Kostenlos und ohne Konto. Zu jeder Frage gibt es den vollständigen Lösungsweg, einschließlich der Begründung, warum jede falsche Option falsch ist. Die Fragen sind eigene Aufgaben von Preparaio auf Englisch, keine offiziellen Fragen des CFA Institute.
1. Ioana Vrânceanu, a rates analyst at Bucegi Bank, is pricing a new three-year plain-vanilla interest rate swap with annual settlement. Annual-compounding spot rates are 3.15% for one year, 3.70% for two years and 4.05% for three years. The fixed swap rate is closest to:
A. 3.63% B. 3.74% C. 4.03%
Lösung und Erklärung Richtige Antwort: C 4.03%
The discount factors are 0.969462, 0.929913 and 0.887715, which sum to 2.787091. A fixed-rate bond paying the swap rate must be worth par, so the swap rate is (1 − 0.887715)/2.787091 = 4.03%. It lies just below the three-year spot rate because the earlier, lower spot rates apply to the earlier coupons.
Rechenweg
DF1 = 1/1.0315 = 0.969462; DF2 = 1/1.037^2 = 0.929913; DF3 = 1/1.0405^3 = 0.887715 Sum of DFs = 0.969462 + 0.929913 + 0.887715 = 2.787091 Swap rate = (1 − 0.887715) / 2.787091 = 0.112285 / 2.787091 Swap rate = 4.03% BA-II-Plus-Tastenfolge
1.0315 1/x STO 1; 1.037 x² 1/x STO 2; 1.0405 y^x 3 = 1/x STO 3; 1 − RCL 3 = ÷ ( RCL 1 + RCL 2 + RCL 3 ) = × 100 = [4.03]
Warum die anderen Optionen falsch sind
A: 3.63% is the simple average of the three spot rates, which ignores how much each payment is worth today. B: 3.74% divides (1 − DF3) by the number of periods, 3, rather than by the sum of the discount factors. Intermediate displayed workings may be rounded; retain full precision until the final answer.
Kernregel: The fixed swap rate is the par rate that gives the swap zero value at initiation: s = (1 − DF3)/(DF1 + DF2 + DF3).
Merksatz: Swap rate = (1 − last DF) / sum of DFs: the par rate of the spot curve.
2. Ottavia Brambilla wants the payoff of a long one-year European call option on shares of Valdorcia Pharma, which pay no dividends, with an exercise price of 80.00. Such calls trade rarely, but one-year European puts with an exercise price of 80.00 are liquid, and Brambilla can borrow and lend at the risk-free rate. According to put-call parity, which position most likely replicates one call?
A. Buy one share, buy one put and borrow the present value of 80.00 B. Buy one share, sell one put and lend the present value of 80.00 C. Buy one put and lend the present value of 80.00
Lösung und Erklärung Richtige Antwort: A Buy one share, buy one put and borrow the present value of 80.00
A share plus a put (a protective put) pays max(S_T, 80.00) at expiration. Borrowing the present value of 80.00 today means repaying 80.00 at expiration, which leaves max(S_T, 80.00) − 80.00 = max(0, S_T − 80.00), exactly the payoff of the call. Brambilla can therefore build the call from the share, the liquid put and a risk-free loan.
Warum die anderen Optionen falsch sind
B: Selling the put and lending reverses two of the signs; at expiration this position pays S_T − max(0, 80.00 − S_T) + 80.00, which does not match the call's payoff. C: A put plus a risk-free bond pays 80.00 + max(0, 80.00 − S_T), which is largest when the share falls, the opposite of a call; it is the fiduciary call that pairs the bond with a call, not a put. Kernregel: From put-call parity, c0 = S0 + p0 − X/(1 + r)^T: a synthetic call is a protective put financed partly by risk-free borrowing.
Merksatz: Synthetic call = long share + long put − risk-free bond paying X.
3. Efua Ansong, a cocoa exporter, and Mabrouk Confectionery agree today that in six months Ansong will deliver 200 tonnes of cocoa beans to Mabrouk at a price per tonne fixed now. Neither party pays anything at the outset. Which description of this contract is accurate?
A. It is a derivative: a forward commitment on a commodity that lets each party lock in its future price. B. It is a contingent claim, because Ansong must deliver only if the spot price then is below the agreed price. C. It is not a derivative, because it is settled by physical delivery of the cocoa beans rather than in cash.
Lösung und Erklärung Richtige Antwort: A It is a derivative: a forward commitment on a commodity that lets each party lock in its future price.
The contract's value to each side will depend on the price of cocoa, its underlying, so it is a derivative. Both parties are obliged to trade in six months at the agreed price and nothing is paid up front, which makes it a forward commitment. It serves a central purpose of derivative markets, transferring price risk: Ansong locks in her selling price and Mabrouk its purchase cost.
Warum die anderen Optionen falsch sind
B: Neither side has a choice: Ansong must deliver and Mabrouk must pay whatever the spot price is in six months, so this is a forward commitment, not a contingent claim such as an option. C: A contract is a derivative because its value derives from an underlying, not because of how it settles; many forwards and futures are settled by physical delivery. Kernregel: A forward contract is a derivative that obliges both parties to transact at a later date at terms set today; its value depends on the underlying, here cocoa.
Merksatz: Obligation on both sides at a price set today = forward commitment, whether settled physically or in cash.
4. Muthoni Kariuki builds a one-period binomial model for shares of Nyandarua Potato Growers and finds that the risk-neutral probability of an up move is 0.55. Equity analysts estimate the actual probability of an up move at 0.68. The difference between the two probabilities most likely reflects:
A. an arbitrage opportunity that Kariuki can exploit by trading the option B. the risk premium that investors require for holding the shares C. an error in Kariuki's model, since q should equal the actual probability
Lösung und Erklärung Richtige Antwort: B the risk premium that investors require for holding the shares
With the same up and down moves, q = 0.55 gives the shares an expected return equal to the risk-free rate. Investors in risky shares expect more than that, and a higher actual probability of an up move, 0.68, is what delivers the extra return. The higher actual up probability produces an expected return above the risk-free rate and therefore reflects the required risk premium. For a fixed up/down tree and risk-free rate, the option's no-arbitrage value uses q rather than the actual probability.
Warum die anderen Optionen falsch sind
A: The option is priced by replication, which needs only u, d and the risk-free rate; a difference between q and the actual probability creates no mispricing to exploit. C: q is not meant to equal the actual probability; it would match only if investors were risk-neutral and the shares were expected to earn just the risk-free rate. Kernregel: The risk-neutral probability q is the probability that makes the share's expected return equal the risk-free rate; it is a pricