# Mandatory convertibles -- math example?

I have to admit, even though I blew away the fixed income sections on the CFA exams I am still pretty ignorant about the mathematical mechanics of convertible bonds. I was speaking to someone about mandatory convertibles today, which are bonds that automatically convert to stock at maturity. He said that he bought bonds issued by Barclays that convert to stock at a 25% discount to current market price, then shorted an equivalent amount of stock against it, and is collecting a 9.5% coupon until maturity in 6 months. He said his only risk was losing the coupon between now and then. So if he locked in a 25% gain on the conversion (or maybe a little less due to slippage, borrow costs, etc.) and at worst is getting a 0% coupon, a) where is the downside risk and b) what is a fair market price for the bond right now? If someone could come up with a math example to relieve me of my mental block it would be much appreciated.

i’m assuming if they convert to stock at a 25% discount the bond would also trade at a significant premium. a bond is convertible into 100 shares of ABC at 25% discount (shares trade at \$10). The bond would trade at \$125ish as it holds the \$25 conversion value plus any premium/discount to the 4.75% coupon that you’d normally get depending on how risk averse the market is. another question is that if a company has 9.5% coupons, what does it distribute to unitholders or common equity? most companies w/ 9.5% coupons will yield about the same or more to unitholders. i’m pretty sure a conversion takes greater than 3 weeks or something so you don’t know the price of the stock until 3 weeks from the date you short an equivalent amount of stock. if the company has 9.5% coupons, they have a high cost of capital so their equity may be diluted horribly depending on what their future capital needs are. for example, assuming the dilution is somewhere near 50%, in a good company the leverage could have brought 50% in upside to the equity with little downside thereby nullifying the benefit of conversion, in a bad company, the price of common equity will already be low, but the converted debt will likely have to be refinanced thereby creating greater risk for the company as a whole, further depressing equity prices. it truly is a case by case analysis, but i’m sure some cases work out very well if the bond is trading at a discount of course.

mandatories comes in several forms (DECS, PERCS, etc) depending on the underwriter and the ones that i familiar with are the preferreds. Most of them have the following characteristics: Upper Strike and Lower Strike, typically 3 years and usually have a yield advantage over the common. most are issued with a premium of 15-30% depending on the environment. I will walk through one and you will see the mechanics and spot the risk: Legg Mason Mandatory 6/30/2011 7% issued at \$50 par now trading at 15.5 vs stock of \$11. Their are two strikes on this the lower being: \$56.3 and the higher is \$67.56. If the stock is trading below the lower strike, it’s convertible into 50/56.3 = .8881 shares. If the stock is trading above the strike of 67.56 then you get 50/67.56 = .7401 shares. If the stock trades between 56.3 and 67.56 then you get basically \$50 b/c the conversion is adjusted to \$50/stock price. So right now if stock stays where is at \$11 then the parity is worth \$11 * .8881 = 9.77. but you also get those fat coupons of 7% of \$50 = \$3.5 for 2.25 more years, thus you get price of about \$15.5 for the mandatory preferred. I prefer the mandatory over the common in this case. Now on a hedged basis you have a lot of negative gamma going on at the 56.3 range of the stock as the number of shares begins to get reduced. Essentially a typical mandatory at issuance is the equilvilent of long the stock + short a call spread + yield pickup

Thanks for the example. So assume the stock never trades anywhere close to \$56.30 prior to expiration. Also, you are delta hedged so you’re short .8881 shares of stock for each bond, so basically you cannot make or lose money on the conversion, right? So basically you’re paying 15.50 for \$15.75 worth of coupons (\$7/year x 2.25 years)? If that is the case what is the attractiveness of buying this bond since it has a YTM of almost 0? I suspect I’m interpreting something incorrectly.

JohnThainsLimoDriver Wrote: ------------------------------------------------------- > Thanks for the example. So assume the stock never > trades anywhere close to \$56.30 prior to > expiration. Also, you are delta hedged so you’re > short .8881 shares of stock for each bond, so > basically you cannot make or lose money on the > conversion, right? So basically you’re paying > 15.50 for \$15.75 worth of coupons (\$7/year x 2.25 > years)? If that is the case what is the > attractiveness of buying this bond since it has a > YTM of almost 0? I suspect I’m interpreting > something incorrectly. it’s a preferred with face of \$50 and 7% coupon = \$3.5/year in dividends. yea no free lunch here. if you are 100% delta hedged then you would be short the .8881 shares (but when you trade with institutional investors they always use lower conversion rate (higher strike) so you say that you were on a 120% hedge…

It’s easiest for me to think about it as ConvertArb states: Mandatory = Stock + (short a call spread, i.e. you negate any gains from your stock between 100-120%), and yield pickup Let’s assume you hold on to the mandatory for 3 years and go short the maximum underlying shares (I think you guys are using 0.8881?). In that case, you are earning the yield but you have to pay through the dividends on your short, meaning your yield pickup is just mandatory coupon minus dividends. If stock price ends at 120% (whatever the upper conversion price is) at maturity, you will have gained nothing thanks to that stupid short call spread position. I think mandatories will usually price such that the gain from your yield pickup over the 3 years outweighs the loss of no upside when compared to common shareholders. I would think the main problem with mandatories (and convertible structures in general) is that there is much less liquidity in these securities. Especially if there is early conversion (rare) or the companies repurchases bits and pieces, liquidity dies down and anyone with a large position is stuck.