Carry Arbitrage Model when there are no Cash Flows
Riskless profit. The term sounds fantastic! Think about how you'd go about creating such a fabulous situation.
You'd probably start with an asset such as a stock and invest in it with borrowed funds. After the stock appreciates, you'd sell it for a great gain and pay off the borrowed amount with interest.
Seems easy right?
Not so fast. What's one issue with that strategy?
Not quite.
Tax implications could be an issue, but this scenario has to do with trading strategy.
You got it!
Price risk is a definite factor. It could ruin your riskless profit by sending your long stock position down below your purchase price. Then you'd lose money. So your position isn't riskless at all.
Here, you had purchased a stock with borrowed funds, essentially creating a long position in the stock. That's an example of the __carry arbitrage model__, which is a no-arbitrage approach where the underlying instrument (the stock) is either bought or sold along with a forward position. So, you're "carrying" the stock. And you would have violated one of the key rules for an arbitrageur, which is: don't take price risk.
As the carry arbitrage model indicates, price risk is where a forward commitment contract comes in. Work through this scenario like an arbitrageur.
Say you're going to buy one share of Golden Holding (GH), a company that pays no dividends, so there are no cash flows. Why did you start with borrowed funds?
No.
There's no rule requiring max leverage.
Not quite.
There's no rule that requires paying interest. That only takes away from your profit.
You got it!
The other rule for an arbitrageur states that you shouldn't use your own funds.
You'd start with borrowed money and complete two simultaneous transactions: buy GH for USD 150, -_S_0, and also borrow USD 150, _S_0, so there's no cash outflow.
So at this point, you'd own GH and be charged interest for the borrowed funds. At a future time, _T_, you'd sell the stock and pay back the borrowed funds.
Say that the annual interest rate is 2.5%, and you hold GH for a year. The future value of the borrowed funds is represented by
$$\displaystyle FV(S_0)$$.
The total payback amount for a one-year period would be
$$\displaystyle FV(S_0) = -150 \times (1+0.025)^1 = -153.75$$.
So GH would need to reach USD 153.75 to break even.
Why's that?
Bingo!
In this example, the only cost is the annual interest charged on the borrowed funds. So, it's 2.5% of -150. But be careful! Since time periods for forward contracts are in fractions of a year, the rate used and time period given don't always match up, so you'll need to make sure both periods are aligned before computing the cost.
Not quite.
That's the total payback amount.
With the cost of carry, you'd need USD 153.75 for GH in one year, but that violates the rules for an arbitrageur regarding price risk, so you'd need to use a forward contract to offset that risk.
So at the time of GH purchase, you'd also sell a forward contract over time _T_ (in this case one year) and, if the transaction results in an arbitrage profit, you'd _borrow_ that amount too.
Borrow the arbitrage profit, which is represented here by taking the present value of the difference between the forward contract less the future value of GH in one year.
$$\displaystyle PV[F_0 - FV(S_0)]$$
So given this transaction, with a stock offering zero cash flows in the future, the expected value at initiation is zero. Why is that the case?
Exactly.
Either answer works. In theory, the transaction is going to be valued at zero because the cash flows during the transaction should effectively cancel out each other. So no arbitrage profit should exist.
In this case, the no-arbitrage forward price is simply the future value of the underlying.
$$\displaystyle F_0 = \text{Future Value of Underlying} = FV(S_0)$$
That's the carry arbitrage model, but there are times when the value of GH isn't going long in the stock, but is actually shorting it.
If that's the case, what's one action you'd take?
No.
You're shorting GH, so you wouldn't buy it.
No, actually.
You'd be receiving funds from selling short.
Way to go!
You'd need to buy the forward contract on GH.
Assuming the same one-year time frame, you'd sell GH short and buy a forward contract to offset the price risk. At the same time, you'd need to invest the proceeds from the short sale at the risk-free rate and borrow any arbitrage profits. This strategy is known as __reverse-carry arbitrage__ because you are short selling the underlying instrument, GH.
To summarize:
[[summary]]
No.
That's the price you bought the stock at.
Price risk
Tax implications
The rule that requires max leverage
The rule of not using your own funds
The rule that requires paying interest
The cost of carry is -150.
The cost of carry is 2.5%.
The cost of carry is USD 153.75.
Law of one price
No-arbitrage approach
Buy GH long
Pay interest on borrowed funds
Buy the forward contract on GH
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