8006 Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition Questions and Answers

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%.

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.

All the statements are correct, and represent the relationships between the values of the Greek variables for calls and puts for the same option. It is important to know why as the PRM exam often asks tough questions about the Greeks.

Statement I is correct because of the put-call parity. According to the put-call parity,

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

Now the delta of the spot is 1, and that of the discounted price is zero. Therefore

Delta of Call - Delta of Put = 1 - 0, or by rearranging we get the equation in statement I.

Statements II and III are correct as the gamma and vega of both the spot price and the discounted price are zero. Therefore using the put-call parity, we can say

Gamma of Call - Gamma of Put = 0 - 0, and

Vega of Call - Vega of Put = 0 - 0

Rearranging, we get statements II and III.

Statement IV is correct because of the following relationship between theta of call and theta of put:

Theta of Put = Theta of Call + rKe^(-rt).

Since rKe^(-rt) can only be a positive number, theta of put can only exceed the theta of a call. However, since theta is generally negative, it often implies that the theta of a call is the larger absolute number.

Additional explanation for the last point: Assume rKe^(-rt) =+1 and theta of a put is -5 (completely hypothetical)

Now Theta of Put = Theta of Call + rKe^(-rt)

Ie -5 = -6 + 1

Now -5 is the larger number than -6. In other words, theta of put exceeds that of the call in a pure mathematical sense, which is what I mean when I say “theta of put can only exceed the theta of call”. But if you ignore the sign, then theta of call is larger at 6 when compared to 5. Therefore the theta of the put is greater than the theta of the call – which is what the answer says.

Treasury bond futures do not specify which bond can be used to effect delivery, but allow the seller to pick from a number of available bonds. As a result, one of these eligible bonds emerges as being the ' cheapest ' to deliver, and this CTD bond is determined by the basis between the cash price of the bond and the futures spot price as adjusted by the conversion factor for this specific bond. (ie, basis = Cash Price of the Bond - Futures Price x Conversion Factor)

The bond with the lowest basis is generally the CTD - therefore Choice ' c ' is the correct answer.

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

The transformation line represents the various possible combinations of the ' risky bundle ' and the risk free asset. Investors can choose different combinations of the risky and risk-free asset depending upon their risk appetite. The transformation line meets the y-axis (portfolio returns) at the point equal to the risk free rate. Choice ' a ' is the correct answer and the rest are incorrect.

The highest possible transformation line, ie the transformation line with the maximum slope, is the transformation line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the ' capital markets line ' . Investors can pick any point on this line according to their risk appetite, and doing so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML.

Remember that the transformation line and the capital markets lines are different from each other - there is only one CML for an efficient frontier, whereas there can be multiple transformation lines.

The key point in this question is that the bond is zero coupon, and can only be called at face value. Since the bond is zero coupon, its value will always be less than its par value at any time during its existence (as any interest rate will be a positive number). Therefore the issuer will never exercise the call. Thus the bond will have a duration equal to what an equivalent non-callable bond would have.

Since zero coupon bonds have a duration equal to their maturity, the bond ' s duration is 10 years.

There are two ways to think about this question: First, because the asset manager thinks that interest rates are going to decline, his profit will be maximized if he buys the bond with the greatest duration. The zero coupon bond has the greatest duration among the alternatives listed. Therefore Choice ' c ' is the correct answer.

The second way to look at this question is to consider what is called ' reinvestment risk ' . Yields to maturity calculations have an implicit assumption that any cash flows received prior to maturity are investible at the ytm rate. Thus, an yield to maturity calculation for a coupon bearing bond assumes that coupon payments get invested at the same rate as the calculated ytm from the time they are received till the time the bond matures. In an environment where interest rates fall, this assumption will not hold true. Reinvestment risk is the risk that coupons will not be able to earn the ytm that the bond was brought at - and the investor will be at a loss to the extent the money does not earn the ytm rate till the end of the investment. One way to reduce reinvestment risk is to minimize coupon receipts so the money stays invested as part of the bond, and the zero coupon bond helps to do exactly that.

Standard deviation of this portfolio can be calculated as SQRT(4000^2*20%^2 + 6000^2*25%^2 + 2*0.6*4000*6000*20%*25%), which is equal to $2,081. Choice ' a ' is the correct answer. The other answers are incorrect.

Investors in CDOs bear credit risk. The SPV is merely a conduit that owns the underlying assets on which the sponsoring institution has bought protection. The investors have sold them this protection, and are on the hook for defaults or other credit events. The reference entity is relevant only to CDSs, not CDOs. Choice ' b ' is the correct answer.

