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Topic Title: Equity Swaps
Created On Tue Oct 08, 02 08:24 PM

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vp913
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Tue Oct 08, 02 08:24 PM

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I am trying to determine how to price the fixed leg of an equity swap where instead of the terms being "pay return on a index (e.g., S&P500)" and "receive fixed", the terms are "pay return on private equity investments" and "receive fixed". Private equity investments being investments in KKR Fund II, etc... Not sure if a traditional approach of modeling the "equity return" as I(n)/I(n+1) where I(j)=level of index at time j can be applied as there is no theoretical index to use. The level of the index itself is not important, i don't think, in pricing the fixed rate as it cancels out in the math, but the level of the index is important in valuing the swap after closing and I'm not sure if I can still apply the same technique here. Any thoughts?

amitabh
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Wed Oct 09, 02 07:06 AM

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hi,
what u can do here is that since it is a fund, i am sure it would have a Monthly NAV , if not a weekly one.Use the NAV as the equivalent of the price level. If u have a lot of historical data points , u calculate the returns and for simplicity assume that the returns are normally distributed , thus the prices/NAV levels will be lognormally ditributed , which should suffice for practical purpose . Now when u want to calculate the receive fixed leg of the swap , u can basically solve for annuity payments .As u have already calculated the total return for the return on a private investment . Lets call that X. Once u have X , solve for the present value a stream of annuity , which would equal X. This embedded rate in the annuity will be the Fixed leg of the swap.

Hope this helps

Amitabh

Johnny
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Wed Oct 09, 02 11:24 AM

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If the fund is traded, it's easy. You can just treat it like any other share.

If the fund is not traded, but you can invest in it, then your hedge is to invest in the fund and then proceed as with any other share.

If the fund is not traded and you can't invest in it then you cannot hedge in such a way as to eliminate the specific risk of the fund. You cannot then justifiably use risk-neutral valuation. i.e. you need to take into account your preferences (and those of your firm) with respect to the risk of holding an unhedged and unhedgeable position in a PE fund. The usual way to do this is to specify a market price of risk. See any good text-book, e.g. Wilmott.

You can try to find a proxy, such as a similar fund, which you can use as a hedge. But notice that this doesn't let you assume risk-neutral valuation. Now you have risk with respect to both funds. So you need to specify market price of risk for both, and hope that they cancel out. This is the reason why you should either do a lot of these trades, or none.

Having a few positions that don't quite diversify is a horrible route to the poor house. In a more formal sense, at this level of unhedgeability the problem becomes non-linear. This means that the value of an additional position (such as the one you are thinking about) depends on what else you have in your trading book already.

Edited: Wed Oct 09, 02 at 11:26 AM by Johnny

Aaron
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Wed Oct 09, 02 09:21 PM

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My first reaction is different from Johnny's. The problem is with many active funds you cannot assume that volatility or index correlations are constant (it isn't true of stocks either, but the active fund adds another layer of uncertainty).

I assume the fund itself has something to do with this product. In that case, it could provide you with data to estimate the future volatility. Also, it is your natural counterparty to the trade. If someone wants to receive the return on the fund, the fund can pay him as if he were an equity investor. If someone wants to pay the return on the fund, the fund can essentially borrow money from her.

-------------------------
Aaron Brown

Johnny
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Wed Oct 09, 02 10:04 PM

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If vp913 was trying to price an option then I would agree absolutely with Aaron's point about volatility. But it's an equity swap, one side paying equity returns and the other side paying fixed, so it's linear in the equity returns and the fund's volatility doesn't enter into it.

Having said that, I agree absolutely with Aaron's instinct in the following respect: with any complex product, see if you can find both sides to the trade and then haggle. That way you (as an intermediary) don't end up taking a lot of model risk or market risk. But in this case I suspect that the fund is closed to new investors. If this is true, then the fund unfortunately won't be available as a counterparty.

Nonetheless, the instinct to find the other side of the trade is the right one. I'm sure you can find someone out there that's a bull of fixed income instruments and a bear of PE funds. Given that govt bond markets are trending up and equities are trending down in the public markets, perhaps you could find a buyer in the trend-following community? ...

... Duck?

Anthis
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Wed Oct 09, 02 10:24 PM

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Question maybe dam but still a question.

Is there any optionality in an equity swap?

