Market Risk - Part A

[humheehum]

Hi David,

I have some questions/clarifications required on Part 1 of the above screencast.

1) Reason for deduction of any income from future's price formula- is this because as we are entering into a derivative contract instead of holding the underlying asset we do not actually get the benefit of the underlying cashflows/products of the asset. Therefore, we are adjusting the price down to reflect this. Eg: - if there are some contracts on say hens (I making this up), if you are the farmer you get the benefit of eggs, but since you are in a futures contract (and you would presuambly not take delivery) you do not get those eggs, and hence price gets adjusted down to reflect this lack of benefit. Similarly cost would get added to reflect the relative "advantage" you get for the maintaince/storage of the commmodity ?

2) Relation between future and forward prices -

a) I would have thought the future price would be higher as futures would eliminate any credt risk and the higher price of a future reflect this ?

b) Could you please elaborate how correlation between Spot and interest rate impacts future/forward price relation. I think this may be due to the margin feature of the futures. If future price decline and interest rates increase - then the long position has a double loss - i) Loss on future ii) Need to borrow money at a higher interest rate to meet any potential margin call. To compensate for the potential double loss compared to the forward, when you will only have loss i, the future price might be lower

c) is there a convergence between spot,future and forward price at expiration ? (the beginning slides of the screencast show a diagram to suggest the above the above)

3) Value of a forward contract - I have not understood this. I would have thought that to place a value of a forward at any time in the future but before expiration the value of your contract should be equal to
= S(t) +/- (PV of remaining Cashflows ie cost/income) - PV[F(0,T)]

where PV[F(0,T)]s the foward price you entered into at t(0) for expiration at time T, discounted till time time t; and
S(t) is the price at time period t.

However, on page 22 of the notes its given as f = [F(0) - K]e^(rt). I have not understood this.

4) Normal Backwardation and Contango - could you please explain how the supply and demand consideration from hedgers and speculators establish the relationship. Does it have something to do with if expected benefits and are greater than expected costs then you would want to hold the spot, thus increasing its expected spot price more than the future ?

Many thanks for your help.

Regards,
Ashim

 

[humheehum]

Hi David,

I have some additional questions on Part1

-Eurodollar futures :-
1) Both the Eurodollar future and the eurodollar deposit effectively derive value from the Libor rate. But isn't there a mismatch as if Libor increases the underlying will increase in value = principal + higher interest. While the derivative would lose value 100 - Libor will lead to a lower quote on the futures contract ? So how is the euro dollar future used to hedge or lock an interest rate?

2) For the Ho and Lee adjustment could you please explain what T1 and T2 are - I have not fully understood the distinction.

Interest rate Swaps
1) When valuing swaps using bonds could you please explain how the fixed rate is decided ? Is it entirely upto the counterparty's or more to do with the libor curve. I ask this because we know the swap should have a value of zero at initiation, we also know that no amount is exchanged between the parties, then there must be a a fixed rate to ensure the above holds. Since the value of the floating would be 100$ or par and we know the future spot libor rates at initiation then it should be the coupon of the bond that solves to give us the fixed rate.

But if the above is the case then the fixed rate is not actually fixed - it will keep on changing as the Libor curve changes - is it therefore for simplicity we have assumed that the Libor curve does not change and the same curve applies at beginning and 30 days after/before the first coupon?

2) Could you please explain the intuition behind using the FRA to value the floating leg of the swap. Is it simply saying that if you are long on a FRA you pay fixed, but if you short a FRA you will pay floating and recieve fixed ?
where the floating part are based on future Libor curve. Since FRA tells us the rate which we lock into say for 6 months starting 3 months from now, we repeatedly calculate this forward rate and compare it against the fixed rate we entered ?

Currency Swaps

1) Why is there no coupon being paid for the final leg of the swap on page 43 of the notes. If all we are now saying is that instead of holding two US$ denominated bonds, we now are long Yen bond and short US $ bond, then we should ideally be showing the coupon payment with the principal in the end.
2) The F/X rate provided as 110 - does this apply at the end of 2 years. Because looking at the notional amounts at the beginning this exchange rate is 1$ = 100 Yen. If so, then this would mean that the yen has appreciated, and since we are recieving yen we stand to gain or have a positive value - as seen in the solution.
But this would only work in case of fix for fix swap as there is no interest rate effect.
Could you please let us know do we need to know how to value currency swaps with floating payments - if so, could i request you to upload a spreadsheet to clarify the calculation.
3) Again I have not understood how to value a swap using FRA - if possible can you do a 5 min screencast to explain its valuation.