Remember that:

- American options can be exercised at any time up to expiry

- European options can only be exercised at expiry

- Bermudan options can be exercised prior to expiry but only at certain dates or during certain specified periods, and not otherwise

- Since an American option can be exercised at any time prior to expiry while a European option can only be exercised at expiry, an American option will always be worth more than a European option

Therefore the first three statements are incorrect and Choice ' d ' is the correct answer.

Commodity futures prices can be expressed as the summation of their spot prices and the carrying costs. Therefore any changes in either of these two would be a risk to the futures prices, and Choice ' b ' is the correct answer. It is common to decompose complex commodity portfolios into underlying equivalent spot positions and the carrying costs, which includes interest, convenience yield and storage costs. For liquid commodities such as gold where changes of a short squeeze are low, interest costs dominate the carryings costs. Choice ' b ' is the correct answer as it is most complete and covers the elements in the other choices. The ' lease rate ' for a commodity is equivalent to (Fwd Price - Spot Price)/Spot Price, and comprises the interest and storage costs and the convenience yield. The other choices do not represent complete answers.

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.

The profit potential from the conversion of convertible bonds into stock is limited by the issuer ' s option to call the security at short notice. Generally, the convertible debt security is convertible into a certain number of shares, and the debt holder will generally not convert the security to shares unless there is a profit to be made. The ' premium ' is irrelevant, because as long as the premium exists, the debt holder has no incentive to convert, as he would be better off buying the shares in the market. It is only when share prices go beyond a level that it becomes advantageous convert the security into shares. However, the prospect of granting cheap shares to the debt holders is not too appealing to the issuer, and as soon as the share price goes beyond a point where the value of the shares exceeds the face value of the debt the issuer has an incentive to exercise its option to call the security.

Therefore the profit potential from the conversion of convertible bonds into shares is limited by the issuer ' s option to call the security, and Choice ' a ' is the correct answer. The ' premium ' , or interest rates, or volatility are irrelevant.

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

This question highlights the difference between zero rates, yield-to-maturity and forward rates. Forward rates are from one point in time to another, for example say from year 4 to 5 in the future. A zero rate is from time 0, or now, to a point in time in the future. The zero rate is dependent on the forward rates for all the different periods from now to the future. The yield curve represents the various zero rates at different points in time, and if it is upward sloping it means forward rates for years further out are greater than the years prior. That is what causes the zero rate yields to increase over time and the curve to slope upwards. So the forward rate from year 9 to 10 will certainly be higher than the 10 year zero rate.

The yield-to-maturity for a bond is the rate at which the payments on the bond discount to be equal to the current bond price. It is therefore the average rate that applies to the bond. This average is based upon the different zero rates for the years in which bond holders receive payments. Since coupons are smaller than the payment at maturity, the zero rate that applies to the payment at maturity will have the most impact on the numerical value of the bond ' s yield-to-maturity. Also affecting the yield-to-maturity will be the values of the coupon payments, which will be discounted at lower rates when the yield curve is upward sloping. So the yield-to-maturity will be an average lower than the 10 year zero. Since the 10 year zero will be lower than the forward rate from t=9-10, the yield-to-maturity will be the lowest rate of the three.

The efficient frontier is plotted on a graph with portfolio return (mean) as the y-axis and portfolio volatility, or standard deviation, on the x-axis. Asset beta and standard deviation of the market portfolio have nothing to do with the determination of the efficient portfolio. Therefore Choice ' d ' is the correct answer, and the rest of the choices are incorrect.

The optimal hedge ratio answers the question of how much of the hedge to buy to hedge a given position. If  103.11.e1  is the standard deviation of the asset to be hedged, and  103.11.e2  is the standard deviation the asset being used to hedge against price movements in x, then the minimum variance hedge ratio is given by the expression  103.11.e3 .  In this question, correlation = 0.00378/(6%*7%) = 0.9. The minimum variance hedge ratio is given by (6%/7%)*0.9 = 0.77

He should therefore hold 0.77 x $10m /$100k = 77 contracts.

[We can also do a direct covar based calculation for the min variance hedge ratio as follows:

103.11.a3

= 0.00378 / (0.07 * 0.07) = 0.7714]

The coupon payments can be considered an annuity which can be valued using the formula for the PV of annuities= annuity . Therefore the value of the five coupon payments is 5 * ((1-1/(1.06^5))/0.06) = $21.06

Similarly the principal payment at the end of 5 years can be valued as 100/1.065 = $74.73

Therefore the total value of the bond today is $95.79

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 ߠ to ߱ 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 equivalent continuously compounded rate in this case can be calculated as ln(1+5.2%) = 5.07%. The other answers are incorrect.

