The two methods illustrated in Homework Assignment #1 are:

I. Forecast the guilder/dollar exchange rate for all future cashflows. Use these exchange rates to convert guilder cashflows into dollars. Discount the dollar and equivalent dollar cash flows at the risk adjusted discount rate for dollars of 16 percent to get the present values. Total to get the NPV for the project.

II. Discount the guilder cash flows at the risk adjusted discount rate for guilders and total to get the present value of the guilder cash flows. Convert the present value of the guilder cashflows into dollars using the known current guilder/dollar exchange rate. Combine this with the present value of the dollar cashflows to get the NPV for the project.

The net present value should be the same using either method. The internal rate of return is the discount rate required to make the NPV equal to zero. It should be the same using either of the two methods.

An Illustration of the Problems of International Investment From Richard Brealey and Stewart Myers, Principles of Corporate Finance.

Outland Steel has an investment opportunity in the Netherlands. Its cash flows in guilders and dollars are:

The riskfree interest rate in the U.S. is 8 percent and the rate of inflation is 5 percent. In the Netherland the riskfree rate is 9 percent.

Outland estimates that the appropriate cost of capital for this investment, considering the risks, is 16 percent dollar return.

Southern Cross Airlines

North Star Airlines, an American company, has the opportunity of creating an airline in the southern hemisphere serving Perth, Australia; Capetown, South Africa; and Rio de Janiero, Brazil. The project will require an initial investment of $10 million. The project involves some leases which run for seven years so the period of analysis is seven years. The following are the cash flows generated in the various countries and their currencies (U.S. dollars, Australian dollars, South African rand, and Brazilian reales) over the life of the project:

At time 0 the exchange rates are 1.4 A$/$, 4 R/$, and 0.5 Rl/$. In the U.S. the risk-free interest rate is 4.0 percent and expected to remain at that level over the life of the project. The expected rate of inflation in the U.S. over the life of the project is 1.0 percent. The risk-free rates in Australia, South Africa, and Brazil are 4.5 percent, 7.0 percent, and 20.0 percent, respectively. Estimate the expected rates of inflation in the three countries. Estimate the rates of changes in the exchange rates.

Exercises:

			   Cash Flows               

	    U.S.        Col          Chile        Arg    

Time      ($ mill)   (pesos bill)  (pesos bill)  (pesos mill)

   0       -15.0          0             0            0

   1           0        1.6           2.0          1.0

   2           0        1.6           2.5          2.0

   3           0        1.6           3.0          2.5

   4           0        1.8           3.0          3.0

   5           0        1.8           3.0          3.2

   6           0        2.0           3.0          3.5 

   7         1.0        2.0           3.0          4.0

At time 0 the exchange rates are 800 CP/$, 400 ChP/$, and 1.0 AP/$. In the U.S. the risk-free interest rate is 4.0 percent and expected to remain at that level over the life of the project. The expected rate of inflation in the U.S. over the life of the project is 1.0 percent. The risk-free rates in Colombia, Chile, and Argentina are 15 percent, 7.0 percent, and 10.0 percent, respectively. Estimate the expected rates of inflation in the three countries. Estimate the rates of changes in the exchange rates.

To run the program you turn on the computer and monitor and let it go through its startup routine. When it is ready to take input type E: to shift it from the C drive to the E drive. On the E drive there a subdirectory for this class entitled econ137b (no space between the econ and the 137B). Type cd econ137B to change the directory from the root directory to the econ137B directory.

Now type callprog and the program will run. Press ENTER to move through the program.

                          Profit if

Run         Profit      held to exp day     

Difference



 1         _________      _________         ________

 2         _________      _________         ________         

 3         _________      _________         ________         

 4         _________      _________         ________         

 5         _________      _________         ________         

 6         _________      _________         ________         

 7         _________      _________         ________         

 8         _________      _________         ________         

 9         _________      _________         ________         

10         _________      _________         ________         

11         _________      _________         ________         

12         _________      _________         ________         

13         _________      _________         ________         

14         _________      _________         ________         

15         _________      _________         ________         

16         _________      _________         ________         

17         _________      _________         ________         

18         _________      _________         ________         

19         _________      _________         ________         

20         _________      _________         ________         



Average    _________      _________         ________         

Value of a one year call option when the strike price is $60 as a function of the current price.

	   Current        Intrinsic         Value if

	   Price          Value             Held to Exp.

