# Calculating the same exchange rate to make an investor indifferent

If we consider an American investor in 2009 with 160 million Dollars to place in a bank deposit in either America or the UK. The (1 yr) interest rate on bank deposits is 6 percent in the UK and 1 percent in USA. The spot rate (which is the immediate exchange rate) is 160 Dollars per UK Pound.

Lets say investor deposits their money in America. He would have around 160,000,000(1.01) = 161,600,000 dollars in 2010.

If said investor deposits their money in the UK, they would alternatively have around (160,000,000/160)*(1.06) = 1,060,000 Pounds in 2010.

What would the exchange rate have to be for the investor to be INDIFFERENT in where they invest their money for a potential return in 2010?

My calculations lead me to believe that I'd simply divide 161,600,000/1,060,000 = 152.45, but I know through trial and error that I can get closer very easily..
Because when I multiply (160,000,000/152.45)(1.06) I get like 1.11 million, which is not equal to 161,600,000/152.45 - which is 1.06 million. To be indifferent, these would have to come out to the same values

..so I'm just wondering how I would approach this. Thanks!

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(1) To be precise, you might want to change "What would the exchange rate have to be" to "What would the forward exchange rate (1 year) have to be" - at least that is how I understand your question – gnometorule Mar 2 '13 at 4:31
(2) I don't understand your last paragraph. You seem to calculate this right, but then comment "I know through trial and error that I can get closer very easily." What are you concerned about? Rounding errors? Continous compounding (which you didn't use in your interest calculation)? Investor preferences/utility functions (in which case you should mention them, but also be aware that this is likely not the best site then for this question)? – gnometorule Mar 2 '13 at 4:32
Because when I multiply (160,000,000/152.45)(1.06) I get like 1.11 million, which is not equal to 161,600,000/152.45 - which is 1.06 million. To be indifferent, these would have to come out to the same values. – tekman22 Mar 2 '13 at 4:36
To clarify, I assume it is your intent to have a very unrealistic exchange rate? (It obviously doesn't matter for the calculation, so long as we state what the assumption is.) I only ask since 1.60 is a realistic exchange rate as opposed to 160. – user23784 Mar 2 '13 at 4:43
Yes that is my intention, don't worry about it being unrealistic it's just for ease of calculating! Haha – tekman22 Mar 2 '13 at 4:44

There's some logical flaw in your reasoning, and, also from my comment (1) above, I'm not sure you completely understand the concept of forward exchange rates. For a reference, I suggest you have a look at this wiki page, and in there focus on the equation I retype from wikipedia:
$$(1+i_d) = \frac{F}{S} (1+i_f),$$ with the obvious notation as there, or, if you add the invested amount $A$ as in your question, $$A(1+i_d) = \frac{F}{S} A(1+i_f) \quad (1)$$ Let's walk through this briefly.

When you do "160,000,000(1.01) = 161,600,000", you calculate $A(1+i_d) := \alpha_1$.
When you do "(160,000,000/160)*(1.06)", you calculate $\frac{1}{S} A(1+i_f) := \alpha_2.$
You then correctly solve for $F$, from (1), using $\alpha_1$ and $\alpha_2$...and you are done. You used what was given to solve for the one unknown, $F$.

When you continue "(160,000,000/152.45)(1.06)", this relates a current amount, dividing it by a forward rate you backed out, then applying to it a current interest rate - so twice today, once tomorrow. It's not a meaningful calculation.

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Note that there are two exchange rates in your problem, the spot and 1 yr forward rate. To be indifferent, you need two alternatives to match:
1. Investing in dollar interest rate.
2. Converting to pounds, earning pound interest rate, and converting back to dollars.

Thus if F is the forward exchange rate, you need:
1.01 = 1/160 * 1.06 * F

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so 152.45 is right..it's just the FORWARD rate? – tekman22 Mar 2 '13 at 4:51
@Jtm22 In brief, yes. – Macavity Mar 2 '13 at 6:07