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).