Here are my example problems.

**Chapter 12.**

**Problem 1**

Grant Steel Company has a mine that will produce a total of 100,000 pounds of steel: 40,000 pounds of copper at the end of the first year and 60,000 pounds of steel at the end of the second year. Extraction costs are always $0.20 per pound. The current forward prices are $0.80 per pound for a one-year contract and $0.60 per pound for a two-year contract. The annually compounded risk-free rates are 4 percent for one-year zero-coupon bonds and 6 percent foe two year zero-coupon bonds.

40,000 pounds copper in year 1

60,000 pounds copper in year 2

$0.20 per pound extraction cost

$0.80 per pound one year forward price

$0.60 per pound two years forward price

4% one year zero coupon risk free rate

6% two year zero coupon risk free rate

What is the mine’s present value?

A risk-free zero-coupon bond payingF1Q1 - K1 in year 1 and a second risk –free zero-coupon bond paying F2Q2 - K2

PV = F1Q1 – K1 / (1 + r1) + F2Q2 – K2 / (1+r2)

Mine Value = [ $.80(40,000) - $.20(40,000) ] / (1 + .05)1 + [ $.60(60,000) - $.20 (60,000) ] / (1 + .06)2

PV = $44,285

**Problem 2**

Pinto Iron Ore Mining’s Australian mine will produce 85 million pounds of iron ore one year from now if economic conditions are favorable. Pinto’s Manager forecasts two possible outcomes for the iron ore prices then: $0.60 per pound if demand is low and $1.20 per pound if demand is high. The year 1 forward price is currently $0.70, implying that a forward contract has a negative future cash flow of - $0.10 per pound next year if demand is low and a positive future cash flow of $0.40 per pound if demand is high. The risk free one year interest rate is 4 percent. The extraction costs are $0.90 per pound; hence if demand turns out to be low, the firm will shut down the time. What is the value of the mine?

Year 0 Year 1

**VALUE ?**

Scenario 1

Cash flow = $0

Scenario 2

Cash flow = $25.5 million

= 85,000,000 ($1.20 – 0.90)

Scenario 1 (low iron ore price = $0.60 / pound): Here the iron ore mine will shut down and be worth zero.

x ($0.60 – $0.70) + y (1.04) = $0

X = pounds of copper purchased forward

Y = dollars invested in zero-coupon bonds today that matures in one year

Scenario 2 (high iron ore price = $1.20 per pound): In this scenario, the iron ore min will be profitable. It will earn $0.30 per pound of iron ore mined and hence it can pay to produce at maximum capacity.

The cash flow in this scenario is $25.5 million = 85,000,000 ($1.20 – 0.90).

The same equation that the same tracking portfolio also yields $8,500,000 if iron ore prices are high is

x ($1.20 – 0.70) + y (1.04) = $ 25,500,000

Simultaneously solving the equations for scenarios 1 and 2 gives the tracking portfolio

x = 42,500,000 pounds of iron ore received from a one-year forward contract

y = $ 40,86,538 invested in zero-coupon bonds.

The value of this tracking portfolio is $40,86,538. Thus, the iron ore mine must also have a value of $ 40,86,538.

**Problem 3**

**Leverage Effect:** A case where leverage increases the price/earning ratio.

Reference: Example 12.9, Page. 447

Assume that Cordell Machinery issues $100 million in debt at the beginning of the fiscal year at a rate of 4 %, and equity is decreased by the same amount through a repurchase of 10 million shares at $ 10 each ( assume that market value equals book value). Note that no taxes and thus no response in stock price per share to increase in leverage. How does the debt issue affect the company? How an increase in leverage can affect Cordell Machinery’s price/earnings ratio.

**Answer:** As a result of debt issuance, resulting in a $100 million increase in liabilities and a $100 million decrease in equity. Here is the expected performance of Cordell in the following year:

Assets $800,000,000

Liabilities $300,000,000

Equity $500,000,000

Number of share outstanding 50,000,000

Equity per share $10 = 500,000,000 / 50,000,000

Expected net income = 16,000,000

= $20,000,000 – (100,000,000) (.04)

Expected EPS $.32 = 16,000,000 / 50,000,000

**P/NI** 31.25 = $10 / $.32

**ROE** 3.2% = NI / P = $16,000,000 / $500,000,000

**Test Questions Reviewed**

**Question 1**

A U. S jewelry importer will buy 2000 watches from a Swiss manufacturer at the end of next year. Each watch is currently priced at 652 Swiss francs. The current exchange rate for Swiss francs is: one Swiss franc is worth 0.85 USD. If inflation is Switzerland totals 3% over the next year that watches will then each sell for an appropriately greater number of Swiss franc. If over the next year the U.S. experiences no inflation and real exchange rate for Swiss francs remains unchanged, how many U.S dollars can importer expert to pay to the nearest dollar?

