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

The transformation line represents the combination of a 'risky bundle' and the risk free asset. Investors can choose different combinations of these assets depending upon their risk appetite. The transformation line meets the y-axis (portfolio returns) at the point equal to the risk free rate. Choice 'b' is the correct answer and the rest are incorrect.The highest possible transformation line, ie the transformation line with the maximum slope, is the transformation line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the 'capital markets line'. Investors can pick any point on this line according to their risk appetite, and doing so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML.

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).

Credit derivatives can indeed be used for reducing credit exposures, earning credit risk premiums and getting market exposure without taking cash market positions. They cannot be used for reducing interest rate risks, as they pay out only when agreed 'credit events' take place. Changes in interest rates are not a credit event. Therefore Choice 'b' is the correct answer.

There was no event such as the CDO squeeze in 2008. (Quite on the contrary, securitized products were selling at distressed prices.).

The silver squeeze of 1979-80 (Hunt brothers), the Chicago fire of 1872 (leading to a short squeeze on wheat), and the wheat squeeze (Hutchingson) of 1866 are real historical events that led to short squeezes in commodity markets. Choice 'b' is therefore the correct answer.

For the PRM exam, you should try to remember the event broadly, and the commodity involved.

The price of an interest rate cap is affected by all of the listed choices except the risk free rate. The risk free rate does not come into play in the pricing of caps, and therefore Choice 'b' is the correct answer.

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.

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.

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.

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.

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

The put-call parity can be expressed as:

Call – Put = Spot – PV of exercise price

Note that a negative sign above means a short position. The 'term PV of exercise price' is the same as a bank deposit placed today equivalent to the PV of the exercise price so that we will have the cash flow on the exercise date to exercise the option.

Therefore only Choice 'd' is the correct answer as rearranging the above gives us Buying a call + Bank Deposit = Buying the stock + Buying a put.

Choice 'd' is therefore the correct answer.

The transformation line represents the various possible combinations of the 'risky bundle' and the risk free asset. Investors can choose different combinations of the risky and risk-free asset depending upon their risk appetite. The transformation line meets the y-axis (portfolio returns) at the point equal to the risk free rate. Choice 'a' is the correct answer and the rest are incorrect.

The highest possible transformation line, ie the transformation line with the maximum slope, is the transformation line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the 'capital markets line'. Investors can pick any point on this line according to their risk appetite, and doing so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML.

Remember that the transformation line and the capital markets lines are different from each other - there is only one CML for an efficient frontier, whereas there can be multiple transformation lines.

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.

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.

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

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

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

An increase in duration increases convexity.

Therefore statements II and IV represent situations where convexity increases.

In terms of beta, his exposure is $10m*0.7 = $7m. This exposure is long.  In order order to neutralize his long exposure, he needs to have an equal an identical short position with the same beta as this long position (of course, in the short direction).  We need to figure out how many contracts will have a beta equal to his held position.  (The beta of a futures contract is slightly different from 1 when compared to spot, but in the absence of other information in the question it is always okay to assume that the beta of the futures contract is 1.  Such precision does not matter because of other errors such as rounding etc that cannot be anyway done away with.)

He needs to short futures contracts on the index with $7m in notional value. The value of each contract is currently 1000*250 =$250,000. He therefore needs to short $7m/$250,000 = 28 contracts to become market or beta neutral.

The put-call parity applies to European options and does not hold for American options because the latter can be exercised at any time up to expiry. Therefore statement I is false.

An American put can not be cheaper than a European put is correct. This is because the American put will always be valued at at least its intrinsic value whereas a European value may be valued at less than intrinsic value if the exercise date is too far away in the future. This is because a long time interval will increase the chance that the intrinsic value may not be realized at expiry. Because the American put can be exercised anytime, it will always be at least equal to its intrinsic value because if it were to be cheaper than that, investors would immediately buy and exercise it, creating a riskless profit. Therefore statement II is correct, and statement IV is correct as well.