 1         _________      _________         ________

 2         _________      _________         ________         

 3         _________      _________         ________         

 4         _________      _________         ________         

 5         _________      _________         ________         

 6         _________      _________         ________         

 7         _________      _________         ________         

 8         _________      _________         ________         

 9         _________      _________         ________         

10         _________      _________         ________         

11         _________      _________         ________         

12         _________      _________         ________         

13         _________      _________         ________         

14         _________      _________         ________         

15         _________      _________         ________         

16         _________      _________         ________         

17         _________      _________         ________         

18         _________      _________         ________         

19         _________      _________         ________         

20         _________      _________         ________         

Value of a one year call option when the strike price is $60 as 

a function of the current price.





Use the Callval program to collect the data on present value.

Intrinsic value is the maximum of (Current Price - Exercise Price) and zero.

					    Present

	   Current        Intrinsic         Value if

	   Price          Value             Held to Exp.

 1         __0________      _________         ________

 2         _10________      _________         ________         

 3         _20________      _________         ________         

 4         _30________      _________         ________         

 5         _40________      _________         ________         

 6         _50________      _________         ________         

 7         _60________      _________         ________         

 8         _70________      _________         ________         

 9         _80________      _________         ________         

10         _90________      _________         ________         

11         100________      _________         ________         

12         110________      _________         ________         

13         120________      _________         ________         

14         130________      _________         ________         

15         140________      _________         ________         

16         _________      _________         ________         

17         _________      _________         ________         

18         _________      _________         ________         

19         _________      _________         ________         

20         _________      _________         ________

• 1. The current stock price, S
• 2. The exercise (or strike) price, X
• 3. The time until expiration, T
• 4. The volatility of the stock, o
• 5. The risk-free interest rate, r_f

The risk-premium of the stock, r_p, will also affect call value. The program CALLS is set up to allow to choose these variables. Use as a base case S=$65, X=$60, T=1 year, r_p=0.08, vol = 0.2, and r_f=0.03. The middle item for each case is the base case. You need to do the base case only once and enter the result for the middle item in each case. Hold all variables except one at these base case values and vary that one variable to levels above and below the base case. Sketch a little graph of the results.

   

Current Stock Price	Call Price	Delta
S		C/S
$60
$65
$70

		Delta      |

 S       C      _C/_S      |

$60     _____   _____      |

$65     _____   _____      |

$70     _____   _____      |___________________

			   60    65     70    S 

Exercise Price             |

			   |

 X       C       _C/_X     |

$55     _____    _____     |

$60     _____    _____     |

$65     _____    _____     |___________________

			   55     60     65   X

Time to Expiration         |

		   Theta   |

   T         C     _C/_T   |

11 months  _____   _____   |

12 months  _____   _____   |

13 months  _____   _____   |___________________

			   11     12     13   T

Volatility    o            |

		    Vega   |

 o       C        _C/_o    |

0.1     _____     _____    |

0.2     _____     _____    |

0.3     _____     _____    |___________________

			   0.1    0.2    0.3  o

Risk-free Interest Rate    |

		Rho        |

 rf      C     _C/_rf      |

0.02    _____  ______      |

0.03    _____  ______      |

0.04    _____  ______      |___________________

			   2%     3%     4%   rf

• 1. The underlying stock does not pay a dividend
• within the period of the option
• 2. The call option can be exercised only on
• expiration day
• 3. The risk-free interest rate is constant over the
• life of the option
• 4. All asset markets operate continuously
• 5. There are zero transaction costs
• 6. There are no taxes
• 7. All assets markets are perfectly efficient
• 8. Short selling is allowed and full use of the proceeds is permitted.

Use the B&S progam in the Econ137B subdirectory to compute the value of a call for the following data (which is the same as in Assignment Six) S=$65, X=$60, T=1 year, o=0.2, and r_f=0.03.

The middle item for each item below is the base case. You need to do the base case only once and enter the result for the middle item in each case. Hold all variables except one at these base case values and vary that one variable to levels above and below the base case. Sketch a little graph of the results.

