PRMIA 8006 Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition Exam Practice Test

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

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

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).

An out of the money convertible security behaves just like a regular debt security as the option is near worthless. Therefore it is affected by changes in interest rates and the credit risk of the issuer, and unaffected by changes in the prices or volatility of the issuer's shares (the underlying).

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.

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.

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 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.

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

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.

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)

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).

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.

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.

We can calculate the duration of the bond as follows:

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

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

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

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

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

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

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.

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.

Since the zero coupon bond has no interim interest payments, its only cash flow is the final payment upon maturity. This would be identical to the spot rate from now till t years, where t is the maturity of the bond.

Forward rates are marginal rates that apply to individual years in a multi-period context. Short rates refer to short term interest rates in money market futures contracts. A zero coupon bond has no 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-ρ2), 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.

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.

In a Dutch auction, the buyers put in their bids and the price at which the entire auction clears is the price at which each bidder receives the security. For example, if Bank A bids 5.5% for a bond, Bank B bids 5.6% and Bank C bids 5.7% and the total amounts bid by these three banks are enough to clear the auction, each of the banks will receive the bond at the price paid by the lowest successful bidder, ie Bank C. This is even though Banks A and B had bid higher (remember that bidding a higher yield means offering a lower price).

In a standard auction, every bidder pays the price they bid.

In a Dutch auction, there is an incentive to bid aggressively if a participant believes the final winning bid will be lower.

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.

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. Therefore statement IV is not true, while the rest are.

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

The 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.

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.)

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

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.

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 dates on which the interest rates applicable to the floating rate leg of an IRS are determined are called 'reset dates'. Choice 'c' is the correct answer, the other choices are incorrect. Settlement date is the date a transaction settles, normally a few days after the trade is done depending upon the market. Trade date is the date the terms of the transaction are agreed. (Settlement normally happens on T+n, where T is the trade date, and n the number of days to settlement as decided by agreement or market convention.)

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

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.

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).

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.

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

The other choices are incorrect.

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

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

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

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

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

Given covered interest parity, the currency with a lower interest rate will trade at a forward premium. Choice 'b' is the correct answer.

For an intuitive reasoning, consider a currency forward contract that matures in 3 months. The seller has agreed to sell, say JPY 1,000,000 in exchange for USD 10,000 in the future. In order to cover himself, he borrows the USD right now and converts it to JPY at spot which he puts in a JPY deposit. Assuming JPY interest rates are less than USD interest rates, he pays more on his USD borrowing than he receives on his JPY deposit. Therefore he has to price the forward contract at a premium to spot to cover the interest rate differential.

A fixed for floating interest rate swap is identical to a series of forward rate agreements coinciding with the settlement periods. For example, a 2 year fixed for floating interest rate swap where the investor pays fixed at 6 monthly rests is equivalent to a long position in 3 FRAs ( a 6 x12, a 12 x 18 and an 18 x 24 FRA). Thus I represents a correct replication of the swap.

An obligation to pay fixed is identical to a short position in a fixed rate bond, and receiving the floating rate is akin to a long position in a floating rate note. Thus II also correctly replicates the swap.

A long position in an FRN correctly replicates the floating cash flow from the swap, but a short position in the FRA obliges the investor to receive fixed and pay floating - which is not equivalent to the flows from the interest rate swap. Thus III is not a correct replication of the interest.

A long position in a cap means the investor will receive the difference between the floating and the cap rate if the floating rate is greater than the cap rate. Effectively, the investor will receive floating. A short postion in a floor at the same strike rate would mean the investor would end up paying the floor rate, effectively equivalent to the obligation to pay a fixed rate. Therefore this replication also correctly replicates the swap, and IV can also be used to replicate the investor's position.

Therefore Choice 'a' is the correct answer and the rest are incorrect.

(For statement IV, let me explain with an example. Recall that the buyer of a cap receives the difference between the current interest rate and the agreed exercise rate if the current interest rate goes above the exercise rate. A cap can be used by a floating rate borrower to limit their exposure to a rise in interest rates. In the same way, the seller of a floor pays the buyer the difference between the current interest rate and the exercise rate if the current interest rate falls below the exercise rate. A floor can be used by a lender to protect against a fall in interest rates.

Now assume the exercise rate is r. Also assume that the current floating interest rate is r'. the holder of the interest rate swap in the example in the question receives r and pays r'. Therefore the net payout is r' - r (pays fixed, ie -r, and receives floating, ie +r).

Now consider the long position in the cap and the short position in the floor.

Case 1: If r > r', the following will happen:

a. The long position in the cap will pay nothing, ie 0.

b. The short position in the floor will require a payment of r - r' to the buyer of the floor, and because it is a payment, the cash flow will be -(r - r') = r' - r

Case 2: If r < r', the following will happen:

a. The long position in the cap will mean the investor receives r' - r.

b. The short position in the floor will pay nothing as the interest rate is higher than the exercise rate.

In both cases, the net cash flow for the investor is r' - r, which is the same as for the swap where the investor pays fixed and receives floating.)

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.

Explanation:

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 forward price for a commodity is nothing but the spot price plus carrying costs till the maturity date of the forward contract. Any increase in carrying costs therefore has the effect of increasing the forward price. Note that carrying costs include interest cost in respect of funding the position, costs of storage, less any convenience yield.

Increase in the carrying costs will not affect the spot prices.

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

Explanation:

If the spot rate is 1.1010 and the forward is 1.1040, the swap rate is 30 "points" (= 1.1040 - 1.1010). The annualized swap premium is the swap premium divided by the spot rate, and therefore =0.0030/1.1010 =0.27%, or 27 basis points.

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.