8013 PRM Exam 1: Finance Foundations Questions and Answers

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.

In an American option, early exercise of the option is permitted. In a European option, early exercise is not permitted. Therefore Choice 'b' is the correct answer.

Open interest refers to the number of outstanding contracts, which is the same as the number of long positions or short positions held by market participants. Note that since for every long futures contract position held there is a seller who holds the short side, the open interest that is long is identical to the open interest that is short. (This is unlike the spot market where one could have long positions without anyone else needing to be symmetrically short).

The total number of contracts traded refers to traded volumes, and not open interest. Other choices are irrelevant in the context.

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.

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.

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.

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 utility function provides a description of an individual or a firm's risk attitude. It expresses how risk seeking or risk averse a firm or an individual is. The utility function would explain differences between risk seeking and risk averse behavior, for example, as an individual becomes richer, he may seek (or shun) risk more than before. A utility function incorporates all of this, and therefore Choice 'd' is the correct answer.

When hedging one fixed income security with another, the question as to how much of the hedge to buy (or sell) (ie the hedge ratio) for a given primary position is determined by their respective basis point values, which in turn are determined by their duration. Therefore, when hedging a long maturity bond with a PV01 of $3 with a short maturity bond that has a PV of $1, we will need to buy 3 times the notional value of the short maturity bond to achieve the same sensitivity to interest rates as the longer maturity bond. Additionally, we may also expect the interest rates on the hedge to move differently from the interest rates on the primary instrument being hedged, and this needs to be accounted for as well as part of the hedge ratio calculation. This is called the yield beta and is calculated as change in yield for primary position/change in yield for the hedge security.

The hedge ratio is determined both by the yield beta and the BPVs of the two securities.

In this case, the yield beta is 1 (as the question speaks of a parallel shift in the yield curve, ie all rates rise or fall together), and the ratio of the BPVs is 5/8. Therefore she should sell 5/8 x 1,000,000 = $625,000 of the 30 year bond.

Choice 'b' 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.

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.

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-ρ 2 ) , where  ρ  is the correlation between the two. Therefore Choice 'c' is the correct answer. Standard deviations affect the hedge ratio, not the effectiveness of the hedge.

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.)

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.

Since $100 is convertible into two shares, the price upon conversion per share is $50, which is the same as the current share price. Therefore there is no premium, and Choice 'c' is the correct answer.

Overcollateralization is when the notes issued by the special purpose vehicle are less in value compared to the underlying pool of assets, thereby providing a buffer to absorb losses. Excess spread implies that the notes issued carry a lower interest rate than the interest rate received on the underlying assets. Cash reserves are reserves intended to take first hits when losses happen. All of these are valid credit enhancements for structured products. Additionally, 'insurance wraps' are also used as a credit enhancement. Choice 'c' is the correct answer.

'Margin requirements' do not mean anything in this context and are not a valid credit enhancement used for credit derivatives.

A higher coupon brings forward the cash flows of a bond, thereby lowering duration. Therefore statement I is true. The lower the duration the lower the convexity. Therefore a higher coupon leads to lower duration and therefore lower convexity. Thus statement II is true.

Bonds do not have negative modified duration, therefore statement III is false.

If the bond is callable, it leads to negative convexity as a fall in interest rates is likely to induce the issuer to call the bond and re-issue at the new lower interest rates (much like refinancing mortgages when interest rates fall), therefore the upside from a fall in rates is limited. This creates 'negative convexity'. Thus statement IV is correct.

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%.

Selling July corn futures would be the cheapest and most appropriate risk management strategy for this corn farmer trying to hedge his risk against falling prices. When June comes, he can close out the futures contracts and any loss or gain on the futures contract would be approximately equal to the change in the spot price of corn from January to June.