Current Stock Price        |

                Delta      |

 S       C      _C/_S      |

$60     _____   _____      |

$65     _____   _____      |

$70     _____   _____      |___________________

			   60    65     70    S 





Exercise Price             |

			   |

 X       C       _C/_X     |

$55     _____    _____     |

$60     _____    _____     |

$65     _____    _____     |___________________

			   55     60     65   X



Time to Expiration         |

		   Theta   |

   T         C     _C/_T   |

0.917 yr   _____   _____   |

1.000 yr   _____   _____   |

1.083 yr   _____   _____   |___________________

			   11     12     13   T





Volatility    o-            |

		    Vega   |

 o       C        _C/_o    |

0.1     _____     _____    |

0.2     _____     _____    |

0.3     _____     _____    |___________________

			   0.1    0.2    0.3  o





Risk-free Interest Rate    |

		Rho        |

 rf      C     _C/_rf      |

0.02    _____  ______      |

0.03    _____  ______      |

0.04    _____  ______      |___________________

			   2%     3%     4%   rf





Gamma:  Gamma is the rate of change of the delta of an option per unit change in stock price; i.e.,





Current Stock Price        

	 Delta    Gamma   

 S       _C/_S    _Delta/_S  

$60-65    _____   

		   _______  

$65-70    _____     

			     

Perfectly Hedged Portfolios:

A portfolio can be created such that its value is unchanged when the price of the stock changes. Suppose Stock A has a current price of $60 and all of the other data of the base case apply. Consider a portfolio containing 100 written calls with exercise price of $60 plus H share of stock A, where H is equal to the delta of the option.

Price of Stock  Value of Stock  Value of Calls Value of Portfolio



    $60           _________        __________      __________

    $65           _________        __________      __________

    $70           _________        __________      __________

Newspapers give the price at which options are traded. These options may or may not fit the conditions under which the Black-Scholes formula is valid. In this assignment I want you to pick out a number of options and compare their market values with the values you get using the Black-Scholes program. The current price, exercise price, month of expiration and market value of the options are given in the newspaper listing. Options expire on the third Friday of the month of expiration. Use 3.5 percent for the risk-free interest rate. We do not have convenient estimates of the volatilities of the underlying stocks and so I want you use the average volatility of all stocks, which is 0.215. Then, by trial and error, find the volatility that make the Black-Scholes value equal to the market value. This figure is called the "implicit" volatility for the option.

Company Current Exercise Time to Volatility B&S  Market Implicit

Name    Price   Price  Expiration         Value  Price Volatility

_____   ______  ______ _______    _____  _____   _____   _______

_____   ______  ______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  ____    _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  ____    _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______   

______  ______ _______ _______    _____  _____   _____   _______

The procedure for generating prices is to use the rate of return, a random variable with a normal rate of return, to compute the price at one time period from the previous time period; i.e.,

p_t = p_t-1 (1 + r_t).

However in comparing prices at different time periods it is essential to deal in discounted values, thus

PVp_t = PVp_t-1(1+r_t)/(1+r_d).

The question is what is the appropriate discount rate r_d. Two possibilities are the risk-free interest rate and the expected value of r_t.

Complex Options A "long" Straddle is created by buying a put and a call on the same stock with the same exercise price and the same expiration date. A "short" straddle is created by selling (writing) a put and a call instead of buying them. If the exercise price of the call is higher than the exercise price of the put, the combination is fancifully known as a strangle.

A spread involves two of the same type of option (put or call) for the same expiration date but different exercise prices and one being a purchase and one being written. The usual case involves call options, say buying a call at an exercise price of $60 and selling a call with an exercise price of $70.

The terms bulls and bears have a special meaning in the stock market. A bull is someone who thinks the price of a stock is going to go up. A bear is someone who thinks it will go down.

A "bull" spread is a combination of options which will produce a profit for the holder if the price of the underlying stock increases. A "bear" spread is a combination that will produce a profit if the price of the underlying goes down.

A butterfly spread involves four options for the same underlying stock and same expiration date but at three different exercise prices X₁, X₂, and X₃, where X₁ < X₂ < X₃. To create a butterfly one buys a call option at an exercise price of X₁, sells two calls at an exercise price of X₂, and buys one at an exercise price of X₃. This would be called a "long" butterfly spread.

A variation on the butterfly combination involving four different exercise prices, X₁234, is called a condor. A condor involves buying one call at X₁, selling a call at X₂, selling another at X₃, and buying one at X₄.

Another combination involving four exercise prices is the box spread. A box spread is created by buying one call at X₁, selling one call at X₂, buying one put at X₃, and selling one put at X₄.