Refer to the tutorial on interest rates for more details on how continuously compounded rates work.

The existence of derivative markets actually drives liquidity out of cash markets into derivative markets. Derivative markets are far more liquid than cash markets. Therefore statement I is not correct.

In times of crises, instruments or financial products bearing greater credit risk lose liquidity rapidly. Therefore statement II is correct.

A higher bid-ask spread indicates lower liquidity, not greater liquidity. Therefore statement III is false.

Normal market size (NMS) refers the ' normal ' transaction size for which a dealer ' s quote is valid. A higher NMS is associated with greater liquidity, therefore statement IV is correct.

This question touches upon a couple of issues that relate to hedging foreign exchange exposures using futures contracts. Firstly, many futures contracts on exchanges are available only at specific maturity dates, for example, the IMM dates. They may or may not coincide with actual liabilities for a running business. Also, futures contracts are standardized, ie their notional amounts are fixed rounded sums, and they can only be traded in whole numbers. This often means business using futures for hedging end up having a close enough, but not perfect hedge.

For our importer in the question, clearly he has to buy USDs so he can make his payment. Since each contract gives him USD 100k, he should buy 11 contracts that will get him very close to the amount he finally needs. Also, the contract expires a month later than his liability is due. This means he should offset the contract by closing it out in August soon as he has made his payment. This will allow him to stay hedged till August. If he does not sell out of the contracts, he will become exposed to a long position for one month till the contract settles, a risk which is unnecessary for him.

Therefore Choice ' b ' is the best answer.

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.

When 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 as shorts try to buy in the spot market to cover their short positions. Similarly, expectations of oversupply in the future, for example due to a bumper harvest may create situations where the forward prices fall below spot prices. 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 costs of carry (which include interest and storage costs, less convenience yields) to being negative.

Since all these factors can contribute to backwardation in the market, Choice ' d ' is the correct answer.

All the choices listed are valid assumptions underlying the BSM option valuation formula except that the BSM formula is based upon the option being exercisable only at expiry. The assumption is that early exercise is not permitted. In other words, BSM applies to European options and not American options. Therefore Choice ' d ' represents the correct answer as it is not an assumption underlying Black Scholes.

A hedge based on portfolio duration alone makes the assumption that the price/yield relationship is linear, and ignores the convexity or non-linearity of the price/yield relationship. As prices change beyond small changes, the non-linear effect kicks in, which can be offset by making a convexity adjustment to the hedge. Therefore statements I and II are correct.

Statement III is incorrect - negative convexity has an adverse effect on bond prices regardless of whether prices rise or fall.

Statement IV is correct, a bond hedge based on duration alone may be mismatched along the yield curve (eg, hedging a 10 year maturity bond with a 3 year futures contract, even though of identical total durations), which is based upon the implicit assumption that all rates will move together (ie parallel shifts).

The price of an interest rate cap is affected by all of the listed choices except the risk free rate. The risk free rate does not come into play in the pricing of caps, and therefore Choice ' b ' is the correct answer.

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.

A CLN is a form of a funded credit derivative. The investors in the note are the protection sellers, and receive the CDS premiums in addition to any risk free rate on their investment. If a default occurs, the money put up by the investors in the CLN is used to make good the party that has brought the protection.

Therefore if a default occurs, the investors in a CLN do not receive their entire investment back, but only what remains after fullfilling their obligations on the credit protection, plus any premiums that might have been earned. In return for this risk, the CLN yields the risk free rate plus the CDS premiums. Only Choice ' d ' is correct, all other options are incorrect.

The TED spread is a bet on the spread between treasury bill futures and Eurodollar futures. T-bill rates are lower than Eurodollar rates, as the former carries no risk. Eurodollars deposits, which are interbank deposits between the highest rated banks, carry very little risk as well. Therefore both these instruments generally trade at very narrow spreads. The spread widens, ie the Eurodollar rates rise in comparison to treasury bill rates when the market has credit risk fears.

A trader is said to be ' long ' the spread when he benefits from the spread increasing, and ' short ' the TED spread when he gains from the spread decreasing. A trader can buy the spread by being long t-bill futures and short Eurodollar futures. Similarly he can be short the spread by being short t-bill futures and long Eurodollar futures.

The put-call parity can be expressed as:

Call – Put = Spot – PV of exercise price

Note that a negative sign above means a short position. The ' term PV of exercise price ' is the same as a bank deposit placed today equivalent to the PV of the exercise price so that we will have the cash flow on the exercise date to exercise the option.

Therefore only Choice ' d ' is the correct answer as rearranging the above gives us Buying a call + Bank Deposit = Buying the stock + Buying a put.

