8006 Sample Questions Answers

Gamma risk is unique to options. Therefore gamma risk can only be hedged using options, and Choice 'a', ie an option position that offsets any existing gamma in the portfolio, is the only way to hedge gamma.

Positions in stock or bank deposits do not carry gamma and therefore cannot be used to hedge gamma risk.

The value of Bond 1 is $900,000 and the value of Bond 2 is $1,100,000, or their respective weights in the portfolio are 45% and 55% respectively. The combined duration is the weighted average of their individual durations, ie (45% x 5) + (55% x 10) = 7.75 years

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. Therefore statements I and II are correct.

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.

Caps, floors and collars provide a hedge against interest rate risks. Therefore statement III is correct too. However. they do not always provide protection against spreads changing, as the rate other than the one referenced by the cap or the floor may change causing the spread to change. Therefore statement IV is incorrect.

A portfolio is considered dominated when it is not on the efficient frontier, which is another way of saying that there exists another portfolio with a better risk reward payoff. Therefore Choice 'a' is the correct answer.If the portfolio has a higher mean and the same or lower standard deviation than any other portfolio, this means it is already on the efficient frontier and is not dominated. The other choices do not make any sense in the context of the question.

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 efficient frontier is a two dimensional plot with volatility and returns on its axes. For a given level of return, it helps identify the portfolio with the lowest volatility. From the other axis, it helps identify the portfolio with the highest return for a given level of volatility.

Therefore Choice 'b' is the correct answer. Note that in the exam you may be asked the question slightly differently.

Professional management that acts as the agent of the shareholders is not likely to protect debt holders from a transfer of wealth to the shareholders. All other mechanisms listed are intended to offset the agency cost for debt holders, ie the expected cost that shareholders and management will transfer wealth away to the detriment of the debt holders.

The put-call parity makes sure that all else remaining equal, an increase in the value of a call option will be accompanies by an increase in the value of the corresponding put option.

Simply stated, the put-call parity can be expressed as:

Value of call - Value of put = Spot price - Exercise price discounted to the present

The two terms on the right hand side of this equation are fixed. Therefore, for the equality to be maintained, any increase in the value of the call option on the left hand side has to be accompanied by the value of the put.

Choice 'c' is therefore the correct answer.

https://riskprep.com/images/stories/questions/102.06.a.png  is the coefficient of risk aversion at x. Its inverse, ie https://riskprep.com/images/stories/questions/102.06.b.png , is called the coefficient of risk tolerance.

Risk aversion or risk tolerance is indicated in a utility function by its curvature. A concave utility function indicates risk aversion and a convex function indicates risk tolerance. The curvature is measured as a ratio of the second derivative to the first derivative of a function. A negative second derivative implies concavity. The expression is the coefficient of risk aversion, and its inverse, which is in the same units as wealth, is called the coefficient of risk tolerance.

Equity futures are by far the most traded futures contracts in terms of value, followed by interest rate products. Choice 'd' is the correct answer.

When hedging an equity portfolio with index futures, the number of futures contracts to hold is given by the [beta x value of portfolio/value of one futures contract]. Therefore Choice 'a' is the correct answer.

The volatility, risk free rate have nothing to do with determining the number of futures contracts to hold. Systematic risk is expressed in the form of beta. Therefore the other choices are incorrect.

Choice 'c' correctly describes a 'holder extendible option'. Choice 'd' describes a 'writer extendible option'. Choice 'a' describes a 'shout option'. Choice 'b' describes a 'knock in' option.

Tailing the hedge refers to adjusting the number of futures contracts required to hedge a position given that futures contracts result in daily cash flows from daily P&L. In the case of forward contracts, P&L is not settled daily but at the end of the contract. Because margin payments (ie the daily P&L) for futures contracts create interest cost or income (depending upon the direction of the flows), the number of contracts required to hedge a given position can fluctuate. This effect is generally small for small positions, or where the time period under consideration is not very long. But it can get large for larger positions that are open for longer time periods, therefore requiring an adjustment to 'tail the hedge'. The adjustment is straightforward - the number of contracts required needs to be divided by (1+r)t, where r is the interest and t the time period (or by ert if continuously compounded rates are being used. Therefore statement I is true and statement II is false.

Statement III is not correct as counterparty risk is certainly an issue for forward contracts, while futures contracts are guaranteed by the exchange, besides having margin requirements and daily P&L movement in place.

Statement IV is not correct as futures come in standardized contracts with standard amounts and a few standard expiry dates to pick, whereas forward contracts are customized and dates, amounts and other details of the contract can be negotiated.

