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

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).

The effect of higher convexity is that when yields rise, the price decrease is lower than the increase in yields, and when yields fall, the increase in price is greater than the fall in yield. In either case, it benefits the holder of the fixed income instrument that carries such positive convexity. The converse is true for someone short a bond - such an investor would prefer lower convexity to higher convexity. Therefore statement I is not true for an investor who is short a bond.

An increase in yields makes bond prices decrease, something that would benefit the short. Therefore statement II is true for an investor short a bond.

Negative convexity has exactly the opposite effect as the one described for positive convexity for statement I above. An investor short a bond would prefer negative convexity (which by the way is exhibited by very few fixed income instruments such as mortgages) to positive convexity, therefore statement III is true for such an investor.

Choice 'b' is the correct answer.

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.

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 incorrect. 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 lower 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 the rest are incorrect.

Statement I is true as standard deviation is the root of the squared deviations, and is always positive, and identical regardless of the positions being long or short.Statement II is false as the expected return of a long position is exactly the negative of the expected return of the short position.Statement III is correct as a correlation of +1 between two long positions implies a correlation of -1 between one long and one short position. This is the basis of hedging, for example, of spot positions with futures.Statement IV is inaccurate. The weight of an asset in a portfolio does not change the correlation of the asset with the portfolio, or with other assets.Therefore Choice 'c' is the correct answer.

The bond equivalent yield for a treasury bill can be calculated as [(Future value - Present value)/Present value x 365/days to maturity]. In this case this works out to ($100-$98)/$98 * 365/90 = 8.28%

[Why do we use 365 days and not 360? And is there a different way to calculate treasury bill yields?

The answer to this question is that there are alternate ways to calculate these yields. For the exam, I would suggest that you calculate according to one method, see if you get an answer that matches, and pick that. I found the below on the NY Fed's website - a very clear example: (http://newyorkfed.org/aboutthefed/fedpoint/fed28.html)

The Discount Yield Method

The following formula is used to determine the discount yield for T-bills that have three- or six-month maturities:

Discount yield = [(FV - PP)/FV] * [360/M]

FV = face value

PP = purchase price

M = maturity of bill. For a three-month T-bill (13 weeks) use 91, and for a six-month T-bill (26 weeks) use 182

360 = the number of days used by banks to determine short-term interest rates (the investment yield method is based on a calendar year: 365 days, or 366 in leap years).

Example

What is the discount yield for a 182-day T-bill, auctioned at an average price of $9,659.30 per $10,000 face value?

Discount yield = [(FV - PP)/FV] * [360/M]

FV = $10,000 PP = $9,659.30 M = 182

Discount yield = [(10,000) - (9,659.30)] / (10,000) * [360/182]

Discount yield = [340.7 / 10,000] * [1.978022]

Discount yield = .0673912 = 6.74%

For the 13-week bill, the same formula would be used, dividing 360 by a maturity of 91 days rather than 182 days.

The Investment Yield Method

When comparing the return on investment in T-bills to other short-term investment options, the investment yield method can be used. This yield is alternatively called the bond equivalent yield, the coupon equivalent rate, the effective yield and the interest yield.

The following formula is used to calculate the investment yield for T-bills that have three- or six-month maturities:

Investment yield = [(FV - PP)/PP] * [365 or 366/M]

Example

What is the investment yield of a 182-day T-bill, auctioned at an average price of $9,659.30 per $10,000 face value?

Investment yield = [(FV - PP)/PP] * [365/M]

FV = $10,000 PP = $9,659.30 M = 182

Investment yield = [(10,000 - 9,659.30) / (9,659.30)] * [365/182]

Investment yield = [340.70] / 9,659.30] * [2.0054945]

Investment yield = .0707372 = 7.07%

For the 13-week bill, the same formula can be used, dividing 365 (or 366) by a maturity of 91 days.

The modified duration of a perpetual bond is given by 1/i, where i is the yield. Since I is a positive number greater than zero, it means the modified duration of a perpetual bond is not infinity. Also note that this formula makes no reference to the coupon rate at all - in other words, the modified duration of a perpetual bond is independent of the coupon. The Macaulay duration of a perpetual bond is given by (1+i)/i, again a formula devoid of any references to the coupon. The price of a perpetual bond can be calculated as Coupon/i, which means that a 15% coupon perpetual bond will be priced at exactly 3x the price of a 5% coupon bond.

Based on the above, statement I is incorrect, but statement II is correct. Statement III is incorrect because Bond A will be priced at exactly 1/3rd the price of Bond B, and not less than that. Similarly, statement IV is incorrect as well.