Choice ' d ' is therefore the correct answer.

The prices must satisfy the put-call parity, and if they do not, it means there is an arbitrage opportunity, The put-call parity is as follows. We plug in the values to check if the parity is maintainted.

Buying a call + Bank Deposit (PV of exercise price) = Buying the stock + Buying a put

Thus the LHS = $10 + $100/(1 + 5%) = $105.24

and RHS = $100 + $10 = $110, ie the equality does not hold.

Since the trader can make a profit by buying low and selling high, and the set of positions on the left should be brought, and those on the RHS should be sold. Thus the trader should buy the call, sell the put and sell the stock to make a risk free profit. (There is no need to explicitly place a bank deposit at the risk-free rate).

Callable corporate bonds need to be priced lower and therefore yield more to investors as they are likely to be called by the issuer when interest rates fall and the issuer finds it attractive to refinance the debt using new cheaper debt.

They are most unlikely to be called when their prices have fallen because falling prices indicate higher rates, which means the issuer is able to fund himself at a cheaper rate.

Because of their callable feature, they are less convex and may actually carry negative convexity.

Therefore Choice ' b ' is the only correct statement and the right 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.

The arbitrage pricing theory is a multifactor model for explaining asset returns as it can be used to incorporate multiple factors, for example inflation, GDP growth rate, employment, interest rates etc to explain asset returns. Choice ' a ' is the correct answer.

The single index model, as the name implies, uses only a single factor, and so does the CAPM which uses excess returns to explain an individual security ' s returns. Neither of these are multi-factor models.

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.

The LIBOR rate to use is the one at the beginning of the period, is 4.5%. The fixed rate payer owes 5%, and the floating rate payer owes 4.5% + 100bps. Thus the fixed rate payer will receive a payment equal to 0.5% for six months on $1m. This works out to $2500.(Recall that a fixed for floating interest rate swap exchanges fixed for floating rate payments, with only the net payment being made by either party.)

The LIBOR rate to use is the one at the beginning of the period, is 4.5%. The fixed rate payer owes 5%, and the floating rate payer owes 4.5% + 100bps. Thus the fixed rate payer will receive a payment equal to 0.5% for six months on $1m. This works out to $2500.

(Recall that a fixed for floating interest rate swap exchanges fixed for floating rate payments, with only the net payment being made by either party.)

Short calls and short puts have negative gamma, ie their delta increases at a decreasing rate as the price of the underlying change. Likewise, remember that long calls and long puts have positive gamma.

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).

The put-call parity applies to European options and does not hold for American options because the latter can be exercised at any time up to expiry. Therefore statement I is false.

An American put can not be cheaper than a European put is correct. This is because the American put will always be valued at at least its intrinsic value whereas a European value may be valued at less than intrinsic value if the exercise date is too far away in the future. This is because a long time interval will increase the chance that the intrinsic value may not be realized at expiry. Because the American put can be exercised anytime, it will always be at least equal to its intrinsic value because if it were to be cheaper than that, investors would immediately buy and exercise it, creating a riskless profit. Therefore statement II is correct, and statement IV is correct as well.

It may be optimal to exercise an American put (unlike an American call - note the difference!) before its expiry on a non dividend paying stock. This would be the case only when the put is deep in the money. If the option is really deep in the money (say, when the underlying ' s price is down to zero), then it may be worthwhile to cash the option out prior to expiry as the upside might be lost if time were to be allowed to pass. Additionally, any further upside in the case of a deep in the money option may be zero or very little, and the time value of money would also make immediate exercise prefereable. Therefore statement III is not correct.

Systematic risk refers to market risk that cannot be diversified away. Specific risk relates to the unique risk from the securities selected in the portfolio, and these can be diversified away by adding other securities to the portfolio. Diversifiable risk is risk that can be diversified away, ie the same as specific risk. Portfolio risk is the total risk of the portfolio that includes both specific and systematic risk.

Generally, it is not a good idea to exercise an option early as any more upside in the remaining period to expiry is being sacrificed. However, if an extraordinarily large dividend is coming due, and this dividend is larger than the interest foregone from holding the option, it may be a good idea to exercise the option early. In such cases, the exercise needs to happen before the ex-dividend date and not afterwards. Choice ' b ' is therefore the correct answer.

Even if the option is deep in the money and delta is approaching 1, and exercise upon maturity is almost a certainty, it would still always be better to sell the option than exercise it . Therefore Choice ' a ' is incorrect. Choice ' c ' is correct in all cases except when a large dividend is coming in. Choice ' d ' is not correct because an early exercise needs to happen prior to the ex-dividend date and not afterwards.