Mortgage backed securities carry prepayment risk as borrowers tend to prepay mortgages when rates fall, and substitute it with newer cheaper mortgages. This creates the issue of 'negative convexity' for mortgages, ie, they lose value when rates rise, but do not gain in value when rates fall.

Prepayment risk can be offset by instruments that also carry negative convexity. A swaption is an option to borrow in the future at an agreed rate, which may be fixed or floating. An option to borrow in the future paying floating and receiving fixed guards against losses when rates fall, as the option can be exercised for a profit when rates are declining and the mortgage portfolio is being prepaid. A callable bond is very similar to an MBS in that the issuer can call it back when rates fall. Thus a long position in an MBS can be offset by a short position in a callable bond. Thus I and III are valid choices.

A cap allows exchanging fixed for floating when interest rates rise above an agreed rate. A long cap position allows borrowing at a fixed rate in exchange for floating, and a short cap implies receiving fixed and paying floating when rates go above the strike rate. Prepayment risk arises from falling interest rates, therefore a short cap will not protect against such a risk as falling interest rates would mean that no payments would be exchanged. Thus II does not help hedge against the risk in question.

A long position in a fixed for floating swap would require paying fixed and receiving floating when rates are falling. This would just make the problem from prepayments worse as the position would pay fixed and receive a falling rate. Thus IV is not an appropriate way to hedge prepayment risk.

Price quotes for futures where one of the currencies is the US dollar follow the convention of using the number of US dollars that one unit of the foreign currency can buy. (For JPY, 100 Yen is used.)

In the forward market, the convention used is the same as that used in the spot market.

In the spot market, prices are expressed as the number of units of the foreign currency that 1 USD can buy, except for GBP, EUR, AUD and NZD where it is the other way round (ie, number of USDs that 1 of each of these currencies can buy).

The delta of an at-the-money call option is close to 0.5. It is close to 1 when it is deep in the money. It is close to 0 when it is deep out of the money. It is never negative. Therefore Choice 'b' is the correct answer.

As a bond approaches maturity, its value will be driven less by the time value of money or its coupons, and more by its redemption value, ie the par. Therefore as maturity approaches, its value converges to its par value and this phenomenon is called pull to par.

Explanation:

The correct answer is the index value times the contract size, in this case 1500 x 50.

One way to think about index futures is this: Consider equity index trading as trading in the shares of a company whose share price is equal to a number of dollars which is the same as the index. If the 'contract multiplier' for a index futures contract is 50, that means the futures contract is for 50 shares of such a fictitious company. Therefore the notional value of the contract will be 15000 x 50, and Choice 'a' is the correct answer.

This question relates to the Black Scholes PDE (partial differential equation). The Black Scholes PDE is derived by combining the results of a stock price following a Weiner process with a mathematical result known as Ito's lemma and holds true for all derivatives whose prices depend upon S, the price of the underlying and upon t, which indicates time.

The derivation of the Black Scholes PDE is beyond the scope of the PRM syllabus (and also of the author of this question), but we need to know what it is. The PDE can be written as:

202.21.q1

Substituting , this can also be written as

202.21.e

202.21.q2

Therefore the first and the second equations are correct, and the other two are wrong. Choice 'c' is the correct answer.

You should know the PDE in case there is a numerical question about it. The two ways of writing it (as I and II above in the question) express the same relationship. The BSM formula itself for valuing a vanilla European option is based upon the PDE. Most other options are also valued using the fundamental relationship expressed in the PDE.

A forward contract is a derivative contract, and has a delta of 1. Therefore Choice 'd' is the correct answer. This is because the value of a forward contract is given by S - Ke^(-rt), where S is the current spot price, r the risk free rate, K the forward price, and t the time to maturity. As S changes from S to (S + δS), the value changes to (S + δS) - Ke^(-rt), ie the change in value is exactly δS in response to a change in the price of the underlying by δS. Therefore the forward contract has a delta of 1.

All other choices are incorrect.

Note that this is different from the delta of a futures contract which is different from 1, and equal to e^(rt), a number greater than 1.

In terms of beta, his exposure is $10m*0.7 = $7m. This exposure is long.  In order order to neutralize his long exposure, he needs to have an equal an identical short position with the same beta as this long position (of course, in the short direction).  We need to figure out how many contracts will have a beta equal to his held position.  (The beta of a futures contract is slightly different from 1 when compared to spot, but in the absence of other information in the question it is always okay to assume that the beta of the futures contract is 1.  Such precision does not matter because of other errors such as rounding etc that cannot be anyway done away with.)

He needs to short futures contracts on the index with $7m in notional value. The value of each contract is currently 1000*250 =$250,000. He therefore needs to short $7m/$250,000 = 28 contracts to become market or beta neutral.

