Davidson asks "My question is this: Today the Fed cut the discount rate and S&P;futures subsequently rose. If the Fed cut the Fed Futures rates wouldn't that imply that futures should decrease if we base it on the cost of carry model? I guess I'm confused because it seems a rate cut should boost the futures price but that seems to contradict the cost of carry model."
I love this thought-provoking question that teases out a couple of key FRM concepts. The correct answer, of course, is "I don't know" why markets do things. But I worry that isn't helpful.
Here is a short answer:
* The cost of carry gives the future price as function of the spot price: f = (spot)exp[(cost - beneft)(T)]
* In this case, where the Fed cut rates, the spot (S&P;500) increased, but the future (future on S&P;index) increased by less - so, by luck, the carry model did work.
* For financial assets, the interest rate (r), impounds many expectations - including the cost of equity capital in addition to the riskless rate
* The delta of futures contract is about 1 [=exp(rt)]. A futures contract is that rare thing: a linear derivative.
And a longer answer....
In my head, as you know from my tutorial on this, I reduce the cost of carry into: future price = (spot) * exp[(cost - benefit)(time)]. That is, the cost of carrying an asset (i.e., interest rate for fin'l asset, storage cost for commodity) tends to increase the futures price; while the benefits of carrying (owning) the asset (i.e., a dividend yield or a 'convenience yield') reduce the futures price. So, this cost of carry LINKS the spot price to the futures price. In the case of a financial asset like the S&P;futures, you only have the rate (the interest you'd pay on money to borrow) which increases the future price and the dividend which decreases the future price. So, in theory, the futures price is a function of the spot price grown (continuously compounded) by the difference between [rate - dividends].
But just a quick digress: Hull refers to rate (r) as the riskless rate, but somewhere he notes it isn't really riskless it is nearer to LIBOR. Culp is semantically more precise; he calls (r) the interest rate or, better, the discount rate. Which gets back to a more primitive point, the futures price is the market's consensus estimate of the EXPECTED FUTURE SPOT PRICE (or if you agree with thy of normal backwardation, which holds up pretty well, it's a little bit less than this to encourage speculators, as if they need encouraging!). So, we can look at a futures price of $11 compared to a spot price of $10, and all we really can say in the moment is "the market expects the spot to be pretty near $11 bucks about then." Then if we want to force-fit the market's sentiment (which day-to-day overwhelms our model), we can say, "the market is pricing a 12% discount rate net of a 2% dividend." Where r = 12%, what Hull calls the riskless rate and Culp the interest/discount rate. I would like to think that the discount rate part of this is the market's COST OF EQUITY, equal to riskless rate plus the equity premium. That is, the market's cost of equity = CAPM = riskless + equity premium.
Neglecting the fact that the Fed cut a pseudo-symbolic rate rather than the fed funds rate, what does a lower short-term rate do to this cost of carry model? Really, it could go either direction because you can point to opposing forces. Yes, it will lower the riskless part of the discount rate. But on the other side, you have the factors that contribute to the empirical tendency for stocks to move in the opposite direction of interest rates. Lower rates presage corporate growth. Importantly, the cost of equity is nominal, so inflation expectations are a big factor. My point is, the nominal (r) in the cost of carry breaks up into a riskless piece, a risk premium, and an inflation piece. So, if the Fed lowers a short-term rate, the impacts on nominal (r) are multifaceted.
I can't say what happened to nominal (r) today, but I do see that, last time i looked, the S&P;500 increased 36% to 1447 while a near term futures increased 25% to 1450. Notice that happens to be consistent with cost of carry where the only change is a lower (r): the future price increased less than the spot. But, okay, day-to-day, I don't read anything into that.
Your question also concerns another section of Hull: the delta of a futures contract. The delta of a futures contract is the change in the futures price given a change in the underlying (spot) stock price. Note the delta of fowards/futures are both pretty close to 1.0; due to daily mark-to-market, the delta of a futures is exp(rT); i.e., a little bit more than 1.0. So, your question could break down into two parts: (1) what is the impact of fed rate on spot prices (of course: I have no idea), and (2) what is impact of change in spot price on future price? Answer: that's delta, so for the S&P;index the future and the spot should move pretty much in lockstep. The cost of carry connects the spot to the future price, and it's ambiguous precisely whether/what impact the fed move would have on that relationship
Sorry for length, Davidson, hopefully you see why I LOVE this question because it relates to two core FRM ideas (cost of carry and delta).