It may be optimal to exercise an American put (unlike an American call - note the difference!) before its expiry on a non dividend paying stock. This would be the case only when the put is deep in the money. If the option is really deep in the money (say, when the underlying's price is down to zero), then it may be worthwhile to cash the option out prior to expiry as the upside might be lost if time were to be allowed to pass. Additionally, any further upside in the case of a deep in the money option may be zero or very little, and the time value of money would also make immediate exercise prefereable. Therefore statement III is not correct.

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

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

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

Tailing the hedge refers to adjusting the number of futures contracts required to hedge a position given that futures contracts result in daily cash flows from daily P&L. In the case of forward contracts, P&L is not settled daily but at the end of the contract. Because margin payments (ie the daily P&L) for futures contracts create interest cost or income (depending upon the direction of the flows), the number of contracts required to hedge a given position can fluctuate. This effect is generally small for small positions, or where the time period under consideration is not very long. But it can get large for larger positions that are open for longer time periods, therefore requiring an adjustment to 'tail the hedge'. The adjustment is straightforward - the number of contracts required needs to be divided by (1+r)t, where r is the interest and t the time period (or by ert if continuously compounded rates are being used. Therefore statement I is true and statement II is false.

Statement III is not correct as counterparty risk is certainly an issue for forward contracts, while futures contracts are guaranteed by the exchange, besides having margin requirements and daily P&L movement in place.

Statement IV is not correct as futures come in standardized contracts with standard amounts and a few standard expiry dates to pick, whereas forward contracts are customized and dates, amounts and other details of the contract can be negotiated.

Average rate options will be the most cost effective in this scenario as they are cheaper than vanilla options. Writer extendible options on commodities will be even more expensive, and correlation products are irrelevant to the manufacturing company's hedging needs.

The portfolio's beta is 0.8, and therefore in order to completely hedge the portfolio (ie reduce beta to 0), the portfolio manager would need to short 0.8 * $25m/(1450*$250) = 55.17 contracts, or 55 contracts. However, the ask here is to reduce the beta to 0.6, and not 0.

The number of contracts required to reduce the beta of a portfolio from ß to ß± is give by (ß - ß±) * Value of portfolio / Value of a single contract. In this case, this calculation works out to (0.8 - 0.6) * $25m/(1450*250) = 13.8, or roughly 14 contracts.

The portfolio manager should short 14 index futures contracts to reduce the total portfolio beta to 0.6.

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.

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

The futures price of the contract will be the future value of the spot price, calculated at a net rate equal to the cost of funding the futures position, less any dividends or other distributions. Also note that when rates are continuously compounded, Future Value = Present Value x (exp(rate x time)).

Therefore in this case the futures price for the S&P = 1000 * exp((5%-2%)*3/12) = 1007.53

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').

Explanation:

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).

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.

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.)

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).

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.

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.

Imagine a bond that provides just two cash flows: $100 in one year and $10 in 10 years. Imagine another bond that pays nothing now but $100 in 10 years. Assume that both have identical yields to maturity. Yet despite the identical YTM, the bonds are quite dissimilar. For the first bond, we face the risk that we would receive the bulk of the present value of the bond in an year's time, and while there is a small amount due in 10 years, the first payment may not find an equally attractive investment opportunity. Reinvestment risk refers to the risk that cash flows from a bond may not be investible at the yield to maturity for the bond. In such cases, bonds that have longer durations are preferable.

A coupon bearing bond with a coupon rate lower than our benchmark bond will carry a lower reinvestment risk as its cash flows are weighted more towards the end. An amortizing bond returns a part of the principle periodically, increasing reinvestment risk. A higher coupon bearing bond will have a higher reinvestment risk, and a zero coupon bond will have the least reinvestment risk.

The OTC market for foreign exchange is indeed much larger than the exchange traded futures market for FX. Therefore statement I is correct.

Delivery-versus-payment (DVP) arrangements make sure that title to a security passes only when payment has been made, and these arrangements, usually implemented through national clearing agencies such as the DTC in the US, help avoid the risk of counterparty defaults.

Electronic Clearing Networks (ECNs) offer cheaper trading costs than exchanges, in fact that is their primary attraction. Exchanges offer other advantages, but lower trading costs is not one of them.

ISDA master agreements provide templates that a large number of OTC market participants use to standardize their OTC trading activities. Therefore statement IV is correct.

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.

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.

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.

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.

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 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.

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.