Mohnish Pabrai Boston College Presentation

[jeffmori7]

Guys, in the end, don't you think that what we have to guess from now on is : Will the appreciation of the stock price be more important than the total amount of dividend being payed over the 5 years by a fair amount ? If so, the warrants wins anyway, no matter the time value of money I would think.

 

At more than 1.2 times the dividend (which is the warrant to strike price ratio) in stock price appreciation, I think you win in all scenario.

 

I calculated it like this :

 

P : the stock price today

D : the amount of dividend being paid

A : appreciation of the stock price

W : Warrant price, let's assume W=P-S

S : Strike price

 

So what you want is basically to solve (P+D+A)/P = (W+A)/W

 

To get the relationship between the dividend overall payment D and the stock price appreciation A.

 

And you get to A=W/S * D

 

Of course I don't calculate a cost of leverage here, but I'm comparing, including a dividend, basically what I need for a warrant to win over the common, assuming the warrant is well priced here.

 

That is basically what Racemize was saying sooner, but without calculating an assumed dividend or final price.

[jeffmori7]

To take into account the mispricing of the warrant, let's say that W is the current warrant value, and W_i is the intrinsic warrant value, W_i = P-S.

 

we can say dW=W-W_i, my previous formula would become : A=(W/(S-dW)) * D.

 

A mispricing with a higher than intrinsic value warrant increase the appreciation needed to overcome the dividend inclusion to the totsal performance while a warrant value under intrinsic value decrease the appreciation needed to equal the performance including the cumulative dividend.

 

 

[ERICOPOLY]

This seems to match txlaw's result:  this time, I'm just taking 18.33 (the strike) and compounding it by the annual missed dividend rate.  The reason why I think it makes sense to compound the strike price is because the first year's missed dividend expense get's compounded 4.5 more times, the second year's missed dividend expense get's compounded 3.5 more times, etc...  So that effectively works in a way that it solves the time value of money problem.

 

He used 1.20 for the annual dividend and therefore 6.5% for the annual dividend cost (1.20/18.33=.065)

He used 68 cents for the option premium

18.33 is the option strike

he used 5.5 years

 

18.33-.68=17.65

 

17.65*1.x^5.5=18.33*1.065^5.5

 

solving for x, I get about 7.19% annualized cost. 

 

That's 6.5%+.69%=7.19%

 

.69% is the annualized cost of the option premium -- he estimated it to be 1%.  It was just an estimate.  I compute it to be .69% using my first equation which calculates it precisely (17.65*1.x^5.5=18.33).

 

So we come to the same result.  Hooray!

 

EDIT:  I reworded the explanation in the first paragraph

[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.

[ERICOPOLY]

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.

 

I believe it solves the problem where the dividends from the common are immediately reinvested into the common stock.

[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.

 

I believe it solves the problem where the dividends from the common are immediately reinvested into the common stock.

 

I guess so, but you have to know the common price at those points to get an actual number, right?

[ERICOPOLY]

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.

 

I believe it solves the problem where the dividends from the common are immediately reinvested into the common stock.

 

I guess so, but you have to know the common price at those points to get an actual number, right?

 

 

I see it like this.

 

For the common, a $40 stock with $1 dividend.  Now it's $1 in cash and $39 in stock.  Reinvest the $1 into stock.  Now you have $40 in stock.

 

So for the common, just pretend like nothing ever happened (instantaneous reinvestment).  $40 in stock is $40 in stock.

 

 

Therefore, we only need to think about what effect the MISSED dividend has on the warrant.  That's what my calculation does, by adjusting the strike price.

[racemize]

How can we verify that this number is correct?

 

Previously, we could look at returns of each vehicle, but with this formula, I'm unsure how we would compare the common to warrant.  For example, the math I used before to test would show that the calculated price was too high with these new costs of leverage. 

[racemize]

I guess if we only had one year, then it would be the same right?  Since there is no compounding? 

 

Assume:

$10 stock

$8 strike

$3 cost of option

$1 dividend

1 year expiry

 

so extra payment is $1, which we subtract from the strike of $8

7*1.x=8*1.125 (which is the same as 8+1)

= 28% cost of leverage

 

My method also returns that. 

 

I think I just made it a tautological statement though...

[ERICOPOLY]

I rather prefer just using my first equation that computes the cost of the option premium.

 

Take that result which is .69%

 

Then add 6.5%

 

Get result of 7.19%

 

Much simpler than modifying that equation like I recently did -- which ultimately just makes it needlessly complicated

 

 

[ERICOPOLY]

I rather prefer just using my first equation that computes the cost of the option premium.

 

Take that result which is .69%

 

Then add 6.5%

 

Get result of 7.19%

 

Much simpler than modifying that equation like I recently did -- which ultimately just makes it needlessly complicated

 

Actually, I got a slightly different result (compared to txlaw) from my modified formula.  They were very close, but a little bit different.

