8006 PRMIA Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition Free Practice Exam Questions (2025 Updated)

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 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.

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).

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

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

The lowest achievable standard deviation will be lowest when the assets are perfectly negatively correlated, ie when correlation = -1. In the case of the above choices, the lowest correlation is -0.33, which is the correct choice.

A rise in interest rates will cause fixed rate bond prices to fall. It will not affect the floating rate notes as their rates are going to float with the changes in interest rates. Inflation linked bonds will only be affected by changes in real interest rates, and even in a situation where the rise in interest rates is happening due to an increase in the real rates, a long position would hurt. As interest rates rise, the coupon payable on the inverse floater would fall, hurting a long position.

Of the given alternatives, the only position that would benefit from a rise in interest rates is the short position in fixed rate bonds. Hence Choice 'd' is the correct answer.

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).

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.

A quanto is any instrument in which one asset is a foreign currency. Examples include any option on a foreign currency asset with the strike price in foreign currency, and an option on a foreign currency asset with the foreign currency risk hedged.

Correlation products refer to credit products on a basket of credits. Options on options represent an option to buy or sell an option in the future, and they are not 'quantos'. Similarly, not every two asset instrument is a quanto unless one of the two assets is a foreign currency.

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

T-bill futures 'discount' can be converted to a price for the bill using the formula Price = [1 - discount * number of days/360]. In this case, this works out to (1- 10% *90/360) * 100 = $97.50. Choice 'd' is the correct answer.

Explanation:

The correct answer is the index value times the contract size, in this case 1500 x 50.

One way to think about index futures is this: Consider equity index trading as trading in the shares of a company whose share price is equal to a number of dollars which is the same as the index. If the 'contract multiplier' for a index futures contract is 50, that means the futures contract is for 50 shares of such a fictitious company. Therefore the notional value of the contract will be 15000 x 50, and Choice 'a' is the correct answer.

We know from the put-call parity that:

Call - Put = Stock - Deposit

In the given situation,

Stock price = 100

call = 35

Put = ?

Exercise price = 100

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

Therefore:

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

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

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

35 - Put = $99.05 - $95.24

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

The buyer of the FRA gets to borrow $100m at 10% per annum for 6 months at the end of 6 months from the contract date. Thus, the interest due is $100m * 10% * 6/12 = $5m. However, at the end of the 6 months (when the notional borrowing period begins), the spot rate is 12%. The interest due on a borrowing at the spot rate would be $100m * 12% * 6/12 = $6m. Since the buyer gets to borrow cheaper than the going rate, he or she has made a gain of $1m on the FRA. However, this amount is not due immediately, it is due at the end of the 6 month borrowing period (ie, 12 months from the date the contract was entered into). The seller of the FRA can make the buyer whole by paying the buyer the present value of $1m due in 6 months time, which is $1m/(1 + 12%*6/12) = $1m/1.06 = $943,396.

Thus, Choice 'd' is the correct answer. Remember that the FRA gets settled at the beginning of the notional borrowing period as all future cash flows, and the applicable discount rates are known with certainty. The buyer and the seller do not need to wait for the entire period to get over first. The cash settlement allows the buyer to borrow at the market rate and still have the same net borrowing cost as he or she had initially contracted for in the FRA.

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.

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.

The stock has a beta of 1.2, therefore intuitively it can be expected to earn more than the broad market index. It will earn the risk free rate, ie 3%, and 1.2 times the equity risk premium of 5% (8% - 3%). The expected returns from the stock therefore are 3% + (8% - 3%)*1.2 = 9%

Remember the four basic axioms underlying the principal of maximum expected utility:

- Transitivity, ie if A is preferred over B, and B is preferred over C, then A must be preferred over C;

- Continuity, ie if A is preferred over B, and B is preferred over C, then B is on a continuum between A and C such that we can be indifferent between receiving B, or a lottery offering either A or C with probabilities p & 1-p respectively.

- Independence, ie choices are not affected by the way they are presented

- Stochastic dominance, ie a gamble that offers a greater probability of a preferred out come will be preferred.

In this case, the first axiom is being violated. Therefore Choice 'c' is the correct answer.

Convexity is nothing but the second derivative with respect to price - or in other words, it is the first derivative of the modified duration. If we could determine two data points for modified duration, we can determine convexity too.

The modified duration is the percentage change in bond price from a 1% change in yield.

In this case, for the up move, the modified duration can be calculated as follows:

Change in price for 10 bps change in yield = $0.60/$95 =0.006316

Change in price for a 1% change in yield = 0.006316 * 10 = 0.06316

Therefore modified duration = 0.06316 * 100 = 6.316

For the down move in price, the modified duration is similarly $0.50/$95 * 10 * 100 = 5.263.

The convexity therefore is the difference between these two modified durations divided by the change in interest rates - note that these two points are separated by 10 basis points and not 20. Therefore convexity is (6.32 - 5.26)/0.1% = 1052.6.

[Note on how duration was calculated: In this case, we can estimate the duration of the bond as follows: we know that a 10 bps increase in rates causes the price to move to $94.50, and a 10 bps decrease causes the price to increase to $95.60. Thus, over the range of the 20 bps, the average change in price per basis point is ($95.60 - $94.50)/20 bps = $1.10/20 = $0.055/basis point, or $0.055* 100 = $5.5 for 100 basis points (ie 1%). We know that modified duration is equivalent to the percentage change in the bond price as a result of a 1% change in interest rates. A 1% change in the interest rates leading to a $5.5 change in a bond priced at $95 equates to $5.5/$95 = 5.79%, in other words the modified duration is roughly equal to 5.79 years.

In fact if we know the price of a bond at any two different interest rates, we can make an estimate of modified duration. Modified duration is just the first derivative with respect to price, and given two prices and the associated yields, we can easily calculate modified duration to be the ratio of the change in price to the change in interest rates. In this question, we are given both an up move and a down move. Using this estimation, only one data point (ie, either the up price or the down price) in addition to the starting point ($95) would have been enough to come to a rough estimate of modified duration. You will notice that the modified duration would be slightly different if we were to use the high point and the starting point (ie $95.60 and $95), and the starting point and the lower point ($95 and $94.50). The difference is due to convexity. The decrease in price is lower than the increase in price - and this is due to the convexity of the bond.]