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Knock-out options start as plain vanilla options and are ' knocked-out ' , ie cease to exist, when the barrier is hit. Knock-in options start lifeless and ' kick-in ' , ie come into play as plain vanilla options when the barrier is hit. Therefore statement I is not correct.

Barrier options are certainly cheaper than equivalent vanilla options because vanilla options have a larger range of prices over which they pay out. Therefore statement II is correct.

Statement III is not correct. Average price options, also called Asian options, are less attractive to a buyer of options and therefore they are cheaper and not more expensive than vanilla options. This is because average prices are less volatile, and also when compared to a strip of equivalent vanilla options, some of the individual vanilla options in the strip may pay out whereas on the average nothing may pay out.

Digital options have a very high gamma close to the strike price as the option payout becomes uncertain and fluctuates sharply between 0 and 1. Therefore statement IV is correct.

The no-arbitrage futures price for the stock is the current spot price plus the carrying cost for one year, which in this case is $100 * 10% =$10. Therefore the no-arbitrage futures price for the stock is $110. ($100*(1+10%), or $100 + $10.)

The minimum variance hedge ratio answers the question of how much of the hedge to buy to hedge a given position. It minimizes the combined volatility of the primary and the hedge position. The minimum variance hedge ratio is given by the expression [ σ(x) / σ(y) ] * ρ(x,y)] . Effectively, this is the same as the beta of the primary position with respect to the hedge.

In this case, the hedge ratio is = 10%/9% * 0.9 = 1

T-bill futures and fed futures are very short term futures contracts, and unlikely to provide much of a hedge against a rise in interest rates. FRAs however are customized and would be the right instrument for the borrower to hedge his interest rate risk. A long FRA position gives him the right to borrow at an agreed rate in the future. This rate will not change regardless of the changes in interest rates, and therefore Choice ' b ' is the correct answer.

An increase in coupon brings the ' average ' cash flows closer, thereby decreasing duration. The higher the coupon, the lower the duration.

An increase in yield discounts the cash flows that are further away more than it does the closer cash flows, so an increase in yield decreases duration.

An increase in coupon frequency brings the bond ' s cash flows closer, thereby decreasing duration. Decreasing the coupon frequency has the opposite effect.

An increase in maturity pushes the payments further out, thereby increasing duration.

An yield curve would be meaningless if it combines, for example, the yield on a AAA bond with the yield on a bond near default on the same curve. Therefore statement I is incorrect as any typical yield curve generally refers to similar instruments, and similarity includes similarity of issuer type and credit risk profiles.

Statement II accurately describes an yield curve, and statement IV explains what the term structure is.

Statement III is incorrect as the liquidity preference theory offers an explanation for an upward sloping yield curve and not a downward sloping one.

When rates are continuously compounded, we can calculate the marginal rate for year 5 as (5*5% - 4*4%) = 9%. (Note that things would be different if the rates were not continuously compounded, and they were annually compounded. In that case we would need to do a calculation of (1.05^5 / 1.04^4) - 1)

The basis point value is equal to $102 x 5 /10000 = $0.051. Therefore Choice ' d ' is the correct answer.

The basis point value, also called the PV01, is the dollar value change in the price of a bond as a result of a 1 basis point (=1/10000) change in interest rates. Since the modified duration is 5, it means a 1% change in interest rates produces a 5% change in the price of the bond, and a 1 bps change will therefore produce a 0.05% change in the value of the bond. Since the bond is currently valued at $102, the change works out to $102 x 0.05% = $0.051.

In an interest rate swap, only the net payment is made. In this case,

- the customer pays the bank 4%*(3/12)*$100m

- the bank owes the customer (2% + 100bp))*(3/12)*$100m

Therefore the customer pays (4% - (2% + 100bp))*(3/12)*$100m. 3/12 represents the 3 month time interval. This is equal to a net payment of $250k from the customer to the bank. Therefore Choice ' c ' is the correct answer and the rest are incorrect.

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

Statement I is correct. If daily returns are not independent, variances cannot simply be added up over the period, and the square root of time rule is not appropriate to use to scale volatility. Statement II is incorr ect. Statement III is correct. If daily returns are positively correlated, it means that a high return on one day will likely cause a higher return the next day, and likewise for low or negative returns. Intuitively, it means that a ' trend ' will be created and volatility will be higher than in a case where daily returns were not correlated. Therefore statement II is not correct. By the same logic, negative correlation between daily returns would mean a higher return on one day would likely be followed by lo wer returns the next day, ie a reversion to mean will result causing the volatility to be lower than the case when the returns are uncorrelated. (The correlation between the daily returns is called the ' autocorrelation coefficient ' .) Statement IV is false because while the random walk of prices does imply independence, it says nothing about the expected value of returns. It does not imply that the returns will have an expected value of zero (or any other value). Thus Choice ' c ' is the correct answer and t he rest are incorrect.