 

I wonder if it's due to this effect:

 

1)  The common stock owner would be reinvesting the dividend into extra shares.  Therefore the dividend needs to be multiplied by the FULL amount of shares.  The total share count grows each year when the dividends are reinvested.

2)  My computation that compounds the warrant strike by 6.5% takes this into account (I think).  It grows the dividend effectively as if it were reinvested into additional shares, each paying 6.5% worth of missed dividend cost.  Each year, the dividend cost is 6.5% bigger than the prior year because there are more shares.

 

Do you agree that makes sense?

 

If you ignore the effect of the missed dividend growing each year (due to the increasing share count of reinvested dividends), then the computation comes out wrong.

 

So I think if you want best accuracy you can't simply add 6.5% to .69%.  It's relatively close though, but for the sake of exactness it misses by a bit.

 

so this is still my best stab at it:

 

17.65*1.x^5.5=18.33*1.065^5.5

 

 

EDIT:  Well, it's better than nothing because it does at least recognize that the share count is rising after each dividend.  However, it might not be adjusting the share count properly because it would depend on how many shares are purchased with the dividend.  A higher stock price would result in fewer added shares, and a lower stock price would result in greater added shares.  This is why the BAC "A" warrant dividend protection formula takes into account stock price when it adjusts the warrants for paid dividends.

[Rabbitisrich]

Are you guys talking about the BAC A warrants with the adjustment feature? In that case, the cost should be pretty close to (cost of carry + div yield - div adjustment). If you calculate the div yield in the same manner as the div adjustment, then they should cancel out, so you just have the negative carry. Think about a warrant trading exactly at fair value (stock price - strike). There is no charge for leverage, so if the company issues a dividend, your warrant return should be neutral for that time frame. This is assuming that the stock price moves only to adjust for the dividend issuance.

 

Hmm, that looks more confusing than it sounded in my head. Appreciate any different viewpoints.

[Rabbitisrich]

Perhaps the comment above is missing the point of the recent thread. It seems like debate is over how to measure dividend yield? You could measure it by the yield from future dividends to the common purchaser at the moment of purchase. But you could also judge yields by the moment of investment, AND every subsequent dividend issuance. With respect to the BAC warrants, the latter yields make more sense because the comparison is between levered common reinvestor and warrant holder. Regardless of the magnitude of future dividends relative to the initial purchase price, you end up with initial stock quantity + future stock quantities purchased with dividends. That gets you somewhere close to the A Warrant.

[racemize]

This was focusing on calls with no protection. For those with a threshold, we only have to worry about dividends below the threshold, as the others are accounted for.

[racemize]

I rather prefer just using my first equation that computes the cost of the option premium.

 

Take that result which is .69%

 

Then add 6.5%

 

Get result of 7.19%

 

Much simpler than modifying that equation like I recently did -- which ultimately just makes it needlessly complicated

 

Actually, I got a slightly different result (compared to txlaw) from my modified formula.  They were very close, but a little bit different.

 

I wonder if it's due to this effect:

 

1)  The common stock owner would be reinvesting the dividend into extra shares.  Therefore the dividend needs to be multiplied by the FULL amount of shares.  The total share count grows each year when the dividends are reinvested.

2)  My computation that compounds the warrant strike by 6.5% takes this into account (I think).  It grows the dividend effectively as if it were reinvested into additional shares, each paying 6.5% worth of missed dividend cost.  Each year, the dividend cost is 6.5% bigger than the prior year because there are more shares.

 

Do you agree that makes sense?

 

If you ignore the effect of the missed dividend growing each year (due to the increasing share count of reinvested dividends), then the computation comes out wrong.

 

So I think if you want best accuracy you can't simply add 6.5% to .69%.  It's relatively close though, but for the sake of exactness it misses by a bit.

 

so this is still my best stab at it:

 

17.65*1.x^5.5=18.33*1.065^5.5

 

 

EDIT:  Well, it's better than nothing because it does at least recognize that the share count is rising after each dividend.  However, it might not be adjusting the share count properly because it would depend on how many shares are purchased with the dividend.  A higher stock price would result in fewer added shares, and a lower stock price would result in greater added shares.  This is why the BAC "A" warrant dividend protection formula takes into account stock price when it adjusts the warrants for paid dividends.

 

So, now how do you adjust the formula when you can get more than one share per warrant?  weeeeee.....

[ERICOPOLY]

I rather prefer just using my first equation that computes the cost of the option premium.

 

Take that result which is .69%

 

Then add 6.5%

 

Get result of 7.19%

 

Much simpler than modifying that equation like I recently did -- which ultimately just makes it needlessly complicated

 

Actually, I got a slightly different result (compared to txlaw) from my modified formula.  They were very close, but a little bit different.