Apologize for the long requests. I think I have only now woken/realized how difficult FRM exam is.

Thanks again.
Ashim

 

[David Harper CFA FRM]

Hi Ashim,

1) Exactly, I love the hen and eggs metaphor, I will be using it! It is just as you say, and why i have framed cost of carry generally as forward = spot *EXP[(costs - benefits)(T)] as the "carry influences" (e.g., convenience, income, financing, storage) are all either (i) costs to hold or (2) benefits of holding (and if a benefit to holding, then forgone "opportunity cost" to the futures holder).

2a) Yes, fantastic point! The assigned reading has not caught up to this idea; i.e., we are studying a model based on market risk and it simply does not incorporate counterpary (credit) risk. I personally think you are (you must be) correct: counterparty risk > convexity bias. So this is simply an omission (simplification, shortcoming) of the model we are looking at. It is not so unusual; Hull's CDS valuation does not incorporate counterparty risk, but could and probably should. (another arguable omission: the forward pricing does not deduct for illiquidity). Counterparty risk is interesting: it is late entering into some of these models, but not anymore after this credit crunch. But it also, IMO, reflects the "silo mentality" where futures/forward fall under market risk and loans fall under credit risk, and recently there is much greater appreciation for their blurred interdependence.

2b) There are two contexts for this, in Hull. In regard to Eurodollar futures/FRA (i.e., intest rate forwards), lower rates will lower the value of the interest rate forward (as the spot:forward will be 1:1 or nearly), so in the case of an Eurodollar futures holder, higher rates are a double-bonus: higher forward plus excess margin is invested at higher rate. Lower rates are mitigated: lower forward, but borrowing at lower rate for margin call.

The other context is regarding the rate (r) used in cost of carry. When Hull uses riskless rate, he is presuming no correlation between spot and market - i.e., no systematic risk. (in other words, probably not realistic). If systematic risk (correlation between spot and market), then the expected future spot price exceeds the futures price (normal backwardation).

3) Page 22 has a typo, apologies, should be: f = [F(0) - K]e^(-rt). Just as you say:
f = PV[spot (t)] - PV [delivery]
f = S(T)e^(-rT) - K*e^(-rT). As E[S(T)] = F(0)
f = F(0)e^(-rT) - K*e^(-rT) = (F - K)e^(-rT)

4) well, supply/demand does not enter the cost of carry model that we are reviewing. There are maybe two ways to look at this:

4a) Hull says, if the underlying and the market are correlated (systematic risk), then F < E(ST); i.e., normal backwardation. WHy? Because with systematic risk, the discount rate attached to the forward must be greater than the riskless rate, so the future price comes down.

4b) A (more intuitive) supply/demand idea is the theory of normal backwardation: assume F = E(ST) is basically true. But, as a speculator (long the hedge), why would I assume risk to make zero profit in the future? I would not, I will only pay F < E(St) because I must have an expected profit to compensate me for the risk (i.e., risk premium compensation)

....new post for next....

David

 

[David Harper CFA FRM]

Ashim,

Eurodollar futures

1) You are right. If rate is 4%, then quote is 96 (100-4) and contract price (defined by CME) is 10,000*[100-0.25*(100-Quote)]: 992,513. If LIBOR increases to, say, 6%, then contract price goes down to 985,000. Loss is $5,000. Loss is $25 per basis point.
Here is brief XLS https://www.editgrid.com/bt/admin/eurodollars ...
... that I used for this screencast:

But it looks like you get that part. Consider the hedger at 4%. And LIBOR goes up to 6%.

Without the hedge: he/she pays the higher LIBOR
With the hedge: he/she pays the higher LIBOR (at 6%) but profits on the futures contract (+$5,000 = $25/bps) and the net cost is the same as LIBOR at 4%.

