Free Practice Questions for PRMIA 8006 Exam

The LIBOR square is a (rare) derivative contract which pays, as mentioned in the question, the square of the interest rate move between two dates. If LIBOR at inception is y, and upon settlement is x, the contract pays (x - y)^2 for x > y; and -(x - y)^2 for x < y.

For any question that involves calculating delta or gamma, and the payoff is described in terms of variables as is the case here, remember that delta is always the first derivative and gamma is the second derivative. For this question, let us calculate the second derivative and see what the gamma is:

If x > y, then the payoff is (x - y)^2

The first derivative wrt x is 2(x - y)

The second derivative wrt x is 2.

ie, the gamma is 2

If x < y, then the payoff is -(x - y)^2

The first derivative wrt x is -2(x - y)

The second derivative wrt x is -2.

ie, the gamma is -2

If x = y, then the payoff is 0. Both the first and the second derivatives are zero. ie the gamma is 0.

Based on the above, we see that the contract can have a gamma of either 0, +2 or -2. 1 is not a possible value for gamma, and therefore Choice 'b' is the correct answer.

Since the equity portfolio has a beta of 1.5, we need to sell short enough number of futures contracts as to have $1 x 1.5 = $1.5m short in notional. The value of one S&P futures contract is 1000 x 250 = $250,000, and therefore in order to be short $1.5m, we need to sell 6 contracts.

Interest rate caps are effectively call options on an underlying interest rate that protect the buyer of the cap against a rise in interest rates over the agreed exercise rate. As with options, the premium on the cap depends upon the volatility of the underlying rates as one of its variables. A floor is the exact opposite of a cap, ie it is effectively a put option on an underlying interest rate that protects the buyer of the floor against a fall in interest rates below the agreed exercise rate.

A cap protects a borrower against a rise in interest rates beyond a point, and a floor protects a lender against a fall in interest rates below a point.

A collar is a combination of a long cap and a short floor, the idea being that the premium due on the cap is offset partly by the premium earned on the short floor position. Therefore a collar is less expensive than a cap or a floor.

The portfolio's beta is 0.8, and therefore in order to completely hedge the portfolio (ie reduce beta to 0), the portfolio manager would need to short 0.8 * $25m/(1450*$250) = 55.17 contracts, or 55 contracts. However, the ask here is to reduce the beta to 0.6, and not 0.

The number of contracts required to reduce the beta of a portfolio from ߠto ߱ is give by (ߠ- ߱) * Value of portfolio / Value of a single contract. In this case, this calculation works out to (0.8 - 0.6) * $25m/(1450*250) = 13.8, or roughly 14 contracts.

The portfolio manager should short 14 index futures contracts to reduce the total portfolio beta to 0.6.

The swap can be valued by valuing the two individual components of the swap.

The fixed rate bond equivalent in the swap is valued at =4/1.05 + 104/(1.06^2) = $96,369,154.

The FRN component will be valued at par as we are at a point where the rate has just been reset, ie $100m.

The investor is paying the fixed rate, and is therefore short the bond. He/she is receiving LIBOR, and is therefore long the FRN. The value of the swap to the investor therefore is +$100,000,000-$96,369,154 = $3,630,846

Detailed explanation:

An Interest Rate Swap exchanges fixed interest flows for floating rate flows. The floating rate leg is tied to some reference rate, such as LIBOR. The parties exchange net cash flows periodically. Conceptually, an interest rate swap is the combination of a fixed coupon bond and a floating rate note. The party receiving the fixed rate is long the fixed coupon bond and short the FRN, and the party receiving the floating rate is long the FRN and short the fixed coupon bond.

An interest rate swap can be valued as the difference between the two hypothetical bonds. FRNs sell for par at issue time as they pay whatever the current rate is, subject to periodic resets. Therefore immediately after a payment is made on a swap, the value of the FRN component is equal to its par value. The bond can be valued by discounting its cash flows. The difference between the two represents the value of the swap. When the swap is entered into, the fixed rate leg is set in such a way that the value of the hypothetical bond is equal to that of the FRN, and therefore the swap is valued at zero. The rate at which the fixed rate leg is set is called the swap rate. Over its life, market rates change and the value of the fixed coupon bond equivalent in our swap diverges from par (whereas the FRN stays at par - at least right after payments are exchanged and the new floating rate is set for the next period). Thus the swap acquires a non-zero value.