 

I wonder if it's due to this effect:

 

1)  The common stock owner would be reinvesting the dividend into extra shares.  Therefore the dividend needs to be multiplied by the FULL amount of shares.  The total share count grows each year when the dividends are reinvested.

2)  My computation that compounds the warrant strike by 6.5% takes this into account (I think).  It grows the dividend effectively as if it were reinvested into additional shares, each paying 6.5% worth of missed dividend cost.  Each year, the dividend cost is 6.5% bigger than the prior year because there are more shares.

 

Do you agree that makes sense?

 

If you ignore the effect of the missed dividend growing each year (due to the increasing share count of reinvested dividends), then the computation comes out wrong.

 

So I think if you want best accuracy you can't simply add 6.5% to .69%.  It's relatively close though, but for the sake of exactness it misses by a bit.

 

so this is still my best stab at it:

 

17.65*1.x^5.5=18.33*1.065^5.5

 

 

EDIT:  Well, it's better than nothing because it does at least recognize that the share count is rising after each dividend.  However, it might not be adjusting the share count properly because it would depend on how many shares are purchased with the dividend.  A higher stock price would result in fewer added shares, and a lower stock price would result in greater added shares.  This is why the BAC "A" warrant dividend protection formula takes into account stock price when it adjusts the warrants for paid dividends.

 

So, now how do you adjust the formula when you can get more than one share per warrant?  weeeeee.....

 

You don't need to.

 

The gap in value is on the initial share, not on the additional shares from dividend reinvestment formula.  In other words, there is no option premium assigned to those additional shares as they are bought with the dividend.

 

So you only need to compute the rate of compounding to close the gap on the initial share purchased.

 

EDIT:  the "gap" I'm referring to is the option premium.  You subtract it off the strike price, then get a number.  At what rate does that number compound to get back to strike?  That's the cost of leverage from the option premium alone.  That rate doesn't change even after the warrant has undergone the dividend reinvestment simulation that results in higher number of shares at conversion.

[racemize]

Well, I'm trying to deal with the case where you don't have adjustments up to a threshold, so in those cases you do (i.e., where the shares per warrant are changing due to other adjustments being made, not within our own formula for catching the dividends that weren't accounted for).  We can ignore that more complicated case for now and just try to focus on this (which will solve the other case at the same time, I believe):

 

Let's consider the citi case with a 10:1 reverse split:

 

Before split:

Warrant price = .63

Strike Price = 10.61

Common Price 5.08

Shares Per Warrant = 1

 

After Split:

Warrant Price = .63

Strike Price = 106.1

Common Price = 50.8

Shares Per Warrant = 0.1

 

The formula needs to be able to deal with that--I know how to adjust it on my formula, since I know which parts the "shares per warrant" affects.  I'm having a bit more trouble with this new formula, however.

[ERICOPOLY]

Well, I'm trying to deal with the case where you don't have adjustments up to a threshold, so in those cases you do (i.e., where the shares per warrant are changing due to other adjustments being made, not within our own formula for catching the dividends that weren't accounted for).  We can ignore that more complicated case for now and just try to focus on this (which will solve the other case at the same time, I believe):

 

Let's consider the citi case with a 10:1 reverse split:

 

Before split:

Warrant price = .63

Strike Price = 10.61

Common Price 5.08

Shares Per Warrant = 1

 

After Split:

Warrant Price = .63

Strike Price = 106.1

Common Price = 50.8

Shares Per Warrant = 0.1

 

The formula needs to be able to deal with that--I know how to adjust it on my formula, since I know which parts the "shares per warrant" affects.  I'm having a bit more trouble with this new formula, however.

 

 

You are making my head hurt forcing fractions into my thinking.  So I'd rather just "unfraction" and reverse the Citi split and then solve.

 

Sort of like if you asked me to compute the rate at which 0.1 grows to 0.2, and if I didn't want to deal with fractions, I would just multiple both by 10 and then solve.  I'd get the same result either way.

 

In math you can multiply both sides of the equation by 10 and you aren't screwing things up -- you just make things easier to think about and work with.

 

[racemize]

Yeah, I was considering just undoing the splits.  I wonder if that will still work for the cases where the shares per warrant is getting adjusted independently as well (e.g., where I'm calculating the missed 0.01 dividend for BAC, but ignoring everything above it since the warrant adjustments take care of it.  Then you end up with strike price adjustments as well as additional shares per warrant).

 

Actually, I should just be able to compare it against my old formula as long as the expiry is one year away, for verification.

 

 

Feels very hacky though.

[ERICOPOLY]

Yeah, I was considering just undoing the splits.  I wonder if that will still work for the cases where the shares per warrant is getting adjusted independently as well (e.g., where I'm calculating the missed 0.01 dividend for BAC, but ignoring everything above it since the warrant adjustments take care of it.  Then you end up with strike price adjustments as well as additional shares per warrant).