The beta of a risky asset is determined solely by the covariance between the asset and the variance of the market portfolio. Recall that β = covariancex, y / variancex, where x is the market portfolio and y is the risky asset. Choice 'c' is the correct answer.

An increase in volatility increases the value of the option, and so do increases in the price of the underlying and the risk free rate. However, since a European option can only be exercised at expiry, an increase in the time to expiry may not necessarily increase the value of the option as it may increase the uncertainty around a more certain payout.

Consider the extreme case of a deep in the money European call option that has 1 day left to expiry, and a payout is certain. Now imagine the time to expiry is increased by say, 6 months. Now the payout is no longer certain as no one knows what the value of the underlying will end up at after 6 months. In such a case, the value of the option would decline. But this applies only to a European option. An American option, which can be exercised any time, will not be affected by this reasoning.

Statement I correctly describes an IRS.

Statement II is correct - in an IRS the parties do not exchange the principal or the notional amounts as they are in the same currency and can be netted off, but in a currency swap the amounts are in different currencies that cannot be netted (not at least without doing a spot transaction simultaneously), and principals are exchanged.

Statement III is incorrect as the floating rate leg actually is subject to no risk as it resets itself to the current interest rate. The true risk is in the fixed leg which is sensitive to changes in interest rates.

Swaps carry counterparty risks just like any other OTC contract, therefore statement IV is false.

Statement I is not correct because the 4% bond with the same modified duration as the zero coupon bond will have a maturity much later in the future than the zero coupon, and the final repayment coming much later will cause its convexity to be greater than the equivalent duration zero coupon bond.

Statement II is not correct as the convexity of a bond increases not in a linear fashion but in proportion to the square root of time as the maturity of the bond is increased.

Statement III is not correct as mortgage based bonds bring negative convexity even in the case of long positions,

Statement IV is not correct as the duration of a zero coupon bond maturing in 10 years has a modified duration greater than that of a coupon bearing bond of the same maturity, which means its convexity is higher and not lower (note this is different from statement I where the modified durations were identical. Here, the maturity is identical which has different implications.)

Therefore none of the statements is correct and Choice 'd' is the correct answer.

A money market security is one that is initially issued with a maturity of less than one year. Treasury bills are short-term government securities with maturities ranging from a few days to 52 weeks. Bills are sold at a discount from their face value, and do not carry a coupon. Treasury notes and treasury bonds are not money market instruments as they are issued for a maturity greater than a year. Treasury notes are issued with maturities of 2, 3, 5, 7, and 10 years and pay interest every six months. Treasury bonds pay interest every six months and mature in 30 years.

Commercial paper is issued by corporations to meet their short term funding needs and is a money market instrument. Bankers' acceptances are short term loans to corporations that are guaranteed by a bank.

Of the given list, since treasury notes are the only instrument that are not money market securities, Choice 'a' is the correct answer.

When the forward prices are greater than the spot prices, the market is said to be in contango. When forward prices are lower than spot prices, the market is said to be backwarded. A short squeeze may contribute to backwardation. Choice 'a' is the correct answer.

To solve this question, we need to revisit how delta is calculated in a single step binomial model:

Consider a portfolio with just two positions: 1 Long Call option, and Δ Short Stock. We do not know what Δ (Delta) is.

Now if the current price of the stock is P, which can take the value P2 (higher value) and P1 (lower value) in the future at time T, then Δ needs to be such that the value of the portfolio is equal in both the cases. If the price goes up, the value of the option will be $1 (given) and the value of the short stock will be -Δ P2. If the price goes down, the value of the option will be $0, and the value of the stock will be -Δ P1.

Therefore, $1 - Δ P2 = $0 - Δ P1.

Solving for Δ, we get Δ = $1 /(P2 - P1) = $1 / ($102-$98) = 1/4.

Since the stock is selling for $100 spot, its 3 month forward price for you would be $100*(1 + 6%*3/12) = $101.50. Since the forward is priced at $104, it is 'rich' compared to the spot and should be sold. A profitable trade would be to buy the spot and sell the forward.

Imagine you borrow $100 and buy the stock, and simultaneously sell the forward. At the end of 3 months, you will receive $104 in return from the forward, and be obliged to return the initial $100 borrowed together with interest of $1.50 for the three months. You would make a riskless profit of $2.50 per share in three months. Thus Choice 'c' is the correct answer.

Note that the treasury yield is irrelevant, what matters is the rate at which you can finance the trade which is 6% per annum.

Repos are collateralized borrowing arrangements. They are used for providing short term funding. They are not used for managing credit risk exposures. They are also used by central banks as part of their money market operations. They are also useful for facilitating short positions, for example, in corporate bonds where the party desirous of shorting the bond may 'borrow' the bond on repo in order to be able to sell it short.

