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1. In 1202 AD, Leonard of Pisa, better known today as Fibonacci (of the Fibonacci sequence of numbers), published a famous book (Liber Abaci) on mathematics that can be described today as “math for merchants”. It showed how to use the Hindu-Arabic system of numbers to solve practical problems and consequently has been viewed as one of the first Western books to demonstrate the superiority of this system to that of Roman numerals.
In the book was the following problem that he described as “On A Soldier Receiving 300 Bezants For His Fief”:
A soldier is granted by the king an annuity of 300 bezants per year for his service, paid in quarterly installments of 75 bezants. The king later alters the payment schedule to an annual year-end payment of 300 bezants. The soldier is able to earn 2 bezants per 100 per month (over each quarter) with his money.
Fibonacci goes on to analyze the effect of the king’s change on the soldier’s annual income. Join Fibonacci in determining the change in bezants in the soldier’s effective annual compensation.

2. A bond has a face value of $1000 and pays coupons annually, with a coupon rate of 5% per annum. There is 1 year remaining before the bond matures, but the probability of repayment is generally known to be only 70%. In the event of default, investors believe bondholders will receive 50% of what they are owed. If the bond’s cost of debt (appropriate discount rate) is 10%, what is the yield to maturity of the bond? Assume the bond is traded in a competitive market.

4. Buzz Beer, a local distributor in Akron, Ohio, currently sells about 10,000 cases of beer per month at $4 per case, which is 15 percent of the Akron market. Buzz’s CFO Natarajan is arguing for a temporary price cut to attract a larger share of the market. He points out that if Buzz lowers beer prices from $4.00 to $3.80 a case, the firm will expand its sales volume by 50 percent. Natarajan estimates that the beer costs will remain at $3.50 per case at the volumes contemplated. Further, Natarajan recommends that the company stick with the lower price for two years and then raise the price to $3.90 per case. He believes that at this higher price they will be able to keep their new customers for the subsequent two years. At that time, a new policy may need to be considered. Ignoring all possibilities beyond Natarajan’s 4 year horizon, and assuming the appropriate discount rate is 1 percent per month, should prices be lowered? Base your recommendation on a DCF analysis.

5. You own a $100,000 portfolio of stocks X, Y, and Z. Your friend suggests that you include stock A in your portfolio. You are considering his suggestion, and decide to compare the historical returns of your portfolio with that of stock A over the past 5 years, as shown below.
                         Annual Returns
Portfolio (XYZ): 15%,    10%    -12%,    -5%,    25%
Stock A:          22%,    15%,   -16%,    - 10%,   27%

In trying to decide on buying A, you do a thought experiment: suppose you had bought stock A 5 years ago, including it in your portfolio, rebalancing your $100,000 so that you had $75,000 in X,Y,Z and $25,000 in stock A, how different would have been the mean (arithmetic) return and volatility (standard deviation) of returns of your $100,000 investment over the 5 years?   

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Practical Problems: Business and Finance
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