     ___________________       __________________________  

	    X1                        X1      

	    X2

     

	       straddle                strangle







     ______________________       __________________________  

	    X1     X2                   X1      

	    X2

   

	     bull spread                bear spread







      ______________________        __________________________

	  X1   X2   X3                  

	  X1    X2   X3   X4

	butterfly spread

                         condor spread

The purpose of this assignment is to gain an acquaintance with the various speculative devices by keeping track of some and computing the rate of return for holding them. The more complex ones are constructed as a combination of simpler ones. Use the options quotations given in the Wall Street Journal or other comparable newspapers.

I. Create a portfolio of blocks of 100 shares. Select any example of each of the following:

     A. Buying a call 



     B. Writing a call 



     C. Buying a put



     D. Writing a put



     E. Buying  shares



     F. Selling short shares

   

     G. Buying a straddle



     H. Writing a straddle



     I. Buying a bull spread



     J. Writing a bull spread



     H. Buying a bear spread



     I. Writing a bear spread



     J. Buying butterfly spread



     K. Writing a butterfly spread



     L. Buying a condor spread



     M. Writing a condor spread



     N. Buying a box spread



     O. Writing a box spread

The selection of the particular stocks and options should not concern you. The goal is to record the details of the options and compute the cost of acquiring the options. Use Khouri's method even though some elements of it are outdated.

Suppose interest rates are not constant over time but vary from year to year. For example, suppose the interest rate rises from 5 percent in the first year to 12 percent in the fourth year and then drops to 10 percent in the fifth year, as shown below in column B.

             Table One

  A        B           C          D

       Interest   Cumulative    Present

Time      Rate       Value    Value of $1    

  0                 $1.0000     1.0000

  1         5%       1.0500     0.9524

  2         8%       1.1340     0.8818

  3        10%       1.2474     0.8017

  4        12%       1.3971     0.7158

  5        10%       1.5368     0.6507

Column C shows the value $1 would have grown to earning those annual interest rates. For Year 1 the value is $1(1.05)=$1.05. For year 2 the value is $1.05(1.08)=$1(1.05)(1.08)=$1.134. So $1 at Time 0 is equal to $1.134 at the end of Year 2. Then the amount of money that would have to be invested at Time 0 in order to have $1 at the end of Year 2 is 1/(1.134)=0.8818. In other words, the present value (i.e. value at Time 0) of $1 at the end of Year 2 is $0.8818. This is also called the discount factor for payments received at the end of Year 2.

Part I: Use a spread sheet, such as Excel or Quattro Pro, to make the calculation of present values and discount factors for the following pattern of interest rates.

Time(yrs) 1  2   3   4   5   6   7  8  9  10  11  12 13  14  15

Interest 

Rate      6% 9% 10% 11% 13% 10%  8% 6% 4%  6%  5%  6% 5%  6%  6%

Consider Table One again. We might ask what was the average interest rate over two years, three years and so on. We want to know what constant interest rate over two years would give the same cumulative value of 1.134. The answer is: the interest rate r such that (1+r) 2  = 1.134. We can find r by taking the square root of both sides to find that 1+r=1.0649 and so the two year interest rate is 6.49 percent. Note that this is not exactly the arithmetic average of the interest rates in the first and second years. The average interest rates for the other periods are as follows:

               Table Two

  A        B           C          D

       Interest   Cumulative    Average  

Years      Rate       Value      Rate    

  1         5%       1.0500      5.00 %

  2         8%       1.1340      6.49

  3        10%       1.2474      7.65

  4        12%       1.3971      8.72

  5        10%       1.5368      8.89

  

Part II: Construct a corresponding table for the interest rates in Part I.

In Assignment Fifteen we had the interest rates that were expected to prevail in a number of years in the future and we computed the discount factors for each of those years. Here we have the discount factors and want to compute the expected future interest rates (these are often called the forward rates).

A pure discount bond is one that pays no interest before maturity. At maturity a pure discount bond pays its face value. Since the pure discount bond sold at a price below its face value the payment made at maturity covers the accrued interest. The ratio of the price of a pure discount bond to its face value is the same as the discount factor for the period of time until maturity. For example, if a $1000 face value pure discount bond which has two years to go before maturity sells for $800, then the discount factor is 0.8. From the discount factor we could solve for the yield rate r we need to.

If we have the discount factor for three years and for two years we can find the forward rate for the third year. Since

________1_________ = DF₃ (1+r₁)(1+r₂)(1+r₃)

_____1______ = DF₂ (1+r₁)(1+r₂)

the ratio of DF₂ to DF₃ is equal to (1+r₃).