2) T1 is maturity of the eurodollar futures contract; T2 in the case is always +0.25 b/c it is the maturity of the underlying rate which is always three month. So, if ED futures is 5 years, then (T1)(T2) = (5)(5.25), if 7 years, then (7)(7.25)

interest rate swap

1) Exactly correct. At inception, PV of fixed leg = PV of floating let. Given the LIBOR forward curve, it is only a matter of solving for the fixed rate that produces the correct PV. Since PV of floating is par ($100 or $1000), the swap rate is the same as solving for the yield to maturity on a the fixed leg bond where price = par (100, 1000). That YTM will give the swap rate or coupon rate for the fixed leg.

"fixed rate is not actually fixed" No. the fixed rate coupons will not change. The yield curve will change and, going forward, this will impact both the floating rate coupons and the discount rates. But going forward, there is no expectation the value will equal zero, so there is no constraint. But still, the static LIBOR is a to keep it simple.

2) You might want to look at the 2nd tab of this EditGrid XLS (four tabs, 2 irate, 2 currency swaps). The valuation in terms of FRA is simpler than it appears:

The LIBOR curve contains a set of implied six-months LIBOR rates
(just like in Tuckman, this is a key concept: any spot rate curve contains embedded a set of forward rates)
Given the six-month forward rates, this "FRA approach" conducts a manual cash flow analysis:
First, what is the series of future fixed cash flows,
Second, what is the series of future floating cash flows [using the forward rates. So the forward rates are to predict the applicable LIBOR rates and get the floating coupons]
Third, net them out (just like the real swap nets) to get a series of net future cash flow
Fourth, discount using the current LIBOR

Currency Swap

1) Agreed, following Hull notice that the last two rows are both YEAR 3 - the coupon is parsed on its own row
2) The notionals are input assumptions, the ratio of notionals does not need to match the FX exchange rate.

Please note, there is a four page XLS duplicating Hull's method: http://www.bionicturtle.com/premium/editgrid/2008_frm_hull_derivatives_swaps/

I will indeed publish up a couple of screencasts on currency swap to match http://www.bionicturtle.com/learn/article/valuation_of_interest_rate_swap_9_min_screencast/


"Apologize for the long requests." Please there is no need, these are thoughtful questions, I thank you for spending the time to post such thoughtful questions so that others might get the benefit.
"I think I have only now woken/realized how difficult FRM exam is." Understood, it is better than to realize this one month later.

David

 

[humheehum]

David - many thanks for the reply. Cleared many mental blocks. However, couple of queries still remain.

First Post

1) Could you please explain why one should expect convergence between future and spot prices at expiration? is this something to do with mark to market feature of the futures contract. If 10 minutes before expiration, spot price (say 5$) and the future price (2$) deviate. Then one could immediately buy the future(2$), take delivery, and sell in the market at a higher price(5$). Since the future contract is marked to market this profit will immediately show in the traders account? Will this be true for forwards as well, given the absence of liquidity and mark to market feature?

2) I still have not understood the valuation of future contract. I don't get the distinction between F and K. Isn't the delivery price the price we have locked ourselves into by purchasing the future, if so, then shouldn't we be comparing the delivery price against spot to arrive at a value? instead of the F i.e. the price of the futures contract?
I think I may not have understood something conceptual about how futures contract work?

3) Could you please elaborate why under systematic risk forward discount rate would be greater than risk free - is this linked in some way to CAPM r = rf + systematic risk(rm-rf) ? so in the presence of beta r > rf ? Then by discounting using a higher disc rate your forward value is lower, than a forward under riskfree discount rate = E(Spot) ??


Second Post

1) So to hedge interest exposure using euro dollar futures we would be holding a short position if we expect interest rates to risk. As Libor increases, futures price declines, since we are short we make a gain offsetting the increase in Libor - thus keeping our net cost the same

2) I think I have understood the calculation using the FRA. Just to ensure I have understood this correctly, could you please correct/agree with the following statements.

Long FRA - pay fixed, receive floating
Short FRA - receive floating, pay fixed

With a FRA you are effectively locking into a forward rate i.e a rate of interest that you agree to pay or receive for say 6 months starting 3 months from now i.e 3f9. Now, Since you are valuing the FRA say 3 months into the contract, our fixed rate remains the same (just like the swap) but given a LIBOR curve, we can calculate a new 6month rate starting 3 months or 9 months from now - and these are the floating rates (as they depend on a changing interest rate curve)

Thanks for your help again.