For example, in an equity swap where the floating leg is a stock index, the floating leg payer receives money only if the index returns plus dividends per tenor term are positive but lower than the fix rate?

Or to put it on other words, if the term fixed rate is 2% and at the same period the index had a total return of ?5%, the floating leg payer should receive a 7% on notional capital or just a 2% on the notional for payment?

If the last is valid I can guess than none would be willing to have the Floating leg position. A bearish investor would be better off to invest in index puts or short index futures than in an equity swap.

Anthis

Johnny
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Thu Oct 10, 02 11:08 AM

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An equity swap is usually an arrangement whereby one side pays total equity returns (capital gains plus dividends including tax adjustments) and the other side pays a floating rate, LIBOR - X bps.

The equity returns need to be paid regardless of whether or not these returns are higher or lower than the interest rate on the other leg. In this sense there is no optionality in an equity swap.

For pedants and hair-splitters, there is a miniscule dependence on volatility. This arises through periodic remargining. The argument here is analogous to the difference between a futures price and a forward price. But it's too small to notice in practice.

Equity swaps are useful in several circumstances. Two examples are:

1. I receive a tax credit on dividends paid on certain shares; you do not. You own those shares; I do not. If you sell me the shares and then we enter an equity swap such that I pay you equity returns and you pay me LIBOR - X, then we can arrange to split my tax credit down the middle. So we're both better off, at the expense of taxpayers.

2. You're a CB arbitrageur. You want to sell stock short against your position but you're worried that your stock loan might be called away from you. So you enter an equity swap where you pay equity returns and receive LIBOR - X. You can see that X is equivalent to the fee you would normally pay to borrow the shares.

Hope this answers your question.

Anthis
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Thu Oct 10, 02 12:09 PM

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Johny Thanks

But i need some clarification.

First. Once you said that there is no netting in the payments between counterparties, and i am the equity leg payer, then what if the index has had negative returns? In my previous example the Libor payer should pay me the Libor Rate or he should pay Libor plus the index loss?

Second you develop your argument in the framework of comparative advantage argument, a commonplace in swaps. If i am just a bearish investor thinkig that the index is overvalued and will be falling for the next couple of years, (as it has really happened for many national markets recently) with no other comparative advantage reason should i be indifferent in enter in a swap transaction compared to shorting index futures or buying index puts (probably longer term contracts)?

Third, lack of liquidity in the underlying equity market, short sales constraints, complete lack or lack of liquid relevant derivatives markets, unstable correlation for cross hedging in other markets, and finally the option to default in a swap agreement are reasons to enter in an equity swap transaction?

Regards

Anthis

Johnny
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Thu Oct 10, 02 01:18 PM

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"First. Once you said that there is no netting in the payments between counterparties, and i am the equity leg payer, then what if the index has had negative returns? In my previous example the Libor payer should pay me the Libor Rate or he should pay Libor plus the index loss?"

I can't remember saying anything about netting. Perhaps I'm getting old ... But yes, in your example, the LIBOR payer should pay you the LIBOR plus the index loss. This is the same as the LIBOR payer receiving (negative) returns.

"Second you develop your argument in the framework of comparative advantage argument, a commonplace in swaps. If i am just a bearish investor thinkig that the index is overvalued and will be falling for the next couple of years, (as it has really happened for many national markets recently) with no other comparative advantage reason should i be indifferent in enter in a swap transaction compared to shorting index futures or buying index puts (probably longer term contracts)?"

Contrast three possibilities:

1. Borrow shares at a fee of X bps per annum, sell them short, deposit the short-sale proceeds at LIBOR.

You win if the shares go down in price. You lose if the dividend is higher than expected as you have to pay a higher dividend to the person from whom you borrowed the shares. You win if LIBOR goes up. You are in trouble if your stock loan counterparty demands his shares back, which he can at any time.

2. Pay total returns on an equity swap, receive LIBOR - X bps. Same pay-off as in (1) except now your counterparty has no right to terminate the agreement arbitrarily. Effectively you have locked in your stock loan.

3. Short an equity forward. Same pay-off as in (2) except now your counterparty takes the dividend risk. If the dividend is higher than expected, he loses, not you. Similarly to (2), your counterparty has no right to terminate the agreement arbitrarily, so again you have locked in your stock loan.