Find the discount factors and from them determine the forward interest rates for years 1 through 7 from the following data.

      Pure Disc   Face   Discount  Forward

Year  Bond Price  Value  Factor    Interest Rate

 1      $952.38   $1000  ______    _______

 2      $865.80   $1000  ______    _______

 3      $773.04   $1000  ______    _______

 4      $702.76   $1000  ______    _______

 5      $650.70   $1000  ______    _______

 6      $608.13   $1000  ______    _______

 7      $579.18   $1000  ______    _______

 

Definitions: A repurchase agreement (usually called a repo) is an arrangement in which one party sells securities with a committment to rebuy them at later time. The most common form of such agreeements is for one day, which is called an overnight repurchase. The interest rate on these is called the repo rate. It is just slightly above the T-Bill rate.

A forward interest rate is the interest rate now for a loan over some period in the future. For example, the interest rate on loans starting on January 1, 1999 which have to be repaid on December 31, 1999 is the forward rate for 1999. A company may know that it needs $10 million for one year starting on January 1, 1999 and want to get a committment on that loan and its interest rate now. The interest rate it gets would be called a forward rate. The duration of a security is the weighted average of the times its return is paid. The weights are the present values of the payments. The maturity is the time until the last payment is made.

A pure discount bond is a bond that pays no interest but pays its face value amount at maturity.

The average interest rate over a period of time is the interest rate on a pure discount bond that reaches maturity at the end of that period. For example, the three year interest rate is the interest rate for a pure discount bond that matures in three years.

The the term structure of interest rates or yield curve is a graph that shows the relationship between the average interest rate over a period of time and the period of time to maturity.

Review Problem:

There is five year bond with a face value of $5,000 and a coupon rate of 6 percent. What are the payments made for this bond?

Year        1      2       3      4       5

Payment   ______ ______ _______ _______ _______

Given the following forward interest rates determine the discount factors for the different years and use them to compute the present value of the payments made for the above bond.

       Forward

       Interest   Discount              Present

Time      Rate    Factor     Payment    Value     

  1         5%    ______     _______    _______ 

  2         8%    ______     _______    _______  

  3        10%    ______     _______    _______

  4        12%    ______     _______    _______ 

  5        10%    ______     _______    _______

The same computations can be made using the average interest rates over different periods. If r_T is the average interest rate over T years then the discount factor for T years is

1/(1+r_T)^T.

Use the following average interest rates to compute the discount factors and verify that they are essentially the same as those computed using the forward rates.

           Average

	  Interest     Discount     

Maturity    Rate        Factor    

  1          5.00%      ______      

  2          6.49       ______      

  3          7.65       ______      

  4          8.72       ______      

  5          8.89       ______      

Find the term structure of interest rates; i.e., plot the average interest rate versus time to maturity.

The previous three assignments dealt with the concepts of forward interest rates and average interest rates over various periods, such as a two year rate, a three year rate and so forth. It was shown that you can get the averages from the forward rates and you can get the implied forward rates from the averages.

The term structure of interest rates is the relationship between the interest rate on pure discount bonds and their duration. In other words, the term structure of interest rates is the relationship between the interest rates on short term securities and and long term securities.

A swap is a transaction that trades one stream of financial obligations for another. The most common swap is an interest rate swap, but foreign currency swaps are also popular, and the market has made available a wide variety of other swaps. The standard interest rate swap, commonly known as a plain vanilla swap, is one in which a floating interest rate obligation is exchanged for a fixed rate obligation. Suppose a company has a $5 million debt with a five year maturity on which it must pay an interest rate that is adjusted to the one year T-note rate plus 2 percent. This means the company's interest payment is variable and it is exposed to an interest rate risk. It may prefer to have a known, fixed financial payment and so it would seek a swap in which some other party (called the counterparty) agrees to pay the floating rate payments in return for receiving a fixed rate payment.

In this assignment you compute the expected present value of the floating rate payments and compare it to the present value of the fixed rate payments. The future T-note rates are unknown but the forward rates are what the market expects the one year rates to be. Assume in the computation that the T-note rates + 2 percent are the market interest rates for the risk class of the company's debt, and use these rates to compute the discount factors.