I love this thought-provoking question that teases out a couple of key FRM concepts. The correct answer, of course, is "I don't know" why markets do things. But I worry that isn't helpful.
Here is a short answer:
* The cost of carry gives the future price as function of the spot price: f = (spot)exp[(cost - beneft)(T)]
* In this case, where the Fed cut rates, the spot (S&P;500) increased, but the future (future on S&P;index) increased by less - so, by luck, the carry model did work.
* For financial assets, the interest rate (r), impounds many expectations - including the cost of equity capital in addition to the riskless rate
* The delta of futures contract is about 1 [=exp(rt)]. A futures contract is that rare thing: a linear derivative.
And a longer answer....
In my head, as you know from my tutorial on this, I reduce the cost of carry into: future price = (spot) * exp[(cost - benefit)(time)]. That is, the cost of carrying an asset (i.e., interest rate for fin'l asset, storage cost for commodity) tends to increase the futures price; while the benefits of carrying (owning) the asset (i.e., a dividend yield or a 'convenience yield') reduce the futures price. So, this cost of carry LINKS the spot price to the futures price. In the case of a financial asset like the S&P;futures, you only have the rate (the interest you'd pay on money to borrow) which increases the future price and the dividend which decreases the future price. So, in theory, the futures price is a function of the spot price grown (continuously compounded) by the difference between [rate - dividends].
But just a quick digress: Hull refers to rate (r) as the riskless rate, but somewhere he notes it isn't really riskless it is nearer to LIBOR. Culp is semantically more precise; he calls (r) the interest rate or, better, the discount rate. Which gets back to a more primitive point, the futures price is the market's consensus estimate of the EXPECTED FUTURE SPOT PRICE (or if you agree with thy of normal backwardation, which holds up pretty well, it's a little bit less than this to encourage speculators, as if they need encouraging!). So, we can look at a futures price of $11 compared to a spot price of $10, and all we really can say in the moment is "the market expects the spot to be pretty near $11 bucks about then." Then if we want to force-fit the market's sentiment (which day-to-day overwhelms our model), we can say, "the market is pricing a 12% discount rate net of a 2% dividend." Where r = 12%, what Hull calls the riskless rate and Culp the interest/discount rate. I would like to think that the discount rate part of this is the market's COST OF EQUITY, equal to riskless rate plus the equity premium. That is, the market's cost of equity = CAPM = riskless + equity premium.
Neglecting the fact that the Fed cut a pseudo-symbolic rate rather than the fed funds rate, what does a lower short-term rate do to this cost of carry model? Really, it could go either direction because you can point to opposing forces. Yes, it will lower the riskless part of the discount rate. But on the other side, you have the factors that contribute to the empirical tendency for stocks to move in the opposite direction of interest rates. Lower rates presage corporate growth. Importantly, the cost of equity is nominal, so inflation expectations are a big factor. My point is, the nominal (r) in the cost of carry breaks up into a riskless piece, a risk premium, and an inflation piece. So, if the Fed lowers a short-term rate, the impacts on nominal (r) are multifaceted.
I can't say what happened to nominal (r) today, but I do see that, last time i looked, the S&P;500 increased 36% to 1447 while a near term futures increased 25% to 1450. Notice that happens to be consistent with cost of carry where the only change is a lower (r): the future price increased less than the spot. But, okay, day-to-day, I don't read anything into that.
Your question also concerns another section of Hull: the delta of a futures contract. The delta of a futures contract is the change in the futures price given a change in the underlying (spot) stock price. Note the delta of fowards/futures are both pretty close to 1.0; due to daily mark-to-market, the delta of a futures is exp(rT); i.e., a little bit more than 1.0. So, your question could break down into two parts: (1) what is the impact of fed rate on spot prices (of course: I have no idea), and (2) what is impact of change in spot price on future price? Answer: that's delta, so for the S&P;index the future and the spot should move pretty much in lockstep. The cost of carry connects the spot to the future price, and it's ambiguous precisely whether/what impact the fed move would have on that relationship
Sorry for length, Davidson, hopefully you see why I LOVE this question because it relates to two core FRM ideas (cost of carry and delta).
