Forward price for an asset with a known percentage yield

[indra]

How can I derive the formula for the forward price of an asset which gives a known % yield? I don't get the logic behind it.
The formula is F = S*exp(rf - q)t ; where rf is the risk free rate ,q is the known % yield, S is the asset's spot price , F is the forward price and t is the time into the future when the forward price is being sought.

 

[ShaktiRathore]

This is basic no-arbitrage principle:
If we assume F>S*exp(rf - q)t then the arbitrageur could easily short the more expensive Forward at value F and long the Asset at price S by getting a loan of S at rate of rf ,after time t at maturity the loan payback is S*exp(rf*t) while the dividends earned on asset at continuous rate of q over time t while there is cost of continuous rate of rf over time t means effective cost in terms of continuous rate rf->rf-q that the effective payment is S*exp(rf-q)t after time t therefore after selling the asset at F ,net cash flow is F-S*exp(rf-q)t >0 therefore the arbitrageur makes a riskless profit.
If we assume F<S*exp(rf - q)t then the arbitrageur could easily long the less expensive Forward at value F and short the Asset at price S and deposit S at rate of rf ,after time t at maturity the payback from deposit is S*exp(rf*t) while the dividends lost on asset at continuous rate of q over time t while there is interest of continuous rate of rf over time t means effective interest earned in terms of continuous rate rf->rf-q that the effective payment received is is S*exp(rf-q)t after time t therefore after buying back the asset at F ,net cash flow is S*exp(rf-q)t-F>0 therefore the arbitrageur makes a riskless profit.
Thus no-arbitrage principle holds and we say that arbitrageur could not make a riskless profit, finacial products are priced that way so that nethier of the above two inequlaities hold and thus we conclude that F=S*exp(rf - q)t.
thanks

 

[David Harper CFA FRM]

Hi @indra Hull briefly discusses this in Chapter 5 (although Kolb and McDonald go into much detail). It's a no-arbitrage argument. To simplify, say the current spot is $100 and the Rf rate is 3%. Cost of carry so far (without storage costs) says the one-year forward price F = 100*exp(3%*1) = $103. Now assume the commodity produces income (yield) of 3% per annum. Now F = 100*[exp(3%-3%)*t] = $100.

The no-arbitrage argument holds that if the forward price is too high, you can "cash and carry" for guaranteed profit. If above, say F(0) = $103 while the asset pays 3% income (no storage cost). We can cash and carry: borrow $100 today in order to buy the commodity at today's spot price; delivery the commodity in one year and collect $103 per the forward contract, which is then used (exactly!) to retire the loan with $103.00. Except we earned the dividend as guaranteed profit! Any forward price that is too high (too low) will allow for (reverse) cash and carry arbitrages, which will move the forward price to the no-arbitrage zone (i.e., within transaction costs).

Ps. Sorry for cross-posting @ShaktiRathore !

 

[indra]

Thanks Shakti and David !!

 

[AnaP]

Hi @David Harper CFA FRM and @ShaktiRathore
I can prove that the relationship must hold using the arbitrage argument, however, I was wondering how to prove the relationship using the principle of equivalence ( EPV(cash inflows) = EPV(cash outflows)) or perhaps setting up two portfolios and then using the law of one price to show that Fo = Soexp(r - q)T.

Thanks in advance.

 

[David Harper CFA FRM]

Hi @AnaP I've only seen the no arbitrage argument to prove COC, sorry. It's not obvious to me how law of one price can be applied (I could be wrong of course!). I'm fuzzy too on how "equivalence" would be really much different. If we are at today, T(0), and we want to own the commodity at time, T(t), then we can borrow, inflow -S(0), to buy the spot commodity, +S(0); or we can enter the long futures contract and we should expect some equivalence. In either case, our net inflow/outflow today is zero and our expected future position is E[S(t)] - [S(0)*(1+c)], where [S(0)*(1+c)] = [S(0)*(1+r)] and is the repayment of the loan. I think the nuance is that we might assume F(0) = E[S(t)] but the theory of normal backwardation asserts that F(0) < E[S(t)], such that the "equivalence" I am thinking is distorted by the risk premium embedded in the futures contract (aka, compensation for risk aversion). Thanks,