Expected Present Value of the Floating Rate Obligation

       Forward  Forward   Expected

       T-note   T-note    Interest   Discount   Present

Year   Rate     + 2%      Payment    Factor     Value

  1      5%       7%      $350,000   0.93458    $327,102.80

  2      8       ___      ________   _______    ___________  

  3     10       ___      ________   _______    ___________

  4     12       ___      ________   _______    ___________

  5     10       ___      ________   _______    ___________

  

Suppose the swap counterparty wants an 11 percent fixed rate payment. What is the annual payment? What is the present value of the fixed rate payments? The fixed rate of 11 percent is a coupon rate and should not be used for determining the discount factors. Use the discount factors you computed in the above problem to make this computation. Since the payment is the same in each year all you have to do to get the present value of the fixed payments is to multiply the fixed payment times the sum of the discount factors.

Plain Vanilla Currency Swaps Suppose Company A in Germany has DM 25 million (Deutsche marks) and Company B in the U.S. has $10 million. Company A has a need for $10 million for three years and Company B has a need for 25 million marks for three years. The current exchange rate is 2.5 DM/$ so they could simply convert their currency for the one they need. But they both know that after the three years are up they will want to convert their foreign currency holdings back to their own currencies. If they simply exchanged their current holding they would be exposed to an exchange rate risk during their three projects. They may want a swap in which they trade their current holdings for foreign currency now and trade them back after three years. During the three years each party to the swap pays the interest that the other would be receiving. The interest rate in the U.S. is 10 percent and in Germany it is 8 percent. The interest rates in both countries are expected to remain constant over the three year period. The problem is to find out if the swap is fair to both parties and if not how much one part should pay the other to make it a fair deal for both.

The first thing to do is to project the exchange rate of marks for dollars. If the level of risk in Germany is about the same as in the U.S. and the tax rates are the same then the difference in interest rates reflects differences in the expected rates of inflation. A 10 percent rate in the U.S. and an 8 percent rate in Germany indicates that the rate of inflation in the U.S. is expected to be about 2 percent higher in the U.S. than Germany. This means the mark will appreciate in value relative to the dollar. More precisely the rate of change of the exchange rate is expected to be [(1+0.1)/(1+0.08)-1]=0.0185; i.e., the marks required to buy one dollar will decline 1.85 percent per year.

           Expected Cash Flows for Company A



			     Expected

			    Exchange  Equivalent Discount Present

Time Dollars      Marks       Rate       Dollars   Factor   Value

    (millions)   (millions)  (DM/$)     ($millions)  @10%  (mill)

  0   10             -25       2.500      0.000      1.000  0.000

  1   -1               2       2.455     -0.185      0.909 -0.168

  2   -1               2       2.410     -0.170      0.826 -0.140

  3  -1-10=-11     2+25=27     2.366      0.412      0.751  0.309

 

						 Total      0.001

The total of the present values comes out to 0.001=$1,000, but this is due to the rounding that took place during the computation. The exact answer is 0, indicating that the swap is fair for both parties.

Compute the net value of the swap for Company A in marks.

Compute the net value of the swap another way.

Evaluate a ten year swap of 50 million marks for $20 million when the interest rates are 12 percent for the U.S. and 10 percent for Germany. The current exchange rate is 2.5 marks per dollar.

Pineapple Computers has an investment opportunity in Germany. It requires an initial investment of $800,000 and gives a return in marks as shown below.

                                       Equiv $     Present

		  Cash Flow  Forecast  Cash Flows  Value

Year   (000s DM)  (000s $)    xr DM/$   (000s $)  (000s $)

   0       0       800        1.60       ______    ______

   1     500         0        ____       ______    ______

   2     600         0        ____       ______    ______

   3     700         0        ____       ______    ______

   4     700         0        ____       ______    ______

   5     700         0        ____       ______    ______

The riskfree interest rate in the U.S. is 6 percent and the rate of inflation is 3 percent. In Germany the riskfree rate is 9 percent.

Pineapple Computers estimates that the appropriate cost of capital for this investment, considering the risks, is 10 percent dollar return.

In order to analyze this project you will have to forecast the exchange rate of marks for dollars. The forward rate exchange rate for one year gives us the "growth factor" (1+g) for the exchange rate since:

E(S_DM/$)/S_DM/$ = F_DM/$/S_DM/$.

If F_DM/$/S_DM/$ is not known, we can find it because we know that it is the same as

E(1_L)/E(1_$)

which is the same as (1+r_L)/(1+r_$).