Buying corn futures is a bad idea because that would require the farmer to buy corn, which would only add to his expected stock of corn expected from the harvest. The same applies to buying a call option on corn. In the same way, selling a put option would be inappropriate because the buyer of the this put would have the right to sell corn to the farmer at the exercise price, which is not what we want. Therefore Choice 'b' is the correct answer.

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.

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.

A 'hard call protection' means the bond cannot be called until a certain date, regardless of what the share price is. Therefore statement I is false. Also note that a 'soft call protection' means that a bond can be called only if the share price reaches a certain threshold.

It is advantageous for the issuer to call its convertible securities when the share price exceeds the conversion price - because these shares can instead be sold in the market at the higher share price than the lower conversion price. Statement II is true.

When the issuer's share price is very high, the convertible bond trades at a discount to the share price because it is almost certain to be called by the issuer and be redeemed at par. Therefore statement III is right. Statement IV is incorrect because convertible bonds need to pay less coupon than equivalent non-convertible bonds because of the value of the option embedded in them.

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.

The tax advantage of debt is an important input into a decision on the financial structure of a firm. Increasing debt increases the fixed expenses of a business that must be paid out of current cash flows, thereby increasing the cost of financial distress. CFOs consider the volatility of their earnings, outlook for the future and the possibility of financial distress at varying levels of debt, so that certainly is an input into the debt-equity tradeoff. Holding higher levels of equity increases financial flexibility for financing new investments or acquisitions, and depending upon their plans for such activities, they may decide to hold a certain level of equity versus debt.

The agency costs of equity refer to the cost to shareholders arising from the management acting in its own interest rather than that of shareholders - and this is irrelevant to a decision on capital structure. Therefore Choice 'd' is the correct answer.

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 requires a calculation of the present value of the future cash flows from the bond. The correct answer is $94.92, calculated as =(5/(1 + 5%)) + (5/(1+ 6%)^2) + (105/(1 + 7%)^3).

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).

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).

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

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.

An m x n FRA is an agreement to borrow money for a period starting at time m and ending at time n at the contracted rate. Therefore, the buyer of the 3 x 6 FRA has committed to borrow $1m at the beginning of 3 months and return it at the end of 6 months, ie a total borrowing period of 3 months at a rate of 5%. Of course, the $1m is never actually exchanged, and at the beginning of the 3 month period when the next three months' interest rate is known (6%), the parties merely exchange the difference in the interest. SInce this interest was only due at the end of the 6 months and is being exchanged at the 3 month time point, it will have to be discounted to its present value.

The correct answer to this question is =(1,000,000 * (6% - 5%) * 3/12)/(1 + (6%*3/12))=$2463.05. Since interest rates rose, the borrower gained as he has the right to borrow at a lower rate, and therefore the seller will pay the borrower.

(Here:

- $1m is the notional

- 6% - 5% represents the difference between the contracted and the realized interest rates

- 3/12 is the 3 month period from month 3 to 6

- Finally, we divide by the current interest rate for 3 months to present value the payment from month 6 to month 3)

Some parts of this question draw upon contingent claims framework to the value of a firm. According to this framework, the relationship between the debt and equity holders of a firm can be viewed as follows: The equity holders have a call option on the assets of the firm with a strike price equal to the value of the debt, and the debt holders have sold them this call. This is so because should the value of the assets of the firm fall below the value of the debt, the equity holders can walk away by handing over the assets to the debt holders in full extinguishment of their claims. If the value of the firm's assets is greater than the value of debt, the equity holders will exercise their call option. At the same time, it is also possible to view the debtholders as holding an asset and having sold a put on the assets of the firm with a strike price equal to the value of the debt. If the value of the assets of the firm were to fall below the value of the debt, they will end up buying the assets at a price equal to the value of the debt.

