Statistics/Simulations- Please see attached

TO HEDGE OR NOT TO HEDGE?

Megatron, Inc. is a large computer hardware company based in Atlanta, Georgia. Megatron manufactures a variety of computer products rang-mg from color printers and scanners to modems and computer screens. In 2014, Megatron had net earnings of $367 million based on revenues of $2.1 billion. The breakdown of revenues by geographical region is shown in Table 1.

Region Revenues ($ million) Percent of all Revenues (%) United States 1,050 50 Europe (Euro) 420 20 Great Britain 336 16 South America 84 4 Asia 210 10

Table 1: 2014 revenues at Megatron from various regions of the world.

Because 50% of Megatron’s revenues are from countries outside the U.S., a significant portion of the company's cash-flows are denominated in foreign currencies. Megatron relies on sustained cash-flows generated from domestic and foreign sources to support its long-term commitment to U.S. dollar-based research and development. If the dollar were to strengthen against foreign currencies, Megatron's revenues (and hence its earnings) would be adversely affected, and the company's ability to fund research and other strategic initiatives would be adversely affected as well.

At the end of 2014, there was widespread internal concern among managers at Megatron about a potential strengthening of the dollar in 2015 against the Euro (EU) and the British pound (BP), and its implications for Megatron's cash-flows. Ramesh Srinivasan, CEO of Megatron, decided to explore the use of currency put options in order to decrease the company's exposure to the risk of currency fluctuations. He asked Bill Sullivan, vice president of finance at Megatron, to look into the problem of currency exchange rate fluctuations and to make a recommendation whether Megatron should hedge its revenues from Europe and Great Britain using put options.

Currency Put Options

In 2014, Megatron had revenues of 322.5 million Euros, which works out to be $420 million, using the exchange rate of 1.3026 US$/ EU ($420 million = 322.5 million X 1.3026). If the dollar were to strengthen against the Euro, then the exchange rate would decrease from 1.3026 US$/ EU to some lesser value, and consequently the value of Megatron's European revenues would be diminished. For example, if the exchange rate were to decrease to 1.2144 US$/EU, then the value of 322.5 million Euros would only be $392 million ($392 million = 322.5 million x 1.2144), which is $28 million less than if the exchange rate were to remain at 1.3026 US$/ EU.

Megatron can address the risk of a future strengthening of the dollar against the Euro by purchasing Euro put options. A put option on the EU (Euro) rate is an option to sell Euros at a guaranteed dollar price. We now explain how a put option works.

A put option on the EU rate is defined by the expiration date, the strike price (k), and the cost (c). Such a put option on the EU rate gives the holder of the option the right (but not the obligation) to sell Euros at the strike price k at the expiration date of the option. The expiration date is a fixed future date, such as one year from now. Let EU1 be the random variable that is the EU rate at the expiration date of the option. Note that EU1 is indeed a random variable, because the EU rate at the future expiration date is an uncertain quantity. Clearly, if the strike price k is greater than the EU rate at the expiration date (EU1), then the holder will exercise the option, and he/she will then have a payoff of k — EU1. Otherwise, the holder will not exercise the option, and his/her payoff will be zero. The cost c of the put option is simply the amount of money that the put option costs to obtain. Therefore, the net payoff of a EU put option with strike price k and cost c is

1 1

1

if Net Payoff

if k EU c EU k

c EU k − − ≤

= − >

This expression can also be written as

1Net Payoff max( ,0)k EU c= − −

Here we see that the put option net payoff is itself a random variable, because it is a function of the random variable EU1, which is the future EU rate at the expiration date. We also see in this formula that the put option net payoff is also a function of the strike price k and the cost c.

