# CFAI Sample, Q18 - Return of foreign investment

You know the formulas: R_d = (1 + R_f) * (1 + R_currency) - 1 in an other place it says: R_d = R_f + R_curreny * (1 + R_f) In the text book, notations are different but these are obsiously the same equations. It helped me (and may be it would help you) to realize that : (1 + R_currency) = S1 / S0 with S1 and S0 being the spots rates at beginning of investment and at the end. When fully hedged, (1 + R_currency) = F / S0 and so: Unhedged : R_d = (1+R_f) * S1/S0 - 1 Principal only hedged: R_d = 1 * F/S0 + R_f * S1/S0 - 1 Fully hedged: R_d = (1+R_f) * F/S0 - 1 I come to the point : the CFAI sample exam, question 18. You may “remember” that the question is about calculating an Austrilian bond return in USD while currency is fully hedged. Data is the following: R_f = 8.50% F = USD 0.67 / AUD S0 = USD 0.69 / AUD So, the exact R_d = (1 + 8.50%) * 0.67 / 0.69 - 1 = 5.36% Formula used by CFAI is R_d = R_f + (F - S0) / S0. This formula is just an approximation of the valid formula that I gave above. The approximation is good when returns are relatively low but here we are talking about 8.50%, the approximation is not no good any longer! So, the right answer should be B) and not C) as mentionned. Happy to hear your comments. MH

mhannebert Wrote: ------------------------------------------------------- > You know the formulas: > R_d = (1 + R_f) * (1 + R_currency) - 1 > in an other place it says: > R_d = R_f + R_curreny * (1 + R_f) > In the text book, notations are different but > these are obsiously the same equations. > I don’t think your second equation is correct. The two equations above don’t give the same answers. It should read: R_d = R_f + R_currency * (R_f)(R_currency)

mwvt9 Wrote: > I don’t think your second equation is correct. > The two equations above don’t give the same > answers. > > It should read: > > R_d = R_f + R_currency * (R_f)(R_currency) Mwvt, I respect you much but I have to say that here you are wrong. My second equation is correct and yours is incorrect… You can rearange mine into R_d = R_f + R_currency + (R_f)(R_currency) which is probably the form you have in mind. MH

I used: hedged return = local currency return + premium (discount) on forward contract On this problem, the forward contract was not priced according to I-rate differentials, but since he was hedging his position, you just used the contract discount/premium. So your return was equal to the local currency return minus the forward contract discount.

Maybe I am. Help me out a bit and show me where I am wrong (I am not too good at math). So if the underlying asset returns 10 local return and the currency returns 5%: 1. R_d = (1 + R_f) * (1 + R_currency) - 1 R_d = (1.1)(1.05) -1= 15.5% 2. R_d = R_f + R_currency * (1 + R_f) R_d = 0.1 + .05 * 1.1 = 16.5% Or are the second two terms grouped? That must be my error, right? like this… R_d = .1 = (.05*1.1) = 15.5% 3. R_d = R_f + R_currency + (R_f)(R_currency) R_d = 0.1 + 0.05 + (.1)(.05) = 15.5%

dubbs your first equation is correct - it accounts for the straight currency gain, and also accounts for the currency gain on your appreciated asset. and having made it to level 3, you’re at a minimum “decent” at math.

ilvino Wrote: ------------------------------------------------------- > dubbs your first equation is correct - it accounts > for the straight currency gain, and also accounts > for the currency gain on your appreciated asset. > > and having made it to level 3, you’re at a minimum > “decent” at math. No it isn’t. mhannebert is right. I meant to put a + before the last term and put a mutliply sign instead.

I am trying to help here mhannebert, not be a pain in the ass. You obviously have a better handle on this than I do. When fully hedged, (1 + R_currency) = F / S0 and so: Doesn’t this assume that the hedge expires at the same time as the end of the holding period? If they hedge isn’t expiring at that time, would the conclusions below still be valid? Unhedged : R_d = (1+R_f) * S1/S0 - 1 Principal only hedged: R_d = 1 * F/S0 + R_f * S1/S0 - 1 Fully hedged: R_d = (1+R_f) * F/S0 - 1

I think I finally see what you are saying. I now remember the question too and recall getting a slightly different answer then them. It was like right between two of their answers. I will stop spamming you post now.

mwvt, that is OK. BTY, R_d = 0.1 + .05 * 1.1 = 15.5%, not 16.5%. You are right, there is the assumption that the futures expires at the same time (and no roll out during the holding period). If not, then everything is a mess, nothing holds true and I will move on to the next question! The equation “hedged return = local currency return + premium (discount) on forward contract” is *just* an approximation ! As said earlier, it is valid when return and premium / discounts are “low”… That was not the case in Q18.

mhannebert Wrote: ------------------------------------------------------- > mwvt, that is OK. > > BTY, R_d = 0.1 + .05 * 1.1 = 15.5%, not 16.5%. > LOL! I still don’t know how to group terms!

mhannebert, I have to say that you’re right because I ended up with the same answer as yours: 1) Invest A\$144.93 (which is \$100 / 0.69) today 2) Sell forward @ 0.67 per A today with A\$157.25mm (which is 144.93 * 1.085) 3) Expect bond to appreciate to A\$157.25mm in 1 yr 4) Convert to dollar A\$157.25 * 0.67 --> \$105.36 Answer is 5.36%

5.36% shall be correct. CFAI’s solution is wrong !