school.

Exercise:

Suppose a competitive firm has as its total cost function:

TC = 24 + 2q2

Suppose the firm’s output can be sold at $66 per unit.

Using calculus and formulas (but no tables) to find a solution, how many units should the firm produce to maximize profit?

 

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

 

MBA Math Monday: Income Statement

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The first and second MBA Math accounting exercises examined the balance sheet, which represents the sources and uses of a firm’s funds at a snapshot in time.  The first exercise focused on balance sheet structure and logic.  The second exercise focused on how transactions during a period in time must be appropriately processed to create a new balance sheet at the end of the period.

This exercise introduces the income statement, which is a financial statement that shows a firm’s revenues and expenses during a period of time, typically a quarter or a year.  The proverbial “bottom line” in business is the profit or loss at the bottom of the income statement.

Before you can interpret an income statement, you need to understand how its structure flows from revenue at the top to profit or loss at the bottom, with various categories of cost in between.

Exercise:

Suppose Lightspeed Industries has the following revenue and expenses (listed in alphabetical order) for 2008:

Revenues of $8,800,000
Cost of Goods Sold of $2,640,000
Depreciation Expenses of $1,200,000
Income Taxes of $1,452,000
Interest Expenses of $50,000
Other Expenses of $400,000
Sales, General, & Administrative Expenses of $880,000

Create an income statement with amounts in thousands

What is the value of Pre-Tax Income?

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Annuities

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

As described in the first and second MBA Math Monday finance exercises, quantitative finance builds incrementally.  Those exercises deal with converting a single amount of money at one point in time into a different single amount at a different time.  This exercise deals with converting multiple cash flows into a single cash flow.  With this step, and the issues that it raises about rates and timeframes, finance supports the efforts of business people everywhere to capture the current value of future financial prospects as well as the efforts of individual and institutional investors to make current investments that will generate future cash flows.

Constant annuities, which are a set of equal periodic payments, are incredibly common, especially in the credit markets.  If you’ve ever taken out a loan to buy a car or house or finance your education with student loans, you’ve sold an annuity.  You get money upfront and the lender receives a stream of constant payments from you.  The inability of the world’s largest financial institutions to correctly value subprime mortgage annuities was the trigger to the ongoing financial meltdown.

Exercise:

What is the present value of an annuity in which $300 is paid each year for 4 years, assuming a discount rate of 8% and the first payment is received one year from now?

 

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Probability

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

Probability is a language for dealing with uncertainty.  Like any language, it has structure and rules that take practice to internalize.

This exercise is at its core a simple logic challenge.

Uncertainty permeates business.  While qualitative reasoning is essential to clarify appropriate strategy and tactics, high quality decision making often requires familiarity with probability to weight the consequences of possible futures associated with various choices.

Exercise:

Let X be a discrete random variable. If Pr(X<5) = 2/9, and Pr(X<=5) = 5/18, then what is Pr(X=5)?

 

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).

MBA Math Monday: Marginal Analysis by Formula

The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.

The first MBA Math economics exercise explained that marginal analysis discovers a firm’s optimal production quantity and profit. Marginal analysis problems can be posed equivalently in terms of tables, formulas,or charts. The first exercise used data in tables. This exercise uses formulas but without invoking calculus.

Economics is tricky because it takes awhile to internalize the necessary chain of reasoning. Of course, it also has its own terminology (qth unit?) that is confusing at first. The goal is generally clear, at least in intro economics exercises. In this exercise, we want to know what level of production yields the highest profit. The challenge is to build a chain of reasoning from the problem as stated to the goal. 

Once you’ve got a clear solution path in your head, the rest is simple algebra.  Starting with algebra before you know where you’re headed is a common form of flailing for beginning (or returning) students.

Working with data in tables is more intuitive for many beginning students but, if you can accept that formulas represent the same information in shorthand, you can get more quickly to an answer with formulas. This one takes about 5 seconds once you know what you’re doing.

The last step in a marginal analysis is to introduce fixed costs to determine whether the optimal strategy results in a true profit or loss. This has been the Twilight Zone world of U.S. car companies recently. However much they’ve (debatably) been closing the gap in quality and marginal production costs, the industry-wide drop in demand combined with their massive fixed cost burdens put them in the position of working like hell to lose the least amount possible.

Exercise:

Suppose that you can sell as much of a product as you like at $92 per unit. Your marginal cost (MC) for producing the qth unit is given by:

MC=10q

If fixed costs are $350, what is the optimal output level?

Solution (with audio commentary): click here

Prof. Peter Regan created the self-paced, online MBA Math quantitative skills course and teaches live MBA courses at Dartmouth (Tuck), Duke (Fuqua), and Cornell (Johnson).