To know whethere there is any arbitrage profit from the market quoted future price and the future price derived from the spot rate, I used the follow the simple principle, if the future price is higher, sell future and buy asset, and if future price is lower, buy future and sell asset.
Nonetheless, in couple of problems in EOC this doesn’t seem to be true, especially for the securities with cash flows. They are comparing spot price + coupons with the market price of future to determine the risk free profits. Could you please let me know what is the rule to know when to buy and when to sell.
When the no arbitrage futures price that you calculated (which includes deducting the PV of interest or dividend payments i.e. cost of carry) is less than the market futures price, there’s an arb opportunity where you would short the forward, borrow at the risk free rate to buy the asset and then upon termination of the future contract, you would sell the asset at the future price (the market futures price and NOT the no arb price).
Are they deducting inbound cash flows to derive at the no arb price?
I’m sorry for being a bit brief on the response. I’m typing this out on the phone.
I know they are doing exactly the same as you have typed but why would you do that? I mean, any reasoning behind it? The basic principle is, “buy low and sell high”. If your future prices are low, you will be selling at a low price in the future, if I follow the rules mentoned in the above comment, it violates the basic principle in finance. I could get all other scenarios right when I follow BLSH princle, but this one somehow came out of the blue. I mean I can understand that even after doing this, arbitrager is able to earn the return. But I would like to know more because this concept is not sticking to my head.
Shouldn’t that be spot price minus coupons with the market price of the future?
If you buy a forward contract on a stock, you agree to buy it at time T. Say the bond has dividends, and the value of the stock today is S. Between now and T, there will be dividends, that is to say, some portion of the stock’s value has been removed, paid out so to speak. So you can’t really say that the future value of the stock, the futures price, is simply the current stock price S invested at the risk-free rate for some period of time, S*(1+Rf). You must adjust for those dividends, that vanished into someone’s pocket at various times between today and time T. To know the value of those dividends today, you use the time value of money, say that the value today is abbreviated PVD. The value you will expect to get once you enter into the deal you’ve agreed to do at time T, that value is simply (S - PVD)(1 + Rf)^T. Think about the DDM, dividend disount model, with fluctuating dividends for a few years, and then you estimate some future terminal value. If you remove those fluctuating dividends you’d also end up with something looking like terminal value = (P - PVD)*(1 + Rf)^T. It is sort of the same logic. Meanwhile, at some intermediate point between the contracts agreement date and the time T, the value is calculated as follows: Suppose you have agreed to buy the stock for the future price FP at time T, that value today (at time t) is FP/(1+Rf)^(T-t). Purchasing is money out, so it’s an obligation, minus sign. If you now, at t, want to secure your financing of the stock you plan to buy, you could imagine a situation where you enter into a NEW forward contract this time to sell, at time T. That new contract is today priced as ( S(t) - PVD(t) )*(1 + Rf)^(T-t) where T-t is the remaining time until T. But what is that NEW contract worth TODAY, at time t? You have to discount the new FP back, with the very same interest rate and time period (1 + Rf)^(T-t) so that term cancels out. It will leave you with Value = S(t) - PVD(t) - (FP / (1 + Rf)^(T-t) where that final term is the value of your obligation to buy, cash out, minus, and the two first terms are the value today of your imaginary NEW offsetting forward contract as of today.
The trick is remembering that you enter into an offsetting transaction to fund your future obligation to buy that asset. Your “value” today is the difference between the two.
I think we are on the same page because you are bringing your future prices as of today and what I am doing, I am calculating my future price and comparing with future price in the market. Both are two different ways of doing the same thing.
Now from what I have seen in one of the problem is that it changes in case of BOND. Other cases are fine. I am referring to the problem 4.B of reading 55. In that, they are selling future even when the future price is low and buying underlying bond. Does anyone know if it changes when the underlying security is bond. Other problems similar to this are 7.B and 8.C which follow the same principle as I listed above.