Given a choice, decision makers will maximize expected utility. The four basic axioms on which the principle of maximizing expected utility is based are:

- Transitivity of choice,

- Continuity of choice,

- Independence of choice, and

- Stochastic dominance.

In the US, the Federal Reserve sets a limit (eg, 2 x, so that only 50% of the money could be borrowed) as a multiple of capital posted to prevent excessive leverage or gearing in the system.

The other choices are incorrect.

According to the CAPM, the expected return from a risky asset is a function of the contribution of the risky asset to the total risk of the market portfolio. Nothing else matters. All assets are priced according to the risk they bring to the market portfolio, regardless of their individual level of risk. An asset that is very volatile on its own, but has a negative correlation to the market may be priced high, ie have low expected return, because of its impact on the risk of the market portfolio. Therefore Choice 'a' is the correct answer, and the other options are incorrect.

Recall that according to the CAPM β = covariancex, y / variancex, where x is the market portfolio and y is the risky asset.

The beta itself is a function of the covariance of the asset's returns with market returns, and therefore only the driver of expected return for an asset is its beta, which is determined by the asset's contribution to portfolio risk. (β = covariance(x, y) / variance(x), where x is the market portfolio and y is the risky asset. )

Energy derivative markets are still not very liquid, and experience high price volatility. This high volatility is responsible for most of the risk in these markets. Choice 'd' is the correct answer.

Backwardation is said to occur when futures prices are lower than the current spot prices. This would happen only when carrying costs are negative. Carrying costs are equal to interest, plus storage costs and less any 'convenience yield'. The existence of large convenience yields may explain backwardation in commodity futures prices. Therefore Choice 'd' is the correct answer.

Contango is the 'normal' market situation where forward prices are higher than spot prices. Storage costs explain contango, not backwardation. Risk free rates, or the cost of funding for the futures position, are always positive and do not explain backwardation.

The Sharpe ratio is the ratio of the excess returns of a portfolio to its volatility. It provides an intuitive measure of a portfolio's excess return over the risk free rate. The Sharpe ratio is calculated as [(Portfolio return - Risk free return)/Portfolio standard deviation].

The Treynor ratio is similar to the Sharpe ratio, but instead of using volatility in the denominator, it uses the portfolio's beta. Therefore the Treynor Ratio is calculated as [(Portfolio return - Risk free return)/Portfolio's beta]. Therefore Choice 'a' is the correct answer.

Jensen's alpha is another risk adjusted performance measure. It considers only the 'alpha', or the return attributable to a portfolio manager's skill. It is the difference between the return of the portfolio, and what the portfolio should theoretically have earned. Any portfolio can be expected to earn the risk free rate (rf), plus the market risk premium (which is given by [Beta x (Market portfolio's return - Risk free rate)]. Jensen's alpha is therefore the actual return earned less the risk free rate and the beta return.

Refer to the tutorial on risk adjusted performance measures for more details

The buyer of a cap who is primarily interested in protecting against a rise in interest rates and does not mind losing some of the gains if interest rates were to fall can reduce her costs by selling a floor with a lower strike rate. Selling a cap would further expose her to the risk of rising rates, exactly what she is trying to protect herself against, and reducing the strike rate for the cap will actually increase the cost of the cap. Similarly, increasing the time period to which the cap applies will also increase the cost of the cap.

The minimum variance hedge ratio is effectively the beta between the asset to be hedged and the asset used as a hedge, calculated as [correlation between the hedge and the asset to be hedged x volatility of the asset to be hedged/volatility of the hedge].The effectiveness of the hedge depends upon the correlation between the two, ie the asset to be hedged and the hedge. The minimum variance hedge ratio can be 1.0 even in situations where the correlation between the asset and the hedge is low, and as such does not mean much unless we know the hedge effectiveness (ie, the correlation).

It is therefore not possible to draw any certain conclusions based on the minimum variance hedge ratio alone. IV therefore is the only correct answer.

Profits and losses on futures contracts are settled daily. (P&L on forward contracts is often settled upon the expiry of the contract, and may even be collateralized.) Therefore Choice 'd' is the correct answer.

A rise in interest rates will cause fixed rate bond prices to fall. It will not affect the floating rate notes as their rates are going to float with the changes in interest rates. Inflation linked bonds will only be affected by changes in real interest rates, and even in a situation where the rise in interest rates is happening due to an increase in the real rates, a long position would hurt. As interest rates rise, the coupon payable on the inverse floater would fall, hurting a long position.

Of the given alternatives, the only position that would benefit from a rise in interest rates is the short position in fixed rate bonds. Hence Choice 'd' is the correct answer.