racemize

I think it makes sense. The problem I have with these types of answers is that I don't see a good way to verify the result. How can we be sure that is the cost of leverage versus the common for example? Do we assume the dividends are reinvested? If so, at what rate do they compound? Or if we were reinvesting, we have to make assumptions about what prices we reinvest at. I guess it is inherently an approximation at this point.

yeah, I think I'm at the point where it is close enough, unless there are major problems with my current solution.

I guess I could calculate what the total returns would be if we reinvested dividends and see if it ended up being txlaw's result. That could take a while though...

That being said, I'm not sure that txlaw's method incorporates the missed compounding of dividends either--it seemed like it was just adding the missed yield relative to the strike to the premium cost of leverage portion. I suspect his method is too high and ours is too low. I'm not sure how to overcome this issue. Perhaps call it close enough, and know that it falls a little short.

I thought your way was an iteration back on Eric's--shouldn't you use Eric's new way to calculate it? I don't think so. They way I do it makes the most intuitive sense for calculating leverage to me. I think if Eric's way is right, our cost of leverage should come out the same (plus or minus NPV effects). I think it might have to do with you guys using a compound interest rate. But my brain is fried at this point. You and Eric are approaching it in a very different manner than I am, so I don't think I can comment too intelligently as to the way you are doing it. Perhaps Eric can.

I guess we could apply some reinvestment rate to the dividends, but then you would have to apply it uniformly. Sounds like a nightmare.

I thought your way was an iteration back on Eric's--shouldn't you use Eric's new way to calculate it?

I do indeed have the suspicion. No idea what to do with it though. But I think your model also has the same mistake. Because you don't do anything with your dividend, you're also effectively just summing it. Instead, you could be reinvesting it in the stock in real time as it gets paid out. I agree, I meant for both of them.

The fact that it is definitely the intersection point of total returns should give us some confidence. I think...

I do indeed have the suspicion. No idea what to do with it though.

Ok, I'll do the same analysis as the prior post for this formula and see what it spits out: 17.75*1.x = 6.6+18.33 = 24.93 ----> 40.4% gross return This is exactly the same as my formula. I think we have a winner. Edit: Your formula is prettier/easier. sadface.

I see the logic behind what you are doing; however, (and I know this will be a shocker), I lean towards the calculation that I perform, which has the advantage of telling you the price point where the warrant was better than buying the common in total returns. Given the feedback I've gotten (or mostly lack thereof), I guess I'm the only person who thinks this way is valuable? I guess it doesn't matter, since I'm the one using it... Moreover, I don't really know how to use your number. For example, if I plug in values for your cost of leverage, then I will come up with a point where the warrant's returns are higher than the common's returns (since yours returns a higher cost of leverage). How would I then use that information in a meaningful way? Let me be more specific, and I'll adjust to what you just posted: Assumptions: GM stock price: 40.16 GM warrant price: 22.41 GM strike price: 18.33 Assumed dividends: 0.3 over the next 22 quarters, for a total of 6.60 in dividends Eric's method amount borrowed = 40.16 - 22.41 - 6.60 = 11.15 gross return needed = 18.33 / 11.15 -1 = 64.3% implied warrant price = 1.643*22.41 = 36.84 implied stock price at expiry from warrant = 36.84+18.33 = 55.17 total return of stock price given the implied stock price = (55.17+6.60)/40.16 = 53.8% As you can see, the implied stock price results in an outcome where the warrant returns more than the total return of the stock price. Thus, I do not think we have solved for the intersection point where the two returns are the same (I believe that was the point from you prior post)? Perhaps I just don't understand what your cost of leverage means at this point, if it isn't that intersection point. My method Calculated Future Stock Price where the returns are equal: 49.81 Total return of common at that price = (49.81+6.60) / 40.16 = 40.46% Total return of warrant at that price = (49.81-18.33-22.41)/22.41 = 40.47% (rounding) Thus, you would annualize 53.8% of gains, and I would annualize 40.46% gains for the cost of leverage. If yours is not where the total returns intersect, then what does it represent?

I believe that's exactly what I'm doing--finding the stock price at which the two vehicles have the same returns, which is the intersection point. It makes a lot of sense to me. The only way I know to calculate it with the dividends included is via the math I posted above. I use a spreadsheet for it, as I calculate for a lot of warrants/options at the same time.