4. Long an equity put. Now you have a completely different pay-off that is a function of volatility and strike price and so on.

So all these different contracts have different pay-offs and should be used for different purposes. They all allow you to take a short position in an equity, so - as you say - the reasons for choosing each instrument are not to do with opinion about market price. The reasons are usually more connected with tax (on dividends), stability of stock loan, hedging dividend risk, and short sales constraints, although the latter is an explicit cheat on the system.

HKQuant
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Fri Oct 11, 02 02:50 AM

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I have a naive question related to the start of this thread - one party pays the performance of the index (or fund, but let's say it's an index to simplify matters) and receives fixed rate. To give a more specific example, say the initial notional is $1 Mil, and the fixed rate (say 5% p.a.) is based on $1 Mil. At the next reset date (say 1 yr), the index has moved to the equivalent of $1.2 Mil, so the net amount received is $200k - 50k = $150k. However, what is the notional after reset, i.e. would the next fixed payment be based on a $1 Mil notional or $1.2Mil notional?

A typical equity swap is often based on Index vs LIBOR - spread (rather than Index vs Fixed), and notional is reset at every reset date. In that case, the theoretical spread (i.e. ignoring the profit margin of the swap counterpart) should just reflect the borrowing cost and the anticipated dividends. If I have sold the swap (i.e. agree to pay the performance of the stock/index), the perfect hedge is to borrow money at LIBOR and buy the index on day 1. At each reset date, I just need to borrow/lend money for the net inflow/outflow. This trade is riskless from the market risk point of view (if dividends are pass through; otherwise there is a dividend risk). The substantial risk is the default risk which should be covered by margining requirements in a separate agreement.

If the equity swap is Index vs Fixed, we need to anticipate the path of the stock, because we need to fund the net inflow/outflow at a fixed rate. In that case, the analysis is not so simple and optionality is definitely involved, because we need to model the correlation between the stock movement and interest rate. I would appreciate if someone could show me how to calculate the fixed rate without using simulating both paths (stocks and interest rates).

Adding to Johnny's arguments as to why one wants to enter an equity swap, one way to look at it is that it is essentially a stock trade based on margin. If the collateral is 0, effectively one party is buying the stock at 0 margin, which of course is very attractive. Even if the collateral is 50% of notional (and requires top up), there is still a 2x leverage. These swaps are often breakable (with maybe a 1% break fee). Hedge funds like these trades (rather than buying/selling stocks directly) because high leverage could be achieved.

Anthis
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Mon Oct 14, 02 09:26 PM

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Johny thanx a lot you ve been very enlightening

But.......
<<Third, lack of liquidity in the underlying equity market, short sales constraints, complete lack or lack of liquid relevant derivatives markets, unstable correlation for cross hedging in other markets, and finally the option to default in a swap agreement are reasons to enter in an equity swap transaction?>>

You have ommited liquidity considerations, and the option to default in a swap reason. Whats your opinion about these?

<<So all these different contracts have different pay-offs and should be used for different purposes. They all allow you to take a short position in an equity, so - as you say - the reasons for choosing each instrument are not to do with opinion about market price>>

I cant understand the reason why, the reasons are not to do with opinion about market price. Heterogeneity of expectations about future market prices is the basis of trading in financial markets, isnt it?

And yes these 4 alternatives have different payoffs but is it because they have different investment horizons and risk profiles? Recall that i wanna invest on my view for a downtrending stock market over then next couple of years (a rather long term view). So the other three alternatives cant be implemented safely (rollover risk) over the whole investment horizon. For example I doubt if you can borrow stocks for more than some months.

Furthermore, a long position in, say ATM, puts is a bet on an increase in volatility as well, plus the belief on assymetric volatility effect, and has quite different leverage. Additionally, if the options are American you run the risk of suboptimal exercise. Its not just a bet on a downtrend in prices.

I would like to know if you agree with my thoughts

Regards

Anthis

Johnny
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Mon Oct 14, 02 11:01 PM

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Anthis

Please accept my apologies - I thought you were still asking about equity swaps. If you just want to get short the market then, yes, I'm sure that index futures and/or puts could be as good as any other way.

Edited: Mon Oct 14, 02 at 11:05 PM by Johnny

Anthis
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Tue Oct 15, 02 01:50 AM

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i am still talking about equity swaps...

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