Intuitively, a perpetual bond is nothing but a perpetual annuity, which will have a present value equal to 1/i * cash flow (which in this case is the coupon). The coupon rate only sets the cash flow dollar amount - there is really no difference between the two bonds described in the question except that buying one of Bond B is economically identical to buying three of Bond A.

Theta measures time decay, ie the change in value of the option with the passage of time. When the option is close to expiry, theta is very low as the value of the option is driven by intrinsic value rather than the time value. Therefore theta approaches zero as the option comes closer to expiry.

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.

All the choices list inputs into the determination of futures prices. Therefore volatility in any of them affects the volatility of futures prices. Of course, the largest contributor to the volatility is the volatility of the spot price of the underlying. 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.

Forward rates can be calculated from spot rates and interest rates using the formula Spot x (1+domestic interest rate)/(1+foreign interest rate), where the 'Spot' is expressed as a direct rate (ie as the number of domestic currency units one unit of the foreign currency can buy). In this case the forward rate will be 1.1239 * (1 + 0.75%*90/360) / (1 + 0.4%*90/360) = 1.1249.

It can be confusing to determine which interest rate should be considered 'domestic', and which 'foreign' for this formula. For that, look at the spot rate. Think of the spot rate as being x units of one currency equal to 1 unit of the other currency. In this case, think of the spot rate 1.1239 as "CAD 1.1239 = USD 1". The currency that has the "1" in it is the 'foreign' and the other one is 'domestic'.

It is also important to remember how exchange rates are generally quoted. Most exchange rates are quoted in terms of how many foreign currencies does USD 1 buy. Therefore, a rate of 99 for the JPY means that USD 1 is equal to JPY 99. These are called 'direct rates'. However, there are four major world currencies where the rate quote convention is the other way round - these are EUR, GBP, AUD and NZD. For these currencies, the FX quote implies how many US dollars can one unit of these currencies buy. So a quote of "1.1023" for the Euro means EUR 1 is equal to USD 1.1023 and not the other way round.

The delivery price of a futures contract is given by Se^(rt), just as in the case of a forward contract. However, a key difference is that a forward is settled at maturity whereas a futures contract pays out the P&L daily. So if the spot price increases from S to δS, the holder of a futures contract immediately receives the change in the delivery price without any discounting to the present. That is, the holder of the futures contract receives (S + δS)e^(rt) - Se^(rt) = δSe^(rt) right away. Therefore the delta of a futures contract is e^rt, which given positive non-zero values of r and t can only be greater than zero.

Therefore Choice 'b' is the correct answer. Note the difference from a forward contract where this difference is not received till the delivery date, therefore making the delta of the forward contract to be equal to 1.

According to ISDA, a credit event is an event linked to the deteriorating credit worthiness of an underlying reference entity in a credit derivative. The occurrence of a credit event usually triggers full or partial termination of the transaction and a payment from protection seller to protection buyer. Credit events include

- bankruptcy,

- failure to pay,

- restructuring,

- obligation acceleration,

- obligation default and

- repudiation/moratorium.

Therefore all four events listed are credit events and Choice 'b' is the correct answer.

The riskiest tranche of a structured product is called the equity tranche. All other choices are incorrect.

The operating free cash flows can be considered a perpetual annuity with a given growth rate.

The value of a perpetuity of a periodic cash flow of 'c', with a discount rate 'r' and growth rate 'g' is given by c/(r - g). In the given case, the company can be considered as providing a perpetual annuity which provides an annual cash flow of $100 which are growing at 2.5% (equal to the GDP's growth rate, as given), and whose cost of capital, or the discount rate to use, is 7.5%

Therefore the value of the firm in this case is given by $100/(7.5% - 2.5%) = $2,000. Recall that the value of the firm is equal to the Operating Free Cash Flow/Weighted Average Cost of Capital (OFCF/WACC). Therefore Choice 'd' is the correct answer.

'In store' delivery is for delivery in a standardized location, and the buyer is handed a 'warrant' that allows him to pick the goods up. This is the cheapest means of physical delivery. The other prices will be higher as they involve more costs for the seller who has to get the goods on board a ship, or to the docks, or insurance and freight as well. Choice 'c' is the correct answer.

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.

Contingent premium options are options where no premiums are due upfront, and are due only at expiry and only if the option finishes in the money. (If this sounds too good to be true, remember that at expiry it is possible that the premiums due exceed the payoffs from the option. So there is no free lunch.) For this reason they are also called pay-later options.