An increase in the volatility of returns on the assets of a firm increases the volatility of the asset value. This means the likelihood that the asset value will go below the value of the debt of the firm will increase. Callable debt can be considered to be a summation of two separate securities: a regular debt, for which the firm pays interest and receives a principal loan; and a call option that the debt holders have sold to the firm allowing the firm to buy back the debt. In return, the firm pays an implied 'premium' to the debt holders who have agreed to buy the callable debt. Therefore the total payments by the company to callable debt holders is equal to the interest payments plus the premium payment for the option. An increase in asset volatility will increase the value of this option as it is likely that the assets will increase in value, and strengthen the company's credit, and lower its spread at which time it would like to repay the debt and refinance/roll over at the new lower rate. Therefore an increase in volatility will increase the 'premium' demanded by the callable debt holders, thereby increasing the total yield and lowering the value of the callable debt. Therefore Choice 'b' (An increase in the value of the callable debt of the firm) is incorrect.

An increase in asset volatility will decrease the value of the firm as it is now riskier than before (higher standard deviation, same expected returns). Therefore Choice 'a' (An increase in the value of the equity of the firm) is false too.

The value of the implicit put in the debt of the firm will increase and not decrease as the volatility of the underlying assets increases. Therefore Choice 'c' (A decrease in the value of the implicit put in in the debt of the firm) is incorrect too.

Choice 'd' (A decrease in the value of the non-callable debt issued by the firm) is correct because higher asset volatility will increase the riskiness of the company's debt, making the required yield higher and decreasing its value.

This is a straightforward bond valuation problem where all variables required 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.

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).

Choice 'b' is the correct answer. The other relationships are not correct.

Delta hedges need to be rebalanced periodically because with changes in prices, delta changes, and the amount of hedge required changes as well, requiring a readjustment. The more frequent the rebalancing, the better the performance of the hedge. However, the greater the rebalancing frequency, the greater are the number of trades required, increasing the costs from trading. The trader has to balance transaction costs against the adverse effects from poorer hedge performance. Therefore rebalancing frequency is a consideration for a trader and statement I is true.

Statement II is not true. When options are granted, their future impact on the price gets priced in into the current price, and no change occurs when these are actually exercised.

Statement III 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, a large negative theta means a large positive gamma and vice-versa. Therefore theta can be used as a proxy for gamma and this statement is correct.

Statement IV is correct. Vega is highest closest to the strike price - look at the graph for vega to see how vega behaves with changes to spot. (Do it yourself or there is a graph in the tutorial too.)

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.)

It is important to note the distinction between the clearing house and the exchange itself. The clearing house does not get involved with physical delivery, nor does it provide any dispute settlement services. It only makes sure that cash is settled as and when due between the members. Therefore Choice 'c' 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.

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.

When hedging one fixed income security with another, the question as to how much of the hedge to buy (or sell) (ie the hedge ratio) for a given primary position is determined by their respective basis point values, which in turn are determined by their duration. Therefore, when hedging a long maturity bond with a PV01 of $3 with a short maturity bond that has a PV of $1, we will need to buy 3 times the notional value of the short maturity bond to achieve the same sensitivity to interest rates as the longer maturity bond.

Additionally, we may also expect the interest rates on the hedge to move differently from the interest rates on the primary instrument being hedged, and this needs to be accounted for as well as part of the hedge ratio calculation. This is called the yield beta and is calculated as change in yield for primary position/change in yield for the hedge security.

The hedge ratio is determined both by the yield beta and the BPVs of the two securities. Choice 'd' is the correct answer.

Investors have differing view of the market, which means differing view of expected returns, correlations and volatilities. Not only do they have differing views, these views change frequently as new information reveals itself. Accordingly, each investor has their own version of the efficient frontier. Once investors have determined their efficient frontiers, they will determine the extent of risk they wish to hold. If they had identical views, they would have held the same portfolio. But they do not, and if they did, there would be very little trading in the markets.

All the above statements are true except statement I which is false due to differing investor expectations.

(Re statement IV: If investors have different risk appetites, their portfolio will vary in the split between the risky and the riskfree assets. But inside the 'risky' assets bundle the proportion of the assets will be identical. Ie, they would express their varying risk appetites by varying how much of the risky bundle and the riskfree asset they hold, but inside the risky bundle the proportion of the different risky assets will be the same.)