Let us see how the formula for the net payoff works out. Suppose that the current EU rate is 1.3026 US$/EU, and that an investor buys a put option today on the EU rate with an expiration date of one year from now, a strike price k = $1.24 US$/EU, and cost c = $0.0274. If the Euro were to depreciate in one year to EU1 = 1.20, then this option would have the following net payoff:

1Net Payoff max( ,0) max(1.24 1.20,0) 0.0274 $0.0126.k EU c= − − = − − =

If, instead, the Euro were to depreciate in one year to EU1 = 1.00 US$/EU, then this put option would have a net payoff of

1Net Payoff max( ,0) max(1.24 1.00,0) 0.0274 $0.2126.k EU c= − − = − − =

If, however, the Euro were to depreciate in one year to EU1 = 1.28 US$/EU, then this put option would have a net payoff of

1Net Payoff max( ,0) max(1.24 1.28,0) 0.0274 $0.0274.k EU c= − − = − − = −

that is, a loss of $0.0274.

A Model for the Euro Rate

Foreign exchange rates are susceptible to potentially large fluctuations. For example, Table 2 shows the average EU rate (against the dollar) over the nine-year period from 2006 through 2014. The last column of Table 2 is the percentage change in the EU rate from the previous year, and is computed by the simple formula:

EU rate in year ( 1)Change from previous year 100* 1 . EU rate in year

t t

 + = − 

 

Based on the eight numbers in the last column of Table 2, we can compute the sample mean of the percentage change in the EU rate as well as the sample standard deviation of the percentage change in the EU rate. Direct computation yields that the sample mean of the percentage change in EU rate was 3.92% over the period from 2006 to 2014, and the sample standard deviation of the percentage change in the EU rate was 9.01%. Let REU be the random variable that is the future annual percentage change in the EU rate. As it turns out, the future change in the EU rate is more accurately modeled if we assume that the future change in the EU rate has a mean of zero, as opposed to estimating this mean based on historical data. However, the estimate of the standard deviation of REU based on historical data is usually quite accurate. Therefore, we will model the future change in the EU rate as the random variable REU obeying a Normal distribution with mean µ = 0.0% and standard deviation σ = 9.0%. The current EU rate is 1.3026 US$/EU. The EU rate next year, denoted by EU1, is then given by the formula:

1 1.3026* 1 .100 EUREU  = + 

 

Note then in particular that EU1, which is the EU exchange rate next year, is a random variable, since it is a function of the random variable REU.

Year Average EU rate (US$/EU)

Change from previous year (%)

2006 0.9828 2007 0.9124 -7.16 2008 1.0736 17.66 2009 1.1579 7.84 2010 1.0652 -7.99 2011 1.1084 4.05 2012 1.2570 13.40 2013 1.2788 1.73 2014 1.3026 1.86

Table 2: EU Rate in US$/EU from 2006 through 2014.

A Model for the British Pound Rate

Let RBP denote the future annual percentage change in the BP exchange rate. We can estimate the distribution of RBP using historical data analogous to that shown in Table 3 and using the same type of analysis as for the Euro. Doing so, we will assume that RBP obeys a Normal distribution with mean µ = 0.0% and standard deviation σ = 11.0%. The current BP rate is 1.434 US$/BP. The BP rate next year, denoted by BPI, is then given by the formula:

1 1.434* 1 .100 BPRBP  = + 

 

Note, just as for the Euro, that BP1, which is the BP exchange rate next year, is a random variable, since it is a function of the random variable RBP.

Moreover, one can also use historical data to estimate the correlation between RBP and REU, which we estimate to be 0.675. That is, we will assume that

CORR(RBP,REU) = 0.675.

Revenue under Hedging

For simplicity, let us first assume that Megatron is only interested in hedging the foreign exchange risk due to future fluctuations in the exchange rate for the Euro.

The marketing department at Megatron has forecast that the 2015 revenues of Megatron in Europe will remain at the same level as in 2014 (in Euros), which was 322.5 million Euros. If Megatron does not hedge against its foreign exchange rate risk, then Megatron's revenue in $million at the end of next year from Europe will be

1Unhedged Revenue 322.5* 322.5*1.3026* 1 ,100 EUREU  = = + 

 

where REU, which is the future percentage change in the EU exchange rate, obeys a Normal distribution with mean µ = 0.0% and standard deviation σ = 9.0%.

If, however, Megatron buys a number nEU (in million) of put options with strike price kEU and cost cEU, then the revenue in $million at the end of next year will be

Hedged Revenue = Unhedged Revenue + Number of Options * (Net Payoff of the put option).