Regards,
Ashim

 

[David Harper CFA FRM]

Hi Ashim,

1) On convergence: yes, exactly as your example shows, it is a no-arbitrage idea. At the instant before expiration/delivery, a difference ought to create a riskless profit (riskless profit = arbitrage). In regard to the forward, it is the same idea: as convergence refers to the what happens as expiration nears, we don't really need the mark-to-market overlay to make the point. So yes should be true of forwards as well. But notes, as with so many financial models, it is a simplified version of reality. In reality, it is called a zone of convergence and fascinating developments earlier in the year where convergence was not happening in corn, wheat, etc. Namely, our model assumes DELIVERY is costless, but it is not (the models are only models!)

2) What you say makes sense, and, yes the delivery price (K) is fixed. Maybe it is just the final formula form that confuses.

In the future, if we are long we will profit by: S(T) - K

We know (K) but we do not know S(T), let us use S(0) to estimate S(T):
S(T) = S(0)EXP(rT), so in the future
f(t) = S(0)EXP(rT) - K

The above matches, IMO, what you are saying. It is not wrong; at contract inception, it should equal ZERO per cost of carry as K should equal S(0)*EXP(rT)

But the other form (above) is the (net) value of the forward contract *AFTER* time has elapsed.
At contract initiation, Forward Price = Delivery (K)
As time passes, delivery will be the same for the long, but the observed forward price will vary to F(0).
and having an updated F(0) means we can replace
f(t) = S(0)EXP(rT) - K with
f(t) = F(0) - K. See how F(0) is just a market price estimate of E[S(T)] which is also estimated by S(0)*exp(rT)?

so then we PV this:
If f(t) = F(0) - K, then f(0) = [F(0) - K]*EXP(-rT)

3) Yes, exactly! The CAPM will do fine here (though it could be any model that links expected return to systematic risk) but Hull even names CAPM so you are spot on. I think you've written it exactly right and this is a nuance in the Hull reading. It is related to the apparent mystery throughout the Hull reading (or at least, was for me the first time i read it years ago): how can he keep using the riskless rate (r)?; e.g., F = S*exp(rT). The answer is he is implicity assuming no systematic risk (to keep it simple) so under CAPM, E(r) = riskless + (0 beta)(ERP) = riskless

1) "So to hedge interest exposure using euro dollar futures we would be holding a short position if we expect interest rates to RISE" Yes, EXACTLY AS YOU HAVE IT. It is easy to forget with hedging that we have two instruments, the original (underlyng) and the hedge (the Eurodollar futures). So, in your example, the short expects to borrow in the future and current rate is, say, 4%. As a borrower, he/she worries the rate will go up so he/she SHORTS the ED futures. If rates go up, to say 5%, indeed, he will borrow at the higher rate. The hedge is a separate instrument, in this case, he is SHORT so the 100 basis point drop will produce a profit of +2,500 ($25*100 bps) per contract. So the total hedge consists of two cash flows: a loss on the underlying mitigated by profit on the hedge instrument (futures) to create a *net effective* rate of 4%. So, on a net basis, the 4% has "effectively" been locked in due to the profit. So your phrase "net cost" is accurate!

2) I agree with this: "With a FRA you are effectively locking into a forward rate i.e a rate of interest that you agree to pay or receive for say 6 months starting 3 months from now i.e 3f9" but let me try a rephrase on the rest:

Under the FRA method to value the i rate swap, we are *merely* using using the implied forward rates (i.e., implied by LIBOR) to "predict" the cash flows that the floater will pay. I do not, stricly speaking, think the FRA mechanics need to enter; we are just predicting the future floating coupons.

Now, it can be "characterized" as FRA agreements because, at inception, the FLOATING RATE RECEIVER is counting on the approximate accuracy of the forward curve. Let's say LIBOR shifts in parallel up (higher rates). The pay floating pays the same (fixed), but recieve more, and so he PROFITS. So the pay floater is essentially similar to a LONG POSITION in the FRA (as the long in FRA will profit from rate increases).

David

 

[humheehum]

Thanks David for the clarifications.

I will continue to post queries on Part A -2 as and when they come up.

Ashim.