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Backwardation can happen in markets where
Convenience yield is the benefit from having access to the commodity - and if the convenience yield is very high, for example in a market where manufacturers must never run out of a particular raw material, then these can switch the total cost of carry (which include interest and storage costs, less convenience yields) to being negative. This causes forward prices to become lower than spot prices, a phenomenon known as backwardation.
Therefore Choice 'b' is the correct answer. If convenience yields are less than other carrying costs, then backwardation will not happen. The sign of convenience yields does not matter, what matters is their relative magnitude when compared to the other costs of carry.
To understand this in an intuitive way, consider that forward prices are nothing but spot prices, plus interest, plus storage costs, less convenience yields. If interest and storage costs are less than the convenience yield, the market will be backwarded.
The theta of a delta neutral options position is large and positive. What can we say about the gamma of the position?
The relationship between the value of an option, and its delta, gamma and theta is given by rV = + rS + 0.5(S)2, where V is the value of the option, r the risk-free rate, S the spot price of the underlying, and , & are the respective Greeks.
For a delta neutral portfolio, = 0 and this equation reduces to rV = + 0.5(S)2. Now rV is generally a small number, which means that if is large and positive, must be large and negative to offset that. Therefore Choice 'b' is the correct answer.
Where futures are being used to hedge a commodities position, which of the following formulae should be used to determine the number of futures contracts to buy (or sell)?
When using a futures contract to hedge a position, the correct way to determine the number of futures contracts to use is to use the following formula. This adjusts for the 'tailing the hedge' adjustment required as a result of the daily settlement feature of futures contracts.
Minimum Variance Hedge Ratio x Dollar Value of Position / Dollar Value of Single Futures Contract
When using forward contracts to hedge a position, no adjustment to tail the hedge is required (as settlement happens only at the end), and the number of units of the exposure when compared to the number of units per contract should be used. The formula in the case of forward contracts is:
Minimum Variance Hedge Ratio x Units in Position Held / Units in Single Futures Contract
In practice however, the difference between the above two formulae is de-minimis; and for the exam you should be okay to use either of the above two to determine the number of contracts - unless the question is directed to testing your knowledge of the 'tailing the hedge' adjustment.
An investor can use which of the following to replicate a fixed for floating interest rate swap where the investor pays fixed and receives floating?
1. Long positions in a series of forward rate agreements (FRAs)
II. A short position in a fixed rate bond and a long position in a floating rate note
III. A long position in a floating rate note and a short position in an FRA
IV. A long position in an interest rate cap and a short position in an interest rate floor at the same strike
A fixed for floating interest rate swap is identical to a series of forward rate agreements coinciding with the settlement periods. For example, a 2 year fixed for floating interest rate swap where the investor pays fixed at 6 monthly rests is equivalent to a long position in 3 FRAs ( a 6 x12, a 12 x 18 and an 18 x 24 FRA). Thus I represents a correct replication of the swap.
An obligation to pay fixed is identical to a short position in a fixed rate bond, and receiving the floating rate is akin to a long position in a floating rate note. Thus II also correctly replicates the swap.
A long position in an FRN correctly replicates the floating cash flow from the swap, but a short position in the FRA obliges the investor to receive fixed and pay floating - which is not equivalent to the flows from the interest rate swap. Thus III is not a correct replication of the interest.
A long position in a cap means the investor will receive the difference between the floating and the cap rate if the floating rate is greater than the cap rate. Effectively, the investor will receive floating. A short postion in a floor at the same strike rate would mean the investor would end up paying the floor rate, effectively equivalent to the obligation to pay a fixed rate. Therefore this replication also correctly replicates the swap, and IV can also be used to replicate the investor's position.
Therefore Choice 'a' is the correct answer and the rest are incorrect.
(For statement IV, let me explain with an example. Recall that the buyer of a cap receives the difference between the current interest rate and the agreed exercise rate if the current interest rate goes above the exercise rate. A cap can be used by a floating rate borrower to limit their exposure to a rise in interest rates. In the same way, the seller of a floor pays the buyer the difference between the current interest rate and the exercise rate if the current interest rate falls below the exercise rate. A floor can be used by a lender to protect against a fall in interest rates.
Now assume the exercise rate is r. Also assume that the current floating interest rate is r'. the holder of the interest rate swap in the example in the question receives r and pays r'. Therefore the net payout is r' - r (pays fixed, ie -r, and receives floating, ie +r).
Now consider the long position in the cap and the short position in the floor.
Case 1: If r > r', the following will happen:
a. The long position in the cap will pay nothing, ie 0.
b. The short position in the floor will require a payment of r - r' to the buyer of the floor, and because it is a payment, the cash flow will be -(r - r') = r' - r
Case 2: If r < r', the following will happen:
a. The long position in the cap will mean the investor receives r' - r.
b. The short position in the floor will pay nothing as the interest rate is higher than the exercise rate.
In both cases, the net cash flow for the investor is r' - r, which is the same as for the swap where the investor pays fixed and receives floating.)
What kind of a risk attitude does a utility function with downward sloping curvature indicate?
A utility function is graphed with utility on the y-axis and the variable driving utility (generally wealth) along the x-axis.
A concave utility function, ie a function with a downward sloping curve, indicates risk aversion. A convex utility function indicates a risk seeking attitude and a straight line (ie no curvature) indicates a risk neutral attitude.