They are simple combinations of a vanilla option and the 'right' number of cash-or-nothing options. The buyer of a contingent premium option has effectively purchased a vanilla option and sold enough number of cash-or-nothing options so that the net premium due at inception is zero.

Therefore statement I, II and III are correct and Choice 'b' is the correct answer.

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.

Choice 'a' is the correct answer as the issue of the floating rate note will provide the utility with the funds it needs, and the interest rate swap would offset the floating rate payment and leave it with a net fixed payment.

Choice 'd' is incorrect as the swap is in the wrong direction.

Choice 'c' is incorrect as buying and selling a floating rate bond would mean the utility will not have any funds that it wants to issue the note for, and combining it with interest rate futures would be just absurd.

Choice 'b' is incorrect as buying a floating rate note would use funds while the utility is trying to raise funds.

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 effectiveness of the hedge is solely determined by the correlation between the position being hedged, and the position 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-ρ2), 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 ρ2, which is essentially the same as √(1-ρ2), both expressions 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.)

Analytical methods are more 'efficient' than numerical methods in terms of computing power required, consistency of results and providing clarity on the inputs driving the results. Therefore 'efficient valuation' is not a reason to adopt numerical pricing methods over analytical methods.

It is much easier to incorporate changing stochastic volatility, discontinuous asset prices or other complex payoffs using numerical pricing methods. So these certainly represent good reasons to use numerical pricing methods for valuing exotics.

Therefore Choice 'b' is the correct answer.

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 correct answer is Choice 'b', as this question is merely asking for the forward rate based on known spot rates. The forward rate applicable to the three month period commencing in 3 months time is given by [(1 + 6%*6/12)/(1 + 5%*3/12) - 1]*4 = 6.91%. Thus Choice 'b' is the correct answer.

Here is a step by step way to think about it: $1 invested now at 6% for 6 months grows to (1 + 6%*6/12)=1.03. At the same time, using the 3 month rate, $1 invested now at 5% for 3 months grows to (1 + 5%*3/12)=1.0125. Effectively, this means that the 1.0125 at the end of 3 months grow to 1.03 at the end of 6 months, implying the rate of interest during the 3 months from 3 to 6 months is (1.03/1.0125 - 1)*4 = 6.91%.

We need to calculate the total dollar duration of the bond portfolio, which is =100*7.5 - 65*12 + 120*6 = $690m. This means a 10 basis point change in interest rates changes the value of the portfolio by $690m*10bps = $0.69m, or $690,000. Therefore Choice 'b' is the correct answer.

[Here is an alternate step-by-step way of thinking about it. Let us consider changes to each position separately.a. A long position worth $100m in a bond with a modified duration of 7.5A 1% increase reduces the value of the bond by 7.5%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.75% = $100m*0.75% = $750,000 decrease.b. A short position worth $65m in a bond with a modified duration of 12A 1% increase increases the value of the position by 12% (remember this is a short position). Therefore a 10bps = 0.1% change in interest rates will change the position by 1.2% = $65m*1.2% = $780,000 increase.c. A long position worth $120m in a bond with a modified duration of 6A 1% increase reduces the value of the bond by 6%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.6% = $120m*0.6% = $720,000 decrease.The net change in the value of the porfolio is therefore:-750,000 + 780,000 - 720,000 = $690,000]

[Here is an alternate step-by-step way of thinking about it. Let us consider changes to each position separately.

a. A long position worth $100m in a bond with a modified duration of 7.5

A 1% increase reduces the value of the bond by 7.5%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.75% = $100m*0.75% = $750,000 decrease.

b. A short position worth $65m in a bond with a modified duration of 12

A 1% increase increases the value of the position by 12% (remember this is a short position). Therefore a 10bps = 0.1% change in interest rates will change the position by 1.2% = $65m*1.2% = $780,000 increase.

c. A long position worth $120m in a bond with a modified duration of 6

A 1% increase reduces the value of the bond by 6%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.6% = $120m*0.6% = $720,000 decrease.

The net change in the value of the porfolio is therefore:

-750,000 + 780,000 - 720,000 = $690,000]

We know from the put-call parity that:

Call - Put = Stock - Deposit

In the given situation,

Stock price = 100

call = 35

Put = ?