We can calculate the duration of the bond as follows:

PV of 1st coupon payment: $5/(1 + 5%/2) = $4.878

PV of final payment: $105/(1 + 5%) = $100

Weighted average of the two = (0.5*$4.878 + 1*$100)/($4.878 + $100) = 0.9767, ie this is the Macaulay duration.

Thus the modified duration is 0.9767/(1 + 5%/2) = 0.9529

(Note that Modified Duration = Macaulay Duration/(1 + y/n), where n is the compounding frequency)

In addition to this calculation, in this particular question we can intuitively arrive at the correct answer by eliminating the incorrect choices. Since the bond matures in a year, its modified duration will be less than a year. Therefore Choice ' a ' and Choice ' b ' cannot be correct. Similarly, 0.700 appears too low as the coupons are not so heavily weighted towards earlier in the year. Therefore only Choice ' c ' can be the correct answer.

This question requires us to use the CAPM.

Recall that according to CAPM the expected return on an asset = The risk free rate + β x Market risk premium.

Using the beta and expected return of security A, we can determine the market risk premium to be (8% - 3%)/1.2 = 4.1667%. The excess return is the return over the risk free rate, and for security B, this is nothing but the beta multiplied by the market risk premium. Therefore the correct answer is 4.1667% x 1.5 = 6.25%. Note that this is the ' excess return ' , and not the total return. (Total return will include the risk free rate).

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.

A zero coupon bond has a Macaulay duration equal to its maturity, ie in this case 5 years. We can calculate modified duration from Macaulay duration by using the following relationship:

Modified duration = Macaulay duration/(1 + y) where y is the yield.

Therefore the correct answer is 5 / (1.05) = 4.76.

Or intuitively, all the other answers appear clearly incorrect: 5 and 5.25 are too high, and 4 is too low. Therefore the only reasonable choice is 4.76.

The principle of maximum expected utility requires maximizing the expected utilities of the different possible outcomes of a gamble weighted according to the probabilities of their occurrence. This is very difficult to apply in practice in the financial markets where utility functions and various other inputs for maximizing expected utility are not known. Markowitz suggested the mean-variance criterion as a simplification of the principle of maximum expected utility, and it can be shown that the mean-variance gives a good approximation when the range of outcomes under consideration does not exceed plus or minus one coefficient of risk tolerance. (Recall that the coefficient of risk tolerance is the value of x where the gambler is indifferent between equal probabilities of winning x or losing x/2.)

Imagine a bond that provides just two cash flows: $100 in one year and $10 in 10 years. Imagine another bond that pays nothing now but $100 in 10 years. Assume that both have identical yields to maturity. Yet despite the identical YTM, the bonds are quite dissimilar. For the first bond, we face the risk that we would receive the bulk of the present value of the bond in an year ' s time, and while there is a small amount due in 10 years, the first payment may not find an equally attractive investment opportunity. Reinvestment risk refers to the risk that cash flows from a bond may not be investible at the yield to maturity for the bond. In such cases, bonds that have longer durations are preferable.

A coupon bearing bond with a coupon rate lower than our benchmark bond will carry a lower reinvestment risk as its cash flows are weighted more towards the end. An amortizing bond returns a part of the principle periodically, increasing reinvestment risk. A higher coupon bearing bond will have a higher reinvestment risk, and a zero coupon bond will have the least reinvestment risk.

The effectiveness of the hedge is solely determined by the correlation between the position being hedged, and the pos ition being used as the hedge. A hedge can be perfect when correlation is 1, and will be less than perfect when it is anything other than 1. The effectiveness of the optimal hedge is given by the formula  √(1-ρ 2 ) , where  ρ  is the correlation between the two. Therefore Choice ' a ' is the correct answer. Standard deviations affect the hedge ratio, not the effectiveness of the hedge.

(Note: In other texts, hedge effectiveness is explained to be measured by  ρ 2 , which is essentially the same as  √(1-ρ 2 ) , both ex pressions being functions of  ρ . You can use either, being aware that one measures the variance explained, and the other is a measure of the unexplained variance.)

The transformation line represents the combination of a ' risky bundle ' and the risk free asset. Investors can choose different combinations of these assets depending upon their risk appetite. The transformation l ine meets the y-axis (portfolio returns) at the point equal to the risk free rate. Choice ' b ' is the correct answer and the rest are incorrect. The highest possible transformation line, ie the transformation line with the maximum slope, is the transformat ion line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the ' capital markets line ' . Investors can pick any point on this line according to their risk appetite, and doi ng so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML.

