Foundations 1b Tutorial
- Thread starter roylauw
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Hi Roy,
FWIW, the XLS is here @ http://www.bionicturtle.com/premium/spreadsheet/1.b.3_stulz_hedging_irrelevance/
So I applied the CAPM; e.g., under the scenario where beta = 0.5, then:
one-period E(St) = spot(0) * (1 + 5% RF + 6% ERP * 0.5 beta)
Please note this has occasionally "frustrated" previous candidates because it is ultimately circular; specifically, i get the price by discounting the Forward at the riskless rate. All I am doing here is illustrating Stulz and Hull's CAPM-based pricing framework; ultimately, the circularity is the premise itself.
To elaborate, the first idea (per Hull & Stulz) is that forward price is a function of the RISK-FREE rate, in this case:
333 spot (0) = $350 forward (0); i.e., per the cost of carry
yet the expected future spot is compounded at > RF rate if the asset has beta > 0:
@ beta = 0.5, expected spot (1) = $333 spot (0) * 1 + 8% = $360; i.e., per CAPM!
to summarize, at beta = 0.5, $360 E(St) is higher than (different than) the $350 forward exactly due to the beta.
and now the "theory of normal backwardation" resolves because it says F(0) < E(sT)--i.e., normal backwardation--must exist where the asset has systematic risk in order to induce the long forward: to go long the risky asset, he/she expects a profit.
In this case, $350 forward * EXP(3% risk premium) ~= $360 expected future spot
i.e., the long forward position is willing to enter long @ 350 b/c he/she expects the profit to compensate exactly for assuming the 3% risk premium.
... and this is just a fancy way of saying the short forward has transferred price risk to the long forward, for a fee
hope that helps, more than you asked but wanted to explain why the math is not inherently satisfying (but, I think the underlying idea is!)
David
FWIW, the XLS is here @ http://www.bionicturtle.com/premium/spreadsheet/1.b.3_stulz_hedging_irrelevance/
So I applied the CAPM; e.g., under the scenario where beta = 0.5, then:
one-period E(St) = spot(0) * (1 + 5% RF + 6% ERP * 0.5 beta)
Please note this has occasionally "frustrated" previous candidates because it is ultimately circular; specifically, i get the price by discounting the Forward at the riskless rate. All I am doing here is illustrating Stulz and Hull's CAPM-based pricing framework; ultimately, the circularity is the premise itself.
To elaborate, the first idea (per Hull & Stulz) is that forward price is a function of the RISK-FREE rate, in this case:
333 spot (0) = $350 forward (0); i.e., per the cost of carry
yet the expected future spot is compounded at > RF rate if the asset has beta > 0:
@ beta = 0.5, expected spot (1) = $333 spot (0) * 1 + 8% = $360; i.e., per CAPM!
to summarize, at beta = 0.5, $360 E(St) is higher than (different than) the $350 forward exactly due to the beta.
and now the "theory of normal backwardation" resolves because it says F(0) < E(sT)--i.e., normal backwardation--must exist where the asset has systematic risk in order to induce the long forward: to go long the risky asset, he/she expects a profit.
In this case, $350 forward * EXP(3% risk premium) ~= $360 expected future spot
i.e., the long forward position is willing to enter long @ 350 b/c he/she expects the profit to compensate exactly for assuming the 3% risk premium.
... and this is just a fancy way of saying the short forward has transferred price risk to the long forward, for a fee
hope that helps, more than you asked but wanted to explain why the math is not inherently satisfying (but, I think the underlying idea is!)
David
Thanks David thats very clear,
Some points:
1. E(St) = spot(0) * (1 + 5% RF * 6% ERP * 0.5 beta) should be E(St) = spot(0) * (1 + 5% RF + 6% ERP * 0.5 beta) ?
2. Are you assuming E(R) = E(S_future) / E(S_current)? in this case you are using arithmetic return instead log reutrn?
Thanks
Roy
Some points:
1. E(St) = spot(0) * (1 + 5% RF * 6% ERP * 0.5 beta) should be E(St) = spot(0) * (1 + 5% RF + 6% ERP * 0.5 beta) ?
2. Are you assuming E(R) = E(S_future) / E(S_current)? in this case you are using arithmetic return instead log reutrn?
Thanks
Roy
Hi Roy,
1. Yes, my typo. I fixed above. Thanks for your detailed eye....
2. Well, notice i deliberately inserted the "~" b/c i was aware of the arith/geo mixing. B/c i tend to think continuously, I was using continuous as in: ln(360 forward/350 spot) ~ 3%. But okay, it's actually = ln(360/350) = 2.817%. And the arithmetic annual "equivalent" is 360/350 - 1 = 2.857%. Now if Stulz used continuous throughout, like Hull, you'd have something maybe more pleasing:
E(St) = 333.3 * EXP(5%) = 361.1
F0 = 333.3 * EXP(8%) = 350.42
such that implied continuous return for the long forward position = ln (361.1/350.42) = exactly 3% premium
For me, it's worth tying this back to Stulz's point. We have showed how systemic asset risk (i.e., beta > 0) implies normal backwardation (i.e., F0 < E(St)) which implies an expected gain for the long forward position (expected future gain = E(St) - F0)).
So, the key "finding" is that the long forward expects a profit and the short forward expects a loss (the fee for transferring risk).
Stulz point is that the firm can reduce it's systematic risk only by incurring a cost (the same cost the short forward expects) so the gains in a reduction in discount rate (lower beta) are offset by the cost incurred to achieve the beta reduction; i.e., no free lunch so far as beta is concerned
(it is a different story for idiosyncratic risk: in that case, CAPM is saying nobody will pay for it; idiosyncratic risk reductions are effectively worthless)
David
1. Yes, my typo. I fixed above. Thanks for your detailed eye....
2. Well, notice i deliberately inserted the "~" b/c i was aware of the arith/geo mixing. B/c i tend to think continuously, I was using continuous as in: ln(360 forward/350 spot) ~ 3%. But okay, it's actually = ln(360/350) = 2.817%. And the arithmetic annual "equivalent" is 360/350 - 1 = 2.857%. Now if Stulz used continuous throughout, like Hull, you'd have something maybe more pleasing:
E(St) = 333.3 * EXP(5%) = 361.1
F0 = 333.3 * EXP(8%) = 350.42
such that implied continuous return for the long forward position = ln (361.1/350.42) = exactly 3% premium
For me, it's worth tying this back to Stulz's point. We have showed how systemic asset risk (i.e., beta > 0) implies normal backwardation (i.e., F0 < E(St)) which implies an expected gain for the long forward position (expected future gain = E(St) - F0)).
So, the key "finding" is that the long forward expects a profit and the short forward expects a loss (the fee for transferring risk).
Stulz point is that the firm can reduce it's systematic risk only by incurring a cost (the same cost the short forward expects) so the gains in a reduction in discount rate (lower beta) are offset by the cost incurred to achieve the beta reduction; i.e., no free lunch so far as beta is concerned
(it is a different story for idiosyncratic risk: in that case, CAPM is saying nobody will pay for it; idiosyncratic risk reductions are effectively worthless)
David
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