Basis risk between spot and futures prices is the risk that the prices of the futures contract will not move in lockstep with the spot prices. Basis risk creates a problem when investors are trying to get exposure to a stock price through a futures contract, or are trying to hedge a position using a futures contract, and over time the prices of these two diverge.

To understand why basis risk arises, we need to think about how futures prices are determined. Futures prices are nothing but the spot prices plus the carrying cost, ie what it would cost for the seller of the contract to buy the spot security and hold it till the time of delivery in the future. During this holing period, the seller of the contract has to pay interest to fund his purchase in the spot market, and also receives and distributions such as dividends. In the case of commodity futures, he may also have to pay storage charges. When determining the futures prices, it is not known with complete certainty as to what the carrying costs will ultimately turn out to be - interest rates may change, dividends may turn out to be more or less, leading to changes in the futures prices without changes in the spot prices. This gives rise to basis risk.

Therefore changes to the risk free rate, which is a proxy for the true funding cost of the futures position and changes in expected dividends creates basis risk. Volatility of the underlying is irrelevant for futures.

Therefore Choice 'b' is the correct answer.

Macaulay duration is the weighted average of the present value of payments received on a bond, weighted by the year in which the payments are received. If the yield curve changes shape, it will change the present values of the coupons and therefore also the Macaulay duration. Statement I is therefore not correct. However, if the bond were to be a zero-coupon bond, changes in the curve will not affect its duration so long as the interest rate at maturity stays constant.

Modified duration = Macaulay Duration/(1 + rate/compounding frequency). Therefore modified duration will change if the compounding frequency changes, assuming the nominal rate of interest for the coupons are identical. Therefore statement II is correct. When rates are expressed as continuously compounded rates, modified duration and Macaulay duration will be identical. This intuitively follows from the formula for modified duration: Modified duration = Macaulay Duration/(1 + rate/compounding frequency). For continuously compounded rates, the compounding frequency approaches infinity, so the denominator approaches 1, leaving modified duration equal to the Macaulay duration. Therefore statement III is correct.

Convexity is higher for a bond that has its payments spread out compared to a bond that has its payments concentrated at a single point in time. For a bond with a higher coupon, the higher coupons have the effect of spreading the present value of the bond over a longer period, when compared to a bond with lower coupons where the payment is comparatively more concentrated at the maturity. Therefore Convexity is higher for a bond with a lower coupon when compared to a similar bond with a higher coupon. Statement IV is not correct.

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.

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.

The BSM formula assumes that prices are lognormally distributed. Therefore Choice 'b' does not represent a valid assumption underlying Black Scholes. All other choices represent valid assumptions.

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 yield to maturity for a bond trading at par will be identical to its coupon. Therefore the yield to maturity for this bond will be 5%

Security A 'dominates' security B, ie it offers the most return per unit of volatility. However, because it is less volatile, its returns are likely to be more concentrated around its mean compared to Security B which will have 'fatter tails' compared to Security A. Therefore Security B will offer a much better chance of making the desired 25% return. Of course, this also means that Security B is much more likely to lose a lot more money than Security A. However, if the portfolio manager is only interested in maximizing his chances of making a 25% or greater return, he should select Security B. The answer can be verified using Excel's Normdist formula, and probability of exceeding 25% will be given by [=1-NORMDIST(0.25,0.1,0.1,TRUE)] for Security A and =1-NORMDIST(0.25,0.1,0.2,TRUE)] for Security B. A combined portfolio of the two will have a return of 10% and variance less than that of Security B (or at the most equal to the weighted average of the two variances, even if their correlation is 1), which means a combined portfolio will also have a lower probability of exceeding a 25% return.

This question reflects how portfolio managers may be tempted to make sub-optimum decisions for their investors by swinging for the fences by taking large bets, for example, when they have had a big drawdown and have to make it up to the high water mark levels before they earn performance fees.

