2012 AM session, question 8: needlessly convoluted solution

CFA Level III Derivatives
#1

I was reading through the solution to question 8 in the 2012 CFA Level III exam and found it to be needlessly convoluted; I wish that they wouldn’t do this.

For those of you who haven’t seen it, the question asks you to change your equity / fixed income exposure, change the beta of the equity, and change the duration of the fixed income, all with derivatives. The solution explains how to compute how many equity futures you need to change the equity exposure (the value of the equity portfolio), then how many you need to change the beta, then how bond futures you need to change the fixed income exposure (the value of the fixed income portfolio), then how many you need to change the duration.

Four transactions; twice as many as necessary. Furthermore, the explanations are likely to lead to more confusion than less.

First, you calculate how many equity futures you need to change the exposure. Then you calculate how many equity futures you need to change the beta. Apparently, when you sell futures in the first transaction it changes the exposure but leaves the beta unchanged, and when you sell futures in the second transaction, it changes the beta but leaves the exposure unchanged. To be clear: the explanation says that _you’re engaging in identical transactions with completely different results _.

Tell me that isn’t confusing.

Similarly, when you buy bond futures in the first transaction it changes the exposure but leaves the duration unchanged, and when you buy bond futures in the second transaction it changes the duration but leaves the exposure unchanged. To be clear: the explanation says that _you’re engaging in identical transactions with completely different results _.

Tell me that isn’t confusing.

And, of course, it’s all completely unnecessary.

The proper way to think of these transactions is that _when you change the equity exposure and change the beta, you’re changing the dollar beta _, and _when you change the fixed income exposure and change the duration, you’re changing the dollar duration _. One equity futures transaction to change the dollar beta , and one bond futures transaaction to change the dollar duration. That’s what you’re really doing, it’s simpler, and it’s not confusing.

Why can’t they just teach it that way?

#2

I agree. I also missed that question, just because I had not seen the way to solve it before.

#3

S2000 , if you don’t mind , could you do the problem the way you suggest , using less steps , and show the calculation steps?

I think the CFA way is clearer , because each step accomplishes just one part of the problem .

exposure change -> beta remains same , notional changes to accomodate new exposure

beta change -> notional remains the new value ( from prev. step ) , beta changes

#4

Equities:

original = 182 * 1.08 = 196,560,000

Target = 154 * 0.9 = 138,600,000

Futures = 129000*0.97 = 125,130

No. of futures = -463.19 --> answer -464

Bonds: Orig = 705,600,000; Target = 756,000,000; Futures = 793,100

No of futures = 756-705.6 / .7931 = 63.5 --> Answer +63

#5

thanks, cp

#6

Evidently I just did, under my pseudonym “cpk123”.

wink

Our original dollar beta is $180,000,000 × 1.08 = $196,560,000.

Our target dollar beta is $154,000,000 × 0.90 = $138,600,000.

The dollar beta for a futures contract is $129,000 × 0.97 = $125,130.

The number of contracts is (target – current) / futures = ($138,600,000 – $196,560,000) / $125,130 = -463.2.

Sell 463 contracts.

If you do, great. (Sincerely.) Always use the method that is clearest to you.

I think that an explanation that says first sell futures that will change the notional but don’t change beta, then sell futures that will change beta but not change the notional is confusing (to at least some candidates): you do the same transaction twice, but the first time it does one thing (changes notional) and the second time it does something completely different (changes beta).

I reiterate: if you don’t find that confusing, great! Others will, and I’m simply trying to help them past this confusion.

#7

Ha - it actually wasn’t confusing until you brought it up, but I see your point. The way they explained it was clear, but it should be obvious that you can’t just compartmentalize a portion of the equity porfolio without affecting the overall beta. Well put.

#8

Then my work here is complete.

#9

I recently took the test. I did it in the opposite order: Changed beta first and exposure later, arrived at values that were pretty close (463 vs 464 for stock futures and 64 vs 63 for bonds, due to rounding.) I seriously thought “this must be a coincidence, I just happened to get the right answers” so I gave myself a ZERO on the answer!