Formula = (number of watches x price in Swiss francs ) x exchange rate for Swiss francs

Answer is 1108400 = 2000x 652 x .85

**Question 2**

An electronic company that uses a substantial quantity of gold entered a forward contract for the gold it is about to use one year ago when gold sold at $630 per ounce with an agreed forward price of $621 per ounce. Presently the price of gold in the spot market is $ 917 and the company needs to determine the appropriate price to use in estimating its costs and setting its product selling price. What price should it use?

It should use $ 971 for estimation its costs and setting its product selling price. We must use $ 971, because we must consider the opportunity cost.

**Question 3**

A firm you are analyzing has two divisions, an onshore drilling operation and an offshore drilling operation. The onshore division accounts for 41% of the firm. You know that the firm’s price to earning to earning ratio is 26 times. Furthermore, you know that typical onshore drilling companies have a price to earnings ratio of 25 times. Estimate to one decimal place of accuracy the price earnings ratio for the firm’s offshore drilling operation.

Answer is 26.7

**Question 4**

• Compute the risk-neutral probability for two stages, one high-demand and the other low-demand. A mine can produce 80 million pounds of copper one year from now. The two possible copper prices then are $0.92 per pound under high demand and $0.3 per pound with low demand. The 1 year ahead forward price from copper is now $0.77 per pound. The risk free 1 year interest rate 5%. Extraction costs are 0.75 per pound. The firm can shut down without cost if needed. Remember the equation for finding pi when you have zero cost forward prices is in chapter 7. The high demand risk risk neutral probability to two decimal places of accuracy is:

Answer: 0.758 + 0.02

**Chapter 22**

**Problem 1**

Kakhi a U.S. based company has a change to enter into the London Market, but the CFO is concerned about the currency risk of such an undertaking. The forecast of estimated sales in London market is £ 3 million (worst-case scenario) or £ 6 million (best-case scenario) over the next 9 months. The possibility that each of these scenarios will occur is equal. The CFO wants to hedge the expected values of these sales, but is not sure which hedging instrument to use.

The following information is provided, and the task is to recommend the best method of hedging, i.e. what is the highest dollar amount you can lock in today.

Current US$/£ spot rate = US$1.36/£

Current forward rate for currency exchanged 9 months from today = US$1.50/£

9-month US$ LIBOR is 6.3 percent per annum

9-month £ LIBOR is 12.6 per cent per annum

To hedge expected value £6m x 0.5 + £3m x 0.5 = £4.5m

MM Borrow £ convert to $ invest $ repay £ use $

4.5/(1.126)9/12 =4.117£

4.117£ x 1.36$/£ = 5.59883$

5.59883$ x (1.063)9 /12 = 5.8613$ money market hedge amount

Repay 4.5£ from earnings

4.5 x 1.50 = 6.75$ Forward contract is best

Able to compare money market & fwd contract

Size to hedge only the sure £3 million, use option for more

MM Borrow £ convert to $ invest $ repay £ use $

3/(1.126)9/12 =2.7445£

2.7445£ x 1.36$/£ = 3.7325$

3.7325$ x (1.063)9/12) = 3.9075$ money market hedge amount

Repay 3£ from earnings

3£ x 1.50$/£ = 4.5$ Forward contract is best

**Problem 2**

What is the T –year futures price for gold delivered one year from now, with a risk free interest rate of 12 percent per year?

Given: Gold trades at $500 an ounce

Equation: Fo = So (1+rf)T

Where

Fo = futures price

So = today’s spot price of the underlying investment

rf = annually compounded yield on a T-year zero coupon bond

$560 = $500(1.12) per ounce of gold

**Chapter 23**

**Problem 1**

Estimate Bond Price Changes with DVO1

A bond position has a DVO1 of $500. Its yield is currently 12 percent. Estimate the change in the bond position if the yield drops to 11.9 percent.

Equation: ∆ P = -DVO1 x (∆bp)

Where

∆ P = the change in bond’s price

∆bp = the interest rate change (in basis points)

Using the above equation a drop in basis points gives a price increase of

∆ P = - 500 x (-10) = $5000

**Problem 2**

The DVO1 of a Portfolio of Bonds

The DV01 of a 30 year U.S Treasury bond is $.25 per $250 face amount

The DV01 of a 10 year U.S Treasury note is $.05 per $250 face amount

The DV01 of a 5 year U.S Treasury note is $.15 per $250 face amount

Compute the DVO1 of a portfolio that has $7 million (face value) of 30-year bonds, $9 million (face value) of 5-year notes, and (a short position) – 15 million (face value) of 10 – year notes.

Answer: Sum of the products of (a) the face amounts over $250 and (b) the corresponding DVO1s per $250.

The result is

DVO1 = $ 7 million / $250 ($.25) + $9 million / $250 ($.05) - $15 million / $250 ($.15)

= $ 7000 + $1800- $9000 = - $200

The negative DVO1 means that the position’s value increases when interest rates rise.