# Long Stock = Long RFR Bond + Long Futures - How ?

Pls. clarify this relationship: Long Stock = Long RFR Bond + Long Futures Let us say, the stock price moves up -> the bond acts like a floor and long futures replicates the gain (the graph is similar to that of long call payoff) But when the stock moves down, say to zero, the long RFR bond still acts like a floor and prevents the synthetic position (right hand side of equation) from going down to zero. Thanks.

portfolio A - long stock at time T, value of portfolio A = S[T]; if S[T] = 0, then value = 0 portfolio B - long bond and stock futures at F = S (1 + RFR) (fairly priced against arb) at time T, value of bond position = S ( 1 + RFR ); value of futures postion = S[T] - F; value of portfolio B = S[T]. if S[T] = 0, the value = 0; portfolio A and portfolio B are indeed replicating each other.

So you think a long futures position can’t go through 0…

Future is symmetric.

randOm, can you explain what happens to the Long RFR bond when Stock goes to 0? The futures would go zero, this I understand, but wouldn’t there be a value left in the RFR bond?

Lets do an example to understand this better. Input Data: Current Price of Stock at time 0 : 100 Risk free rate : 10% Time period of the futures contract : 1yr From the given data, futures would be trading at 100(1+10%) = 110. Let us consider the scenario in which after one year, the stock is trading at 0. Portfolio A: Contains 1 stock Initial Outflow: 100 Final value of the portfolio: 0 Loss on portfolio A is (0-100) = -100 Portfolio B: Contains 1 RFR Bond with notional 100 and one long futures contract with exercise price of 110. Initial Outflow: 100 for RFR bond + 0 for Futures. Final Value of portfolio: (+110) + (-110) + 0 = 0 +110 for the inflow you get for the RFR bond. -110 for the outflow to buy the stock as promised in the futures contract. 0 for the value of the stock at time of expiry S(T) So, total loss on portfolio B is (0-100) = -100 So, even in the scenario in which the stock goes to 0, the portfolios A and B are equivalent. I hope this helps.

So what happens if the stock drops to 0 on the first day?

You get your money back as you have a 48 hour return policy, but you need your trade statemetn.

CareerChange Wrote: ------------------------------------------------------- > randOm, can you explain what happens to the Long > RFR bond when Stock goes to 0? > > The futures would go zero, this I understand, but > wouldn’t there be a value left in the RFR bond? in this case, the long rfr position accumuates just enough (F) to cover the loss in the future postion (-F).

So I’ve never traded a futures contract on a single stock but in the US they are deliverable contracts so “you’ll get the difference, but if it’s lower than the price you’ll have to pay the difference” is not really true. But we’re getting messed up here on other stuff. First, Long Stock = Long RFR Bond + Long Futures is sort-of true, but it doesn’t mean that if I buy a bond and a futures contract in portfolio A and let it ride I will have exactly the same amount of money as if I had a stock. The problem is the margin account. Next, Long Stock = Long RFR Bond + Long Forward is exactly true and I will have the same amount of money in either account and I should be indifferent to either side of “=” (modulo taxes, transactions costs, liquidity, etc). Alas, the futures account is marked-to-market daily and I get interest on the margin balance. That means the RFR bond doesn’t even work here, but let’s say it’s a risk-free money market account and there is no difference in the interest rate on the overnight money market account and the forward contract rf bond (which is wrong). If my stock plummets to 0 on the first day, I’ve got a big problem with the futures contract (bigger even than holding the stock). My futures contract was at a price higher than the current stock price and now my margin account gets dinged for the entire amount. I gotta come up with more money than my “RFR bond” is worth or they will take my house and my boat (canoe, actually). On the other side, if my stock triples in price on the first day I’m now collecting three times as much interest as I thought I would. That equation doesn’t mean you will have the same thing; it means that the futures contract is priced fairly this way (but that’s another matter). Further, as bruce points out the dividends need to be included in the futures price where they are just a direct offset to the interest rate.

bigwilly Wrote: ------------------------------------------------------- > You get your money back as you have a 48 hour > return policy, but you need your trade statemetn. I am rolling on the floor…

Just to take a break from the boring qualitative portion of level 3, Here is the mathematical proof: Left Hand Side: Long Stock: Value at time t = S(t) - S0 ------------------- (1) S(t) = stock px at t; S0 = initial stock price Right Hand Side: Starting with same initial amount of S0, Long RFR Bond Value at t = S(T)/(1+Rf)^(T-t) - S0 S(T) = S0*(1+Rf)^T --> Value = S0*(1+Rf)^t - S0 ------------ (2) Long Future Value at t = S(t) - FP/(1+Rf)^(T-t) FP = S0*(1+Rf)^T (no arbitrage) --> value = S(t) - S0*(1+Rf)^t ------------------ (3) As you can see (1) = (2) + (3) Proves that long stock = long RFR bond + Long Futures - bn

Can we think of the relationship from the concept of CAPM?

Let’s simplify it away from equations. Long stock has two return components - capital gains and dividends. Dividends are nothing more than a series of periodic cash flows. This is like a bond. Capital gains (and losses) are similar to taking a futures position since futures do not pay out cash flows but are capital gains and losses. Therefore if you add a bond (dividend) and a future (capital gain) together you get a synthetic long position in a stock.

jasonysy Wrote: ------------------------------------------------------- > Can we think of the relationship from the concept > of CAPM? ? CAPM has nothing to do with this. CAPM isn’t even true.

BN Wrote: ------------------------------------------------------- > Long RFR Bond Value at t > > = S(T)/(1+Rf)^(T-t) - S0 > S(T) = S0*(1+Rf)^T > not ture, if S(T) = So * (1 + RF) ^ T, who would invest in stock?

rand0m Wrote: ------------------------------------------------------- > BN Wrote: > -------------------------------------------------- > ----- > > Long RFR Bond Value at t > > > > = S(T)/(1+Rf)^(T-t) - S0 > > S(T) = S0*(1+Rf)^T > > > > not ture, if S(T) = So * (1 + RF) ^ T, who would > invest in stock? Right - It’s not true. It’s F(t) = S0*(1 + r)^t