A short squeeze results when short sellers are trying to cover their short positions by buying in the spot markets, which do not have adequate supply. This results in sharp spikes in spot prices, which further forces any other shorts to try cut their losses. The result is a sharp rise in spot prices.

Choice ' a ' is the correct answer, the other choices do not describe a short squeeze.

Once the price of the stock exceeds $100, the investor would forego any additional gains as the call would be exercised. Therefore the maximum gain to the investor is $10, plus the premium earned, which is $3, making for a maximum gain of $13.

In the event the stock price declines, the investor would lose on his long stock position an amount equal to the decline, less any premium earned. The worst case price for a stock is 0, and in this case the investor would lose his $90 in the stock, offset by $3 in premiums earned (the written call would expire worthless), thereby limiting his total loss to $87.

Choice ' c ' is therefore the correct answer.

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.

Implied volatilities are the same for calls and puts with similar exercise and strike prices. If not, it would offer an arbitrage opportunity. Therefore Choice ' b ' is the correct answer.

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.

The buyer of an FRA is considered ' long ' , and a 3 x 6 FRA allows him or her to borrow funds at the agreed rate starting at the end of month 3 till the end of month 6. (ie, 3 x 6 indicates that the borrowing period commences at 3 months, and ends at 6 months, for a period of 6 - 3 = 3 months). The seller has exactly the opposite position, ie he or she is committed to lend funds at the agreed rate for 3 months starting at the end of 3 months from today (for a 3 x 6 FRA). [Note that the obligation to borrow or lend in and FRA does not mean that either parties will actually do that, instead they will just exchange cash flows to get to get to an identical economic situation.]

Since the seller, or the short, is committed to lending in the future starting 3 months from now, it is akin to borrowing now for 3 months, and investing the borrowed amount for 6 months. During the first 3 months, the amounts borrowed and lent ' cancel ' out (conceptually) and after the 3 months the short returns the amount borrowed, and is left with just the net amount lent, ie the FRA.

Thus the correct answer is Choice ' d ' . Choice ' b ' describes the position of the FRA buyer. Choice ' a ' and Choice ' c ' are nonsensical.

Note that intrinsic value of an option is the difference between the value of the underlying and the strike price of the option.

European options can only be exercised at expiry, and Bermudan options only at certain dates during the life of the option. Therefore the option may be valued at less than intrinsic value if the earliest possible exercise date is not very close. An American option however can be exercised at any time prior to expiry, which means that its value can never fall below its intrinsic value. Because if it did, arbitrageurs would buy the option and immediately exercise it to get a risk free profit. It does not matter whether the option is a call or a put - therefore the correct answer is Choice ' c ' .

All the statements given are true, except that standard deviations cannot be averaged to get the portfolio standard deviation unless the assets are perfectly positively correlated. Therefore Choice ' d ' is the false statement, and the correct answer.

For a portfolio of uncorrelated assets, ie correlations being equal to zero, variances can be added together to get portfolio variance. Also, regardless of correlations, portfolio returns are always the weighted average of asset returns, and just averages will do in this case as the portfolio is said to be equally weighted across the assets. A correlation of zero produces a risk level less than that possible with positive correlations.

In this case, we need to determine the value today of an FRA where the bank has to pay 4% from year 2 to 3 in exchange for the then prevailing LIBOR. We do this by using the forward rate from year 2 to 3, and comparing it to the fixed rate. The forward rate is determined from the zero rates as =(1.03^3 / 1.025^2) - 1 = 4.0073%. The bank is committed to paying 4%, therefore the value of the FRA at the end of year 3 = (4.0073% - 4%) * $10m = $732.90. But this is the value at the end of year 3, and needs to be discounted to the present using the 3 year zero rate. Therefore the value of the FRA is $732.90/(1.03^3) = $670.70.

The loss to the fund manager is $20*100 = $2,000. The initial margin placed was $5,000. After the loss is charged to his account, the margin balance is only $3,000 ($5,000 - $2,000). The margin to be maintained is $5,000. Therefore the margin call is $2,000.

Note that there would have been no margin call till the balance in the margin account had reduced to the ' maintenance ' level of $4,000. So if the price had fallen by $5, and the loss had been $500, there would have been no margin call. But when the margin call would be made, it would require the balance to be brought up to the initial margin of $5,000 (and not the maintenance margin of $4,000).

A cash or nothing call is one that pays a fixed amount if the underlying exceeds a certain threshold, or nothing. An asset or nothing call is the same as the cash or nothing call except that instead of paying a fixed amount, it ' pays ' the asset. Therefore the payoff profile of a long asset or nothing call has the same payoff as a long cash or nothing call and a long vanilla call.

