8006 Questions and Answers

A fixed for floating interest rate swap would be the most appropriate to the company's needs. It will allow it to receive a fixed rate (which will offset the fixed payment it has to make on the loan) and pay a floating rate which it expects will be lower than the fixed rates. Choice 'a' is the correct answer.

Answer:

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

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

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

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

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

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

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

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.

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

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.

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

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

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.

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.

To answer this question, we need to look at the put-call parity, which can be expressed as:

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

or, Value of call - Value of put = Stock - Bond with a future value equal to exercise price

Therefore, a long call and a short put is equivalent to a long stock position and a short bond.

Choice 'a' is therefore the correct answer. (Alternatively, we could also have constructed a graph of the payoff profiles to arrive at the same answer).

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

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

When assets are uncorrelated, variances are additive. But volatility (which is standard deviation) is not. In the given situation, the total variance of the portfolio will be equal to the the square root of the sum of the variances of the assets in the portfolio weighted by the square of their weights. Its volatility will be the square root of this variance. Thus Choice 'c' is the correct answer.

(This is because V(cA + dB) = c^2 V(A) + d^2 V(B) - refer tutorial on combining variances.)

Choice 'a' is incorrect as it describes the calculation of variance, not volatility. Also, the presence of a large number of uncorrelated assets does not create a portfolio with volatility equal to zero or that of the market. The other choices are therefore incorrect.

Explanation:

Notional principals are not exchanged at the start and the end of an FRA. In fact, if the notional principals were to be exchanged, it would increase credit risk and not decrease it by introducing settlement risk. Therefore Choice 'd' is incorrect.

All other choices correctly describe various aspects of an FRA.

An option on a futures contract is like any other option contract, and only the option premium is due upfront. If the option is exercised, then the futures contract comes into existence and futures margins become due in the normal way. Therefore Choice 'b' is the correct answer.

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

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

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

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

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

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

Caps, floors and collars provide a hedge against interest rate risks, but do not protect against changes in credit spreads unless the reference rate already includes the spread (eg, by reference to the corporate bond rate), and they certainly do not have anything to do with gamma risk. Therefore Choice 'c' is the correct answer.

Choice 'b' represents a correct description of a bullet bond. All other choices are incorrect.

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.)

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 investor would benefit from the risk reversal and the bull spread as both these strategies have a payoff profile that benefit from rising prices of the underlying. The collar is the opposite of risk reversal, and benefits during a bear market, and the butterfly spread benefits when prices remain range bound. Therefore Choice 'c' is the correct answer.

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

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.

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

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

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

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

Choice 'd' represents the mean-variance function to be maximized for selecting between mutually exclusive gambles. The other choices are incorrect.

(The mean-variance criterion is a fairly complex subject, and this question is only intended to make sure that you know, and can identify the function that is being maximized. A complete explanation/derivation of the mean-variance criterion, that links together expected returns, volatility and the risk tolerance of the investor to arrive at the efficient frontier is beyond the scope of the PRM syllabus.)

The change in the price of the bond will be equal to $100* (-11*0.1% + 0.5* 80 *(0.1%)^2) = -1.096. Therefore the new price will be $100 - 1.096 = $98.904. Thus Choice 'd' is the correct answer, the rest are incorrect.

Note that the change in the price of a bond if you are given modified duration is convexity is given by the formula P * (-Duration*δi + 0.5*Convexity*(δi)^2); where δi is the change in interest rates. 10 basis points equal 0.1%, or 0.001.

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.

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

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

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

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

Choice 'd' is the correct answer. Continuously compounded returns are given by the log of S2/S1. The expression (S2 - S1)/S1 gives discrete returns, not continuously compounded returns.

Note that the question is only asking for the return over the given time period, not annualized returns. So t is not relevant here. If we had to find annualized returns, and t were expressed in years, then the annualized return would have been (1/t)*ln(S2/S1).

Also refer to the tutorial on interest rates and compounding for more information on continuously compounded rates.

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.

Choice 'b' lists all the factors that affect the forward price of a physical asset and is the most complete answer. Forward prices for physical assets are not affected by volatility (only options are), nor are they arbitrarily decided by any prevailing 'views'.

The objective of decisions relations to capital structure is to maximize the value to shareholders. The management of the company is entrusted with the job of maximizing shareholder value, and this objective is generally achieved by attempting to maximize enterprise value. Enterprise value includes the value of both debt and equity, and since debt has a fixed claim on the assets of the business, maximizing enterprise value also maximizes the value of equity.

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

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

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

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

Choice 'c' is therefore the correct answer.

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

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

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.

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.

Over the counter and electronic communication networks match buyers and sellers. However, there is no market making function, ie, in periods of stress liquidity may completely disappear from these markets. Exchanges normally have market makers that are required to present two way quotes on the securities they are making the market for. Therefore Choice 'a' is the correct answer.

A covered call is an attempt to earn premium income by selling calls while protecting against the downside by holding long the underlying. In case prices rise, the holder of the covered call can give away the held underlying. In case prices fall or stay below the strike price of the sold call, the holder of the covered call gets to keep the premium.

The holder of the covered call will suffer a loss if the price of the underlying declines due to the long position held in the underlying. However, these losses will be partly offset by the premiums earned on the calls sold.

Choice 'd' does not represent a correct statement as in the event of a rise in the price of the underlying the holder of the covered call will lose any upside, but will not have unlimited losses. In the event of a decline in stock prices, his losses will be equal to the loss on the underlying, minus the premium earned.