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.

A short position in a fixed rate bond and a long position in an FRN has the effect of paying fixed and receiving floating. The floating received offsets the floating payment on the borrowing, leaving the borrower with just a fixed rate outflow. Therefore the combination identified in statement I can be used to achieve the objective of paying fixed.

A collar is equivalent to a long position in an interest rate cap combined with a short position in an interest rate floor. This has the effect of setting a range within which the investor's borrowing rate will vary. In the case where the cap and floor rates are the same, the combination of a collar and a long FRN effectively produces an outcome where the holder of such positions pays a fixed rate. Therefore, an interest rate collar can be used to convert the fixed payment to a floating rate payment. [Example: Assume current interest rate is 3%, and therefore the borrower has a liability of 3% on the FRN. Assume that the borrower now buys a collar at the strike rate of 4%. Now the borrower receives 0% (=Max(3 % - 4%, 0)) on the cap part of the collar, and pays 1% on the floor part of the collar (=Max(4% - 3%, 0)). The net borrowing cost therefore is 3% paid on the FRN plus 1% paid on the collar, equal to 4%. Now if interest rates rise to say 6%, the borrower pays 6% on the FRN, and receives 2% from the collar (=Max(6% - 4%, 0) - Max(4% - 6%, 0)), creating a net cost of 6% - 2% = 4%.

A collar is often issued with an FRN to convert floating flows to fixed. Therefore combination II is an acceptable choice.

A short position in a fixed rate bond and a short position in an FRN produces a cash flow that does not produce a net fixed cash outflow when combined with the borrowing. Therefore statement III is not a valid combination.

An interest rate swap where the investor pays fixed and receives floating, when combined with a floating payment on an FRN leaves a net fixed payment, Therefore statement IV is a valid way to achieve the borrower's objective.

The security market line is generally shown graphically with returns on the y-axis and the asset's beta on the x-axis. The correct answer is Choice 'a'.

Note the difference between the SML and the CML (the capital markets line). The CML 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, and expresses the relationship between risk and return.

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.

The maximum value for delta of a call option is 1 and not infinity. The value of theta for a deep out of the money call option does indeed approach zero as the value of the option is hardly affected by the passage of time. Vega measures the sensitivity of option prices to the volatility of the underlying, an increase in vega increases the price of the option regardless of whether it is a call option or a put option. (Vega is identical and positive for both calls and puts). Gamma changes very rapidly and reaches its maximum ( & not zero) close to the strike price, including for a digital option. Therefore statement II is true and the rest are incorrect.

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.

Futures initial margins are determined by the exchange. SPAN is the name of a framework the CME uses to determine margins. Only Choice 'd' is correct.

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

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%.

The return on an asset is a function of the covariance of the asset's return to the returns of the market portfolio. They do not depend upon the standard deviation of the asset itself. Therefore Choice 'b' is correct and Choice 'c' is incorrect.

The expected returns on an asset are equal to the risk free rate plus the beta times the market risk premium, therefore Choice 'a' is correct.

Portfolios on the efficient frontier will all have a different Sharpe ratio, which is the ratio of excess returns to portfolio volatility.

Choice 'c' is the correct answer (note that the question is asking for an identification of the INCORRECT statement).

The duration of a perpetuity is 1/running yield. Therefore the correct answer is 1/.05 = 20 years

Straddles and strangles are strategies that would benefit from sharp movement in option prices, regardless of direction. These comprise a long call and a long put, which would benefit regardless of whether prices rise or fall. The only time they would lose money would be when prices stay constant.

A collar would gain when stock prices fall, and not when they rise. Since our investor does not have a view on the direction of the movement, this strategy will not work for him.

A butterfly spread or a condor would gain when prices stay range-bound, so that cannot be a useful strategy.

Therefore Choice 'd' is the correct answer.