This then leads to the following formula:

1 1Hedged Revenue 322.5* *max[( ,0) ]

322.5*1.3026* 1 *max[( 1.3026* 1 ,0) ]. 100 100

EU EU EU

EU EU EU EU EU

EU n k EU c R Rn k c

= + − −

   = + + − + −       

For example, suppose Megatron buys nEU = 250 million put options on the EU rate with an expiration date of one year from now, a strike price of kEU = $1.24 US$/EU, and at a cost of cEU = $0.0274. The current EU rate is 1.3026 US$/EU. If the change in the EU rate over the next year turns out to be

23.23%− , then the observed value of the random variable REU would be rEU = 23.23.− If Megatron did not hedge their revenues by buying any put options, then Megatron's total revenue in 2015 would be

23.23Unhedged Revenue 322.5*1.3026* 1 $322.502 million. 100

− = + =   

If Megatron had purchased nEU = 250 million put options on the EU rate, then their total revenue in 2015 would be

23.23Hedged Revenue Unhedged Revenue 250*max 1.24 1.3026* 1 ,0 0.0274 100

$322.502 $53.148 $375.650 million.

  −  = + − + −       = + =

Notice that even though the EU rate would have dropped by 23.23% in this case, the total revenue dropped by only 10.6%, that is, from $420 million to $375.650 million. This is what makes put options so attractive.

If, however, the change in the EU rate over the next year turns out to be 10.84%, then the observed value of the random variable REU would be rEU = 10.84. If Megatron did not hedge their revenues by buying any put options, then Megatron's total revenue in 2015 would be

10.84Unhedged Revenue 322.5*1.3026* 1 $465.626 million. 100

 = + =   

If Megatron had purchased nEU = 250 million put options on the EU rate, then their total revenue in 2015 would be

10.84Hedged Revenue Unhedged Revenue 250*max 1.24 1.3026* 1 ,0 0.0274 100

$465.626 $6.85 $458.776 million.

   = + − + −       = − =

The Hedging Problem

Bill Sullivan felt that the main source of foreign exchange risk in 2015 would be from fluctuations in the Euro and the British pound, because revenue from Europe and Great Britain represents the major portion of the company's revenue from foreign markets (a total of $756 million out of $1.05 billion from all foreign markets). Although Megatron’s marketing department has predicted that 2015 revenues in Europe and Great Britain will remain at their 2014 levels (in Euros and British pounds, respectively), the revenues in dollars could fluctuate due to uncertainty in foreign exchange markets. Ramesh Srinivasan and Bill Sullivan wanted to ensure, as much as possible, that possible dollar losses due to foreign exchange rates be limited to a maximum of $50 million.

Based on conversations with his CEO, Bill Sullivan decided to focus on minimizing the likelihood that next year's revenue from Europe and Great Britain would be less than $706 million, that is, $50 million less than the current revenue of $756 million. Bill Sullivan instructed his staff to obtain quotes on put options on the Euro and the British pound with one-year expiration dates. The results of his staff’s inquiries are shown in Tables 3 and 4. Tables 3 and 4 show the strike prices and costs of nine different put options on the EU and BP rates respectively.

Assignment:

Construct a simulation model to help Bill Sullivan select a hedging strategy using put options to minimize the risk of exchange rate fluctuations on the Euro and British pound.

Strike Price for EU (in $)

Cost (in $)

1.1 0.002802 1.18 0.012582 1.2 0.016776

1.22 0.021702 1.24 0.027422 1.26 0.034002 1.28 0.04159 1.3 0.064382

1.32 0.17171 Table 3: Strike prices and cost of nine different put options in the EU exchange rate, with a one-year

expiration date.

Strike Price for BP (in $)

Cost (in $)

1.10 0.000285 1.15 0.001318 1.20 0.003808 1.25 0.009133 1.30 0.018760 1.35 0.032937 1.40 0.052355 1.45 0.096025 1.50 0.159427

Table 4: Strike prices and cost of nine different put options on the BP exchange rate, with a one-year expiration date.