 

Actually, I should just be able to compare it against my old formula as long as the expiry is one year away, for verification.

 

 

Feels very hacky though.

 

 

The reason why it works for the BAC warrants is because you have two independent costs.  The option premium has nothing to do with the costs from lost dividends.  And with BAC, the only lost dividend is 1 cent per quarter.

 

So the formula works almost perfectly at approximating BAC "A" warrant cost of leverage, except that it will be off slightly for not taking into account the prices at which the lost 1 cent dividend quarterly dividend would be invested in to the common.  I'm compounding that lost dividend cost (against the strike price) which is slightly wrong because it depends on how many new common shares could be purchased with that lost 1 one cent dividend.

 

But...  it will be damn close.  That 1 cent dividend can't purchase many shares so there is little to be skewed.

 

Now, if the BAC class "A" warrant had a 0 cent adjustment hurdle (rather than 1 cent), then I believe the calculation would be perfect.

 

After all, you start out with the option premium being your only cost -- and that's the only cost truly because there is no lost dividend.

[ERICOPOLY]

You can think of BAC common stock as being logically identical to a warrant with 0 strike and full dividend protection (adjustment).

 

My formula computes the cost of leverage of BAC common to be zero.  This is the correct results.  It's correct because there is no option premium to the common, and there is no loss from dividends.

 

It does not matter at all how many shares this common (with full dividend protection) converts to in the end.

[ERICOPOLY]

Another simple way to think about it is that the lender is letting you borrow $13.30 per class A BAC share.  His conditions are that the interest rate is fixed and that you prepay him all of the interest.

 

So your cost of interest on that $13.30 is a static cost.  It's simply  the interest you owe for borrowing $13.30.

 

Now, compare that to borrowing $13.30 from your local banker.  Does he charge you a higher interest rate if you invest it in BAC shares and then reinvest your BAC dividends into more shares?  Of course he doesn't -- he charges you interest rate on that $13.30 no matter how you invest it.  You could give it to the Pope and it still is going to be the same amount of money you owe in interest.

[racemize]

Oh, I agree with all that, I'm just trying to account for the lost 0.01 per quarter for BAC.  While it isn't as big a deal there, it is for say, WFC, who's threshold is $0.34 a quarter.  I'll keep messing around with it until I have an adjustment I'm happy with.

[ERICOPOLY]

Oh, I agree with all that, I'm just trying to account for the lost 0.01 per quarter for BAC.  While it isn't as big a deal there, it is for say, WFC, who's threshold is $0.34 a quarter.  I'll keep messing around with it until I have an adjustment I'm happy with.

 

It would be the same as with the GM formula.  You just treat 34 cents as the missed dividend cost expressed as a percentage of the strike price.

 

You'll get a more accurate leverage cost estimate as compared to GM.  After all, GM won't keep a dividend at 1.20 for each year -- they would raise it a bit each year.

 

But WFC's lost dividend cost is capped at 34 cents -- so it doesn't have the complexity of the GM case.

 

So here is what we used for the GM formula (an approximate result it spits out):

17.65*1.x^5.5=18.33*1.065^5.5

 

You would just rework it with WFC's numbers.  using the maximum lost WFC dividend yield on the right hand side of the equation.

 

So if the WFC strike price were $38 (I'm not sure if it is, but just for an example) then the missed dividend yield would be 3.57%.

 

Thus, you would have for the right side of the WFC equation:

=38*1.0357^5.5

Assuming 5.5 years left on the WFC warrant and 38 for the strike (I think both numbers are different, but I'm too lazy to look them up)

[racemize]

Oh, I agree with all that, I'm just trying to account for the lost 0.01 per quarter for BAC.  While it isn't as big a deal there, it is for say, WFC, who's threshold is $0.34 a quarter.  I'll keep messing around with it until I have an adjustment I'm happy with.

 

It would be the same as with the GM formula.  You just treat 34 cents as the missed dividend cost expressed as a percentage of the strike price.

 

You'll get a more accurate leverage cost estimate as compared to GM.  After all, GM won't keep a dividend at 1.20 for each year -- they would raise it a bit each year.

 

But WFC's lost dividend cost is capped at 34 cents -- so it doesn't have the complexity of the GM case.

 

Yes, but what I'm trying to adjust for, is in say 3-4 years when the shares per warrant has changed for WFC (e.g., from 1 -> 1.1).  At that point, I can't ignore that the shares per warrant is different and have to adjust for it when calculating the cost of leverage.  Thus, while it isn't terribly useful now, it is something that will have to be taken into account when the shares per warrant starts changing (already has for a few of these).

 

It is a minor difference at the 1.1 level, but a big one at the reverse split level.  Current thought is to just normalize the numbers to 1 when I do the calculation itself, but that is rather inelegant.