This is best considered by building payoff diagrams which will show how the two positions are economically equivalent. Choice ' d ' is the correct answer, and the rest are not.

Since the note pays daily LIBOR, there is no interest rate risk associated with this instrument. Note that interest rate risk for a fixed income instrument arises when the current interest rate changes to a rate different from that the instrument pays. In this case, the rate floats, and is reset daily, therefore there is no risk and the duration is 0. Choice ' c ' is the correct answer.

Swaps rates are calculated such that the swap ' s value at inception is zero. Therefore statement I is correct and statement II is incorrect. OTC swaps are not standardized, in fact they are customized by the parties to suit their needs, which is why they are over-the-counter. Therefore statement III is incorrect. It is correct that interest rate and commodity swaps are the most traded, and statement IV is correct.

Therefore Choice ' c ' is the correct answer.

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 relationship between the value of an option, and its delta, gamma and theta is given by rV = θ + rSΔ + 0.5Γ(σS)2, where V is the value of the option, r the risk-free rate, S the spot price of the underlying, and θ, Δ & Γ are the respective Greeks.

For a delta neutral portfolio, Δ = 0 and this equation reduces to rV = θ + 0.5Γ(σS)2. Now rV is generally a small number, which means that if θ is large and positive, Γ must be large and negative to offset that. Therefore Choice ' b ' is the correct answer.

This question is based upon a weather derivative contract traded on the CME in the US. For each day, ' Heating-Degree-Days ' (HDD) is calculated as equal to 65 degrees Fahrenheit minus the daily average temperature. The daily average temperature is based upon the temperature reported by the Earth Satellite Corporation using a specified automated weather station. Based upon daily values of HDD, an aggregated number called the ' CME degree days index ' is calculated for each contract month. In other words, the index for a particular month is just the aggregation of the ' HDD ' value for each of the days of that month. Each contract settles at the end of the month at a value equal to $20 x Degree Days Index. (In a similar way, ' Cooling Degree Days ' are also calculated and a futures contract offered, except that CDD is equal to the average daily temperature minus 65 degrees). (Source: CME ' s website at CMEGroup.com)

In the given question, we are interested in hedging against the possibility of the temperature being too low. This means we should buy the HDD futures contract (the lower the temperature, the higher the difference of the average temperature from 65 degrees, and the higher the settlement). Therefore Choice ' b ' is the correct answer. The lower the actual temperature turns out to be, the higher the payout to the grower. It would not be wise to wait till January to buy the contract as by then the prices of the contract would have already risen if the grower ' s fears of a colder January appear to be coming true. He can hedge his exposure by immediately locking in the January prices.

The Sharpe ratio is the ratio of excess returns to risk. Excess returns are measured as the returns over the risk-free rate, and risk is measured in terms of volatility, ie standard deviation. Two portfolios with identical Sharpe ratios will certainly have the same ratio of risk and return, though the absolute levels of the return and the risk may vary. Therefore Choice ' c ' is the correct answer.

The square root of time rule can be used to convert, say a 1-day volatility to a 10-day volatility, by multiplying the known volatility number by the square root of time to get the volatility over a different time horizon. However, there are key assumptions that underlie the application of this rule, and statements I to III correctly state those assumptions. If serial correlations (whether negative or positive) exist, then asset returns are not independent as they are affected by the prior day or prior period ' s returns, and we cannot use the square root of time rule. Therefore Choice ' d ' is the correct answer.

In order to use the ' square root of time ' rule, asset returns should be iid, volatility should stay constant (ie there should be no volatility clustering), and no serial correlations (ie the returns of one day should not be affected by the returns of the prior periods). Choice ' d ' is the correct answer.

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.

Since the 10 year rate has stayed the same, but the curve has gone from being flat to upward sloping, it means the shorter term interest rates have gone down. Recall that Macaulay duration is the PV of the payments received on a bond over its life weighted by the year each payment is received in. Also remember that a bond ' s current price is nothing but the summation of the different PVs of its various payments. It effectively gives the real average life of the bond - for example if a hypothetical (and impossible) bond derives half of its value from a payment today and the other half from a payment in 10 years, the investor can look at the average as being 5 years (0*50% + 10*50%). That is what is what we are calculating when deriving Macaulay duration.

In this case, a reduction in the near term interest rates increases the PV of the payments received prior to the payment at 10 years. This increases the value of the bond, and also increases the weightage of the earlier payments, decreasing the Macaulay duration (by shifting it closer to now and further behind from the maturity date).

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.