Thanks S2000 (and cpk) for exposing the logic behind the numbers. I did not even think about “dollar beta” akin to “dollar duration”.

Also note that your method needs LESS rounding (once instead of twice for each asset class) so the answer is better.

EDIT: BTW 1. do you have to multiply by 0.01 for “dollar beta” like you do for “dollar duration”? And 2. is there a “dollar delta”? If yes, can you show with an example?

#10

you actually do not have to do a 0.01 on the dollar beta in this example - because both numerator (for the Portfolio itself) and the denominator (for the futures) would both get multiplied by 0.01 and hence cancel out.

#11

There’s no harm in doing that, but not much benefit. Usually you’ll be interested in the change in your portfolio return (in dollars) for a 1% change in the market return, so you can put the 0.01 in the dollar beta, or in the Δmarket return with equal results. Note, too, that for this derivative stuff, it doesn’t matter because any such factor will cancel: it’s in the numerator and the denominator both.

Let me ponder that one. I have to review some Level II questions for a class tonight, a short Level III exam for Windsor on Friday morning, some Level III essay questions for a webinar for Singapore/Malaysia Friday and Saturday nights, and a Level III Mock exam for Toronto on Sunday. But I’ll get back to you.

#12

I knew they would cancel out, but when defining, do you define it like dollar duration (DD = MD or ED * MV of bonds * 0.01)?

I think you don’t have to multiply by 0.01 because for duration, the change in yield is 1% whereas for equity beta, the underlying change could be 1 (i.e. 100%). For example, assuming an Rf = 0, if the market returns 100% then a company with a beta of 2 would return 200%.

The underlying point is: fixed income duration is known to be valid for only a small change in yield, because of convexity effects. So a duration that makes sense for a 1% change in the interest rates is known to be invalid for a 100% change in interest rates. There is no such restriction on beta because to be honest, nobody knows how equity prices work, certainly not with the precision of fixed income. So for large or small changes, a linear relationship between the market risk premium and the company’s risk premium is as good as any other. There is no convexity (fixed income) or gamma (options) for the beta (equities.)

#13

For options, the following would work, when hedging (or changing duration for bonds):

For bond options:

Dollar delta

= delta * bond duration * MV of bonds

= option duration * MV of options

= delta * bond duration * price of bond * number of bonds

= option duration * price of options * number of options

= delta * dollar duration of bonds * 100

For stock options:

Dollar delta

= delta * MV of stocks

= MV of options

= delta * price of a share * number of shares

= price of option * number of options.

#14

Looks good to me.

My whole thought process on this is that there’s no difference (as far as risk goes) between a $100mm equity portfolio with a beta of 1.0, and a $200mm portfolio with a beta of 0.5, or between a $50mm bond portfolio with a duration of 6.0 years and a $40mm bond portfolio with a duration of 7.5 years. Losing a dollar is losing a dollar, either way.

#15

I guess what bothers me is that if you sell futures and someone asks what you did – did you reduce your portfolio’s value or did you lower your portfolio’s beta? – you cannot answer them, yet CFA Institute’s explanation tries to make it sound as if you can.

Ah, well.

#16

In CFA solution when the 222 contracts are sold to

reach beta goal

is it still a USD portfolio worth 154 million

Thanks

#17

Actually, it’s still a USD portfolio worth $182 million.

#18

That’s the problem, right there.

#19

But equity portion of portfolio worth less than 154 goal?

#20

Again, that’s exactly the issue: the equity portion of the portfolio hasn’t changed. You’ve taken the short position on some futures contracts and waved your hands (remember: I’m a magician; when I say stuff like this, it’s an expert, professional opinion): “_ These _ futures contracts reduced the value of my equity portfolio, but _ those _ futures contracts didn’t. _ Those _ futures contracts reduced the beta of my portfolio, but _ these _ futures contracts didn’t.”

Hogwash!

Taking the short position in (all of) the futures contracts changed the dollar beta of your portfolio, but didn’t affect the value at all: you still have a $182 million equity portfolio.