Futures contracts carry no gamma. Only options have gamma. Choice 'a' is the correct answer. Any instrument whose price varies in a linear fashion with respect to the underlying will have gamma equal to zero.

The swap can be valued by using the new swap rate of 5%. The investor is paying fixed and receiving LIBOR, and can effectively get out of his position by entering into a swap to receive 5% and pay LIBOR. This will leave him/her with a net cash flow of 1% for two years, ie $1m for 2 years that can be discounted to the present using the rates provided, ie =(1/1.05 + 1/(1.05^2)) = $1,859,410.

Detailed explanation:

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

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

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

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.

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.

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.

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.

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.

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.

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 implied volatility for a call and a put is identical if the two have the same strike and expiry. Therefore Choice 'b' is the correct answer and the remaining choices are incorrect.

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 underlying asset in a treasury bond futures contract is any long term US Treasury bond with a maturity of more than 15 years and not callable within 15 years. The underlying asset in a Treasury note futures contract is any long term Treasury note with a maturity of no less than 6.5 years and no more than 10 years.

Note the difference between what the underlying asset is for treasury bond futures and treasury note futures. In either case, adjustments will be made for the coupon and maturity of the actual bond delivered (the concept of the adjustment of 'cheapest-to-deliver').

https://riskprep.com/images/stories/questions/102.07.a.png  is the coefficient of risk aversion at x. Its inverse, ie  https://riskprep.com/images/stories/questions/102.07.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.

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. Choice 'c' is the correct answer.

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

The underlying asset in a Treasury note futures contract is any long term Treasury note with a maturity of no less than 6.5 years and no more than 10 years. The underlying asset in a treasury bond futures contract is any long term US Treasury bond with a maturity of more than 15 years and not callable within 15 years.

Note the difference betwen what the underlying asset is for treasury bond futures and treasury note futures. In either case, adjustments will be made for the coupon and maturity of the actual bond delivered (the concept of the adjustment of 'cheapest-to-deliver').

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.

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.)

In this case, we need a portfolio that is "dollar duration" neutral, ie the net effect of any change in interest rates creates offsetting gains and losses in the asset and liability portfolios. The $125m asset portfolio with a duration of 15 years has a dollar-duration of $125*15/100=$18.75m, implying that a 1% change in interest rates will create a $18.75m change in the value of the asset portfolio. Similarly, the liabilities of $100m with a duration of 20 years mean that a 1% change in interest rates will lead to a $20m change in their value. Since $18.75m < $20m, the pension fund is short duration, and needs a strategy that will give it more duration. It can do so by buying more bonds, or swapping out some of its shorter duration bonds with longer duration bonds, or doing other things that will increase the duration of the portfolio so it becomes duration neutral.

Entering into an interest rate swap to receive fixed and pay floating is like a long position in a bond, which will increase duration. Thus Choice 'b' is the correct answer.

A swap to pay fixed and receive floating will have the opposite effect, and so will selling bond futures. Choice 'c' and Choice 'a' are therefore incorrect. As we have seen, Choice 'd' is incorrect as the fund does have interest rate exposure.

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.

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.

A repo, or a repurchase agreement, is the lending of securities in return for cash, with an agreement to buy the securities back at a later date at the borrowed amount plus interest. It is a form of collateralized borrowing. A 'reverse repo' is exactly the opposite of a repo transaction, ie where cash is lent and securities borrowed. Therefore Choice 'b' is the correct answer. In any repo transaction, the counterparty will therefore always have a 'reverse repo' position.

A reverse repo is a useful transaction - not merely for the purpose of lending short term funds, but more importantly to enable short positions. For example, if an investor wishes to short a bond, he can borrow the bond on a 'reverse repo' and sell it. Of course, he will have to return the bond when the reverse repo matures, but hopefully by that time prices of the bond would have fallen to allow him to do so profitably. Short positions in physical bonds are nearly always facilitated by reverse repos.