question info: a us portfolio manager holds a portfolio of European stocks, currently worth €300,000. The spot exchange rate is currently 1.10/€. The portfolio manager enters into a 3-month future contracton on the euro at 1.15/€. In one week, the value of portfolio is €320,000, the spot exchange rate is $1.20/€, and the futures exchange rate is $1.23/€. The question ask what’s the hedged return in dollar terms. The answer given in the back calculates as: loss on future: €300,000*($1.15/€ - $1.23€)=-$24,000 profit on unhedged portfolio: (€320,000*$1.20/€)-(€300,000*$1.10€)=$384,000-$330,000=$54,000 in net, the investor made a dollar return of (-$24,000+$54,000)/330,000=9.1% I’m getting really confused about this question, I originally thought I can use the formula provided in the book: (1+return on foreign asset) \* (1+ return on FX) - 1 to get the answer. So I went ahead and calculated the return on the asset in percentage(320,000/300,000=1.067) and since the curency is hedged at 1.15/€, I calculated the return on curency (1.15/1.1=1.045). Now I think I have all I need to solve this problem, so I went ahead and did 1.067 * 1.045 - 1= 11.15% but this isn’t the answer, can someone explain to me why this formula wouldn’t work?

edit: I had assumed different information, see my post below after reviewing the posted question

Sorry, I typed out the question, can you review and see?

ok I see where you are getting tripped up.

the approximation for a hedged return = unhedged return in LC + spot return - return on hedge

the unhedged return in this case is 20k/300k = 6.7%

spot return = 1.20/1.1 - 1 = 9.09%, return on future = 1.15-1.23 / 1.10 = -7.27%

adding together and dropping the cross product term the approx = 6.7 + 9.09 - 7.27 ~ 8.5%. the cross product term accounts for about 0.60%. (NOTE this is only a quick approx and the actual calc is explained below - conceptually think of it this way, do not calc tlike his on the exam)

your initial calc arriving at the 11.50% is not correct as you need to properly handle the spot and futures returns. note that the futures return in this case almost offsets the spot return (but not exactly).

in order to use your original equation your fx component is not 4.5%, it is 9.09%. and you need to add the return on the futures as computed as (Fo - F1) /So = - 7.27%. so (1.067)(1.0909) - 1 - 7.27% = 9.1%

ignore my comment before, I had assumed different information. the proof for the above augmented formula is as follows:

return with NO hedge = return on asset in LC + return on spot + (cross product of the two). this is what you have above but your spot return is incorrect.

with the hedge we must add in the return on the hedge as calc’d by (Fo - F1)/So. note it is So that is the denominator and not Fo since the return on hedge = (PoFo - PoF1)/PoSo where Po is notional hedged and PoSo is the domestic currency value and note the numerator is the DOMESTIC $ return on the hedge.

does this make sense?

to summarize my long post the components for a hedged return are as follows:

the unhedged return of the ptf by comparing the current LC ptf value in domestic currency to the initial domestic currency position. This takes into account investment return AND spot returns. this is the 54k in your original post divided by $330k

the return on any hedged position which is the change in futures rates multiplied by the initial LC hedged and then divided by the DOMESTIC value of the hedge at inception (so LC notional * initial spot). this is the -24k in your original post divided by $330k (notional hedged was E300, spot was 1.1)

You then add these returns together. Alternatively, you could compute the spot return and LC return of the ptf separately and there would be three returns you add

I guess I’m confused as why would you use the spot rate a week later $1.20/€ instead of the froward rate you have “locked” in at $1.15/€ when you calculate your return for the currency portion, isn’t the whole purpose of hedging is that you lock at a rate and it doesn’t matter if the rate fluctuate you will be having the return you desired? So even a week later spot rate at $1.20 is irrelevant since you “locked” yourself in at $1.50/€? Also does this mean the formula of Rdc=(1+Rfc)(1+Rfx)-1 only applies to unhedged foreign investment? And if hedging is involved, you need to add another piece to this equition to calculate the return on the forward contract?

it was locked in at 1.15 E/$ 3 months later. you are however calculating the hedged position One month later.

So you have to

a. take impact of the spot rate + portfolio growth into account

= 320 * 1.2 - 300 * 1.1 = 54

b. you sold the notional amount (to hedge the portfolio) when the forward rate was 1.15. Now the forward rate is 1.23

so -300 * (1.23-1.15) = -24 = loss on forward position.

entire portfolio has gain of 30

gain = 30 / (300 * 1.1) = 9.1%

U r a genius, thx for explaining!