There are two ways to value a swap. If interest rates for the future are known, the bond and the FRN can be valued and their difference will be equal to the value of the swap. Sometimes, the current swap rates are known. In such a case, the swap can be valued by imagining entering into an opposite swap at the new swap rate, which will leave a residual fixed cash flow for the remaining life of the swap. This residual cash flow can be valued and that represents the value of the swap. For example, if a 4 year swap was entered into exchanging an annual fixed 5% payment on a notional of $100m for a floating payment equal to LIBOR, and at the end of year 1 the swap rate is 6%, then the party paying fixed can choose to enter into a new swap to receive 6% and pay LIBOR. All cash flows between the old and the new swap will offset each other except a net receipt of 1% for the next 3 years. This cash flow can be valued using the current yield curve and represents the value of the swap.

Forward premium or discount can be easily calculated as {(Forward rate - Spot rate) / Spot rate x 365/number of days]. In this case, it can be calculated as =((1.1652 - 1.1763) / 1.1763 ) * 365/91 = 3.785%, which is a discount as it is a negative number. It can also be interpreted as a discount as the forward price is lower than the spot price.

Statement I is true. Consider the Black Scholes PDE 202.23.e1

and substitute delta = 0 (as this is a delta neutral portfolio), and we get the result 202.23.e.

Since r is generally small, and σ and S are positive, theta will be negative when gamma is positive and vice versa.

Statement II is incorrect. The gamma of a call and a put are equal and do not add to 1. The relationship described applies to delta, and not gamma.

Statement III is correct because positive gamma means that the portfolio gains both from an increase and a decrease in the value of the underlying, given delta neutrality. (Generally, a positive gamma is a good thing, but like everything else, it does not come free. A trader can choose to keep their portfolio gamma positive, but in doing so he or she would be giving up premiums from options they would have sold to neutralize the gamma.)

Statement IV is correct because the only way to hedge gamma and vega is through other options positions as only options have gamma and vega. If only one tradeable option is available, it would be possible to hedge either the gamma or the vega, but not both, as achieving neutrality in one will upset the neutrality of the other. The only way to simultaneously hedge the two would be to use two different options on the underlying, and determine the number of options to be traded (using a system of simultaneous equations).

Each of the choices describes various scenarios related to the issue of bonds. A when-issued market is a market in government securities where securities are traded as forward contracts prior to their issue. Choice 'c' is the correct answer.

Choice 'd' refers to a 'bought deal'. Choice 'b' refers to the 'grey market', usually in corporate bonds. Choice 'a' refers to a fixed price re-offer mechanism.

The lowest achievable standard deviation will be lowest when the assets are perfectly negatively correlated, ie when correlation = -1. In the case of the above choices, the lowest correlation is -0.33, which is the correct choice.

This is a two step problem:

First, calculate the bond price using the yield information, then

Second, once you know the bond price, calculate the 18 month zero rate using the bootstrap method.

Step 1: Bond valuation: All variables required for pricing the bond are known. The coupon payments will be $1.50 in 6 months and 1 year from now, and a final paymento of $101.50 will be received in an year's time. This can be discounted at the yield provided as follows, and summed together to get the bond price of $97.14.

Step 2: Boot strapping: Discount the 6 month and 12 month coupons at the zero rates for those periods, and subtract the total of these PVs from the bond price. These work out to 1.5/(1+2%/2) = 1.485, and 1.5/(1+3%/2)^2=1.456. That gives us the present value, $97.14 - $1.485 - 1.456 = $94.203, which grows to the final payment of $101.50 at the end of 18 months. The zero rate inherent in this price can then be worked out as we know that (1+r/2)^3 = 101.5/94.203, or r = 5.03%.