Hi Hyten, if it trades below the price that comes out of the formula at expiry, then the common was better, and if it is above, then the warrant was better.

txlaw, that calculation looks like it will be at least close to the right cost of leverage. However, my answer comes in slightly lower (some of this has to do with your 1% rounding from 0.65% I think). Although I derived the formula earlier, here are the basics for what I am doing: I determine the common price at which the total return of the common matches the total return of the warrant, which I believe should be the equation that gives the same cost of leverage that Eric is calculating. However, it is possible that it is answering a slightly different question than Eric's, but returns very similar results. In any event, the initial equation is: Total Returns of Common = Total Returns of Warrant @ some future stock price, OR: (Future Stock Price - Current Stock Price + cumulative dividends) / (Current Stock Price) = (Future Stock Price - Strike Price - Current Warrant Price) / Current Warrant Price Solving for Future Stock Price gives you: Future Stock Price = (Current Warrant Price * Cumulative Dividends + Current Stock Price * Warrant Strike Price) / (Current Stock Price - Current Warrant Price) So, this is the number that tells you the stock price that makes the total return of the common = the total return of the warrant, from today's prices, based on the entered cumulative dividends. If I plug in your value of $1.20 per year, I get: Cumulative Dividends = 6.6 (22 quarters left @ 0.3 per quarter) Now, using Current Stock Price = $40.34 Current Warrant price = $22.57 Warrant Strike = 18.33 Future Stock Price = $49.99 Checking my work: Total Return for Stock = (49.99 + 6.6 - 40.34) / 40.34 = 40.3% return Total Return for Warrant = (49.99-22.57-18.33) / 22.57 = 40.3% return Annualizing that number = 6.25% cost of leverage. (from 0.63% with no dividend) Would appreciate any faults with the above, if there are any.

Yes, that's what I meant, haven't been getting enough sleep lately... I guess I'm wondering, if people stop buying the Leaps because they get too expensive (maybe?) then it stops being a floor for the warrants. Do you agree?

Yes, I was trying to figure that out this morning. I'm not sure I've reached any solid conclusions. Let's say the BAC has a $1 dividend and the leaps start having a cost of leverage of 7% due solely to the missed dividends. Would we expect it to trade at a higher cost of leverage than 7%? Perhaps, but I guess we could ignore it now for now. So if someone is looking at Leaps with a 7% dividend-adjusted cost of leverage, and the warrants are trading at 5% dividend-adjusted cost of leverage, then clearly the buyer would go for the warrants. (Unless they wanted less raw gains and a shorter duration). I guess I was thinking (and I think this is what you are saying) that the warrants should trade at least at the same cost of leverage as the Leaps? How much is there a possibility that the cost is just too high and people stop writing LEAPs? Probably, people would still be writing them in some amounts, but that's a lot of cost, since it is 7% for dividends plus any additional expected gains.

Interestingly, a floor also gets established to support the cost of leverage when a dividend is paid for a call without dividend protection. Let's say GM issues a dividend at 3% and the cost of leverage is at the 6% area due to the dividend. That means that the cost of leverage can never fall below the 6% floor (assuming it is caused solely by the dividend) because if it did, it could be immediately aribitraged. So, it would seem that floors are established for both dividend protected and unprotected leveraged vehicles when a dividend is paid. However, it is obviously better to get the floor from getting the dividend protection than not. I guess you were already saying that, but in the reverse way. i.e., the A-warrants get the 7.5% of value because of the dividend floor that just got added to the calls without protection. This is also about to be true for both the WFC and JPM warrants, as they are right at their dividend thresholds. As the dividends increase, the warrants get increasingly more attractive than the calls.

Would you mind posting a cost of leverage example with the dividend after you have thought about it? (Perhaps you already have). txlaw and I have come up with different costs for various dividend scenarios and are working that out at the moment. I'm curious to see if it matches the total return formula I derived earlier in the thread. I would think they would match or at least that they will be fairly similar.

