Webinar #1 - Log Return example problem

[dthigale]

As I watched Webinar #1 to review my math skills, I completed the log return problem differently than David did. I wanted to share this way because after all, we all do things differently and we all learn differently so maybe this will help someone out there.

We learned early on in the webinar that the formula for compounding continuously is FV = PV * e^r*n.

Our log return question (about one hour into the webinar if you want to find it) was "What is the log return of an asset that grows from $8 to $16 over 4 years?"

I simply filled in the formula with the information given.

FV = 16
PV =8
r = my unknown (what I am trying to find)
n = 4

16 = 8(e^4r) ---------> This creates an exponential problem that we discussed in the 10 minute log break.

First we must do a little algebra to get rid of the 8. So divide both sides by 8.

16/8 = 8(e^4r)/8
2 = e^4r ----------> Remember how we learned that ln and e "undo" each other.

If we add ln to both sides, it will "undo" the e. That means the ln and e on the right side cancel out.

ln 2 = ln e^4r
ln 2 = 4r ---------> Now back to algebra. Divide by 4.

ln 2/ 4 = rate per year

.173 = 17.3% is the rate per year.

Hope this helped someone out there! It sounded a LOT more confusing putting it in the forum than when I worked it out on my paper!!

Bridget

 

[David Harper CFA FRM]

Bridget,

Thanks for sharing your helpful way of solving this! I'd like to just note that, as we both end up dividing by (4), that illustrates the virtuous property of continous compound/discounting (i.e., log returns are continuous, by definition). As I originally solved for a 69.3% four-year log return (i.e., = ln2), you show how easy it is to create an annual log return: because your exponent is 4*r, you can divide by 4. To go from four years to annual is not so trivial with discrete returns, but these continous returns are "time additive:" 4r = r + r + r +r. This is their chief advantage. David

 

[dthigale]

David,

Is it safe to say that we could use the calculator ( input: N, PV, PMT and FV) to find out I/YR descrete returns: Most of Fixed Income problems

While to calculate continuous returns we use the exponentioan formula.: Most of Derviatives /Options problems.

Thanks.

 

[David Harper CFA FRM]

Bridget (or Dinesh?)

Yes, your division has historically been roughly correct. This has been due to the assignements:

* Tuckman (and now Fabozzi) tend to use discrete returns (in particular, semi-annual compounding under the assumption of semi-annual coupons) while
* John Hull is central in regard to derivatives/options and Hull, consistent with an academic treatment, tends to use continous (but not always, his swaps/CDS have occassionally both, which can be a source of confusion; e.g., as the instruments have discrete coupons, the question sometimes includes discrete assumptions that require conversion)

I am looking forward to the AIMs b/c I had recommended GARP give a little more prospective clarity (e.g., continous as a default) but I am not expecting it. There is no way around that fact that being facile with the conversion is the best strategy.

e.g., the exercises here:
http://www.bionicturtle.com/learn/article/fourteen_compound_frequency_conversion_practice_exercises_practice_frm_quan/

if you heard me on the webinar about this already, apologies, but i think too few candidates master this key building block (i.e., to be fluent in the conversion back and forth such that you are indifferent to the frequency)...

David

 

[David Harper CFA FRM]

Ohhh, thanks! Well, the good news is, i was going to recruit that smart poster, but now i don't need to !!! David