Exercise price = 100

However, we cannot use the stock price as-is, we need to adjust it for dividends that will be received during the period for which the option is valid. Dividends are $1, which need to be discounted to the present. In financial theory, dividends are often assumed to be continuously paid, though in reality they are single discrete payments at points in time. In this situation, for simplicity we assume that the dividend is paid at the end of the year, and needs to be discounted to the present and deducted from the spot price.

Therefore:

Stock price adjusted for dividends = $100 - ($1/1.05) = $100 - $0.95 = $99.05

Similarly, the bank deposit amount, which is the PV of the exercise price, can be calculated as $100/1.05 = $95.24

We can now calculate the price of the put by plugging the numbers above in the put-call parity:

35 - Put = $99.05 - $95.24

Therefore the value of the put = 31.19, which is closest to 31.20 which therefore is the correct answer.

Choice 'a' correctly describes a holder extendible option. Choice 'd' describes a writer extendible option. The other choices are meaningless.

We can use the put-call parity to estimate the value of the call option in this case.

The put-call parity can be expressed as:

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

Substituting the values we know:

Value of call - $5 = $100 - $110/(1 + 10%); which implies the value of a call is also $5.

Choice 'a' is therefore the correct answer.

Note about interest rates: How do we know if the interest rate provided in the question is continuous or discrete? For short periods of time, the difference between discrete and continuous exchange rates tends to be not material. For ease of computation, use discrete rates (as above), and it should allow you to answer the question.

From the put-call parity, we know that Call - Put = Stock - Bank deposit. Since the bank deposit has a zero Gamma, and the Gamma of the Stock itself is also 0, we get the relationship Gamma of Call - Gamma of Put = 0. Therefore, if the Gamma of a call option is 0.08, the Gamma of the corresponding put option is also 0.08.

An easy way to analyze the impact on convexity is to think about the impact on duration. If the change causes duration to increase, it would cause the convexity to increase too, and if it causes the duration to decrease, it would reduce convexity as well.An increase in yields reduces duration, and therefore reduces convexity.

An increase in maturity increases duration, and therefore increases convexity.

An increase in the coupon rate reduces duration, and therefore reduces convexity.

An increase in duration increases convexity.

Therefore statements II and IV represent situations where convexity increases.

Everything else remaining the same, an increase in the rate of dividends causes the value of call options to fall and the value of put options to rise. Therefore, Choice 'b' is the correct answer. (In the exam, the question could address either a call or a put option, so be aware of the answer in either case).

To understand this, consider how dividends are accounted for when valuing an option using the Black Scholes model. Future dividends are discounted to the present using the risk free rate and this discounted value is reduced from the spot price used in the BSM valuation. Effectively, this reduces the spot price used in the BSM formula. When the spot price reduces, and the exercise price remains the same, then the value of the call option goes down. In the same way, when spot price is reduced by the present value of dividends (and the exercise price stays the same), obviously the put option becomes more valuable. Therefore an increase in the rate of dividends increases the value of the put option.

There is another intuitive way to think about this: A call option is like a long position in the stock, but the holder of the call option is not entitled to receive dividends (unlike the holder of the stock). Since the holder of the call option has to forego the dividends, he is willing to pay less for the option; or in other words, the value of the call reduces.

In the same way, a put option is like having a short position in the stock. The holder of the short position has to borrow the stock in order to get into the short position in the first place. When dividends are paid, the holder of the short stock position has to make good any dividends that might be paid to the lender of the stock. The holder of a put option does not have to make any such payments. Therefore the put option is more valuable, and the existence of dividends (or an increase in dividends) increases the value of the put option.

(Try this out using the Black Scholes Excel model given under the tutorials by varying the spot price.)

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.

:

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.

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.

The duration matched hedge is the best hedge for the bond manager. Matched durations in the bond portfolio and the hedge positions will ensure that any decline in the bond portfolio due to interest rate changes is matched by a rise in the futures position. Hedging dollar-to-dollar will not work unless the bonds and the futures contracts are identical in duration terms. For example, if the duration of the portfolio is 7 years, and each futures contract has a duration of 3.5 years, then selling $1m in value will only cover half the risk. Choice 'b' is the right answer.

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.

This is a trick question, no great science to it. Remember that the 'present value of a basis point' refers to PV01, which is the same as BPV (basis point value) referred to in the PRMIA handbook. In other textbooks, the same term is also variously called 'DV01' (dollar value of a basis point). Remember these other terms too.

PV01, or the present value of a basis point, is the change in the value of a bond (or other fixed income security) from a 1 basis point change in the yield. PV01 is calculated as (Price * Modified Duration/10,000).

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 line 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 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.