Interestingly, a floor also gets established to support the cost of leverage when a dividend is paid for a call without dividend protection. Let's say GM issues a dividend at 3% and the cost of leverage is at the 6% area due to the dividend. That means that the cost of leverage can never fall below the 6% floor (assuming it is caused solely by the dividend) because if it did, it could be immediately aribitraged. So, it would seem that floors are established for both dividend protected and unprotected leveraged vehicles when a dividend is paid. However, it is obviously better to get the floor from getting the dividend protection than not. I guess you were already saying that, but in the reverse way. i.e., the A-warrants get the 7.5% of value because of the dividend floor that just got added to the calls without protection.

After updating my spreadsheet (a few mental glitches in the process), it is pretty fascinating to look at the relationships. GM is one of the ones affected the most, since the strike is ~0.5 of the current stock price. Most of the other ones are much closer in terms of strike/current common price.

This is the best example. I need to think about it, but it does require that the price actually drops by the dividend amount which is rational, but I'm not sure that the prices actually do that, as the dividend is coming out of earnings (for ordinary dividends). That being said, these models never work unless you assume the stock drops by the dividend amount. I'm just going to go ahead and spell out my math/model, while still thinking about the above. No dividend Current Stock Price 40.87 Warrant Strike Price 18.33 Current Warrant Price 23.00 Solving for when the total return of stock = the total Return of Warrant at expiry. Variables are current stock price (CSP), current warrant price (CWP), future stock price (FSP), warrant strike price (WSP); solving for FSP: (FSP - CSP) / CSP = (FSP - WSP - CWP) / CWP CWP*FSP - CWP*CSP = CSP*FSP - CSP*WSP - CSP*CWP CSP*CWP = CSP*FSP - CWP*FSP CSP*WSP / (CSP-CWP) = FSP Ok, so now we plug in the nubmers to get FSP: 40.87*18.33 / (40.87-23) = 41.91 Thus, at 41.91, buying the common and buying the warrant will yield the same result, which is 41.91/40.87 - 1 = 2.54%, which can then be annualized. With ordinary dividends new variable: cumulative ordinary dividends = COD = 1.00 Again, solving for total return of stock = total return of warrant at expiry: (FSP - CSP + COD) / CSP = (FSP - WSP - CWP) / CWP CWP*FSP - CWP*CSP + CWP*COD = CSP*FSP - CSP*WSP - CWP*CSP CWP*COD + CSP*WSP = CSP*FSP - CWP*FSP (CWP*COD + CSP*WSP) / (CSP - CWP) = FSP Plugging these numbers in to get FSP: (23*1 + 40.87*18.33 / (40.87-23) = 43.21 (43.21+1) / 40.87 - 1 = 8.17% return, which can then be annualized So, adding a 2.45% dividend has caused the breakeven growth to increase from 2.54% to 8.17%, or a difference of 5.63% So yeah, now we can plainly see with my own math that Eric was right and I was wrong. I'll go back to my cave now. Thanks for sticking with me and making me work this out correctly (hopefully correctly anyway).

It is a cost. But, in my comparison, the total return of the common is what defines that cost. The math of the common total return indicates a loss of only the dividend yield. If I compared a vehicle that did get the dividend yield, then it would get 2x the dividend, as you say, but I am not comparing that vehicle.

How come the expected return from a $20 increase in stock price is higher in one vehicle versus the other, if you only have $18 invested in each one? I'm sure you will agree that it's because you are leveraged to twice as many underlying shares. Then why don't you agree that you are also leveraged to twice as many dividend payments? Only, you are getting none of them :( You seem to believe that you are entitled to twice as much upside in stock price (even though you only have the cash for half that much), but not twice as much dividend. Why the distinction? We'll, I'm simply not entitled to the dividends--that's what the contract says. Like I said, I'm comparing total returns--if I do it your way, then the total returns are not the same.

Again, I am measuring something different--I'm comparing total returns and not margining. I agree with all of your examples if I was thinking of it in your way, but I am simply not. Or said another way, I am comparing what price the common has to get to for my total return of the option to match the total return of the common. I cannot consider borrowing simply because I am not comparing common to margin, but common to calls. The math in my dividend example shows exactly why I only missed the yield when comparing common to calls. Perhaps I am the only one who thinks this way, but it makes a lot more sense to me. Is there any flaw with my example?