Once we know the growth factor for the exchange rate (1+g) then we can estimate the exchange rate new year by:

E(S_DM/$) = (1+g)S_DM/$.

For the second year in the future we apply the same growth factor to the estimate of the exchange rate next year, and so on to the third, fourth and beyond years.

Determine the NPV of the project.    __________________



Is the investment worthwhile?     ___________  What is the Internal Rate of 

Return (IRR) of the project?  ________



What should the appropriate cost of capital in Germany is be?  ____    

Compute the NPV in marks. _____________



What is the dollar value of this NPV converted at the current spot rate of 

DM/$? _____________  Is the project worthwhile? _________

________________________________________________________________



In the above problem what is the real rate of interest in the U.S.? 

_____________



If this is also the real rate of interest in Germany, what is the expected 

rate of inflation in Germany?  _________



If 10 percent is the risk adjusted cost of capital for Pineapple, what is 

the risk premium? ____________



If the same risk premium applies to the cost of capital in Germany and the 

riskfree rate is 9 percent, what is the appropriate risk-adjusted cost of 

capital for the marks cash flow?__________

     Expected Present Value of the Floating Rate Obligation

	       

       Forward  Forward

       T-note   T-note    Interest   Discount   Present

Year   Rate     + 2%      Payment    Factor     Value

  1      5%       7%      $350,000   0.93458    $327,102.80

  2      8       10       _500,000   0.84962    _424,808.84  

  3     10       12_      _600,000   0.75859    _455,152.32

  4     12       14_      _700,000   0.66543    _465,799.16

  5     10       12_      _600,000   0.59413    _356,478.95





				     Total     $2,029,342.07

10.674%   af=3.80235

Explanatory Notes for Some of the More Complex Options

Horizontal (Time) Spreads: The simultaneous purchase and sale of the same type of option (put or call) with the same exercise price but different expiration dates. Consider the purchase of a 90 day call and a 120 day call on a stock for which the Black and Scholes method is valid which is now selling for $100 with an exercise price of $120. The rate of interest is 12% per year and the standard deviation of the rate of return is 0.3. The o t variable is (0.3)(.25) = 0.15. The present value of the exercise price is 120/(1.12)^1/4= 116.65 so the ratio of current price to the PV of the exercise price is 0.86. The Black-Scholes table gives the value of the call equal to 1.3 percent of the current price or $1.30 per share. For a call option with a maturity of 120 days the values for the B&S table are .17 and .87. Interpolating in the table gives a value for the call option of 2 percent of current price, about $2 per share. If the second call option is written then the net value of the combination is ($2.00-1.30) or $0.70 per share.

If the current price of the stock is $120 then the ratio of the current price to the present value of the exercise price for the 90 day option is 0.97 so the value of the call is about 4.65 percent of 120 or $5.58. The value of the call sold is about 5.65 percent of 120 or $6.78. Thus the net value of the combination is $1.20.

For a current price of $140 the value of the 90 day call option is $24.36 whereas the 120 day call has a value of $25.20. The net value of the combination of buying the 90 day option and selling the 120 day option is $0.84.

We may also excess the value of the combination as a function of the price of the stock on the expiration date of the 90 day option. On that day the call option will have a value of zero for any price at or below the exercise price. At that time the other option will have 30 days(=0.0833) years left before expiration. Its value is determined from o t = 0.3 0.0833 =.087 and P/(PVX). At a current price of $100 this ratio is 0.84. The value is about 1.7 percent of share price; i.e., 17 cents. Since this is the option that was sold (written) the net value of the combination is -$0.17.

Swaps

A swap is a transaction that trades one stream of financial obligations for another. The most common swap is an interest rate swap, but foreign currency swaps are also popular, and the market has made available a wide variety of other swaps. The standard interest rate swap, commonly known as a plain vanilla swap, is one in which a floating interest rate obligation is exchanged for a fixed rate obligation. Suppose a company has a $5 million debt with a five year maturity on which it must pay an interest rate that is adjusted to the one year T-note rate plus 2 percent. This means the company's interest payment is variable and it is exposed to an interest rate risk. It may prefer to have a known, fixed financial payment and so it would seek a swap in which some other party (called the counterparty) agrees to pay the floating rate payments in return for receiving a fixed rate payment.

For a current price of $160 the value of the 90 day option is $42.72 and that of the 120 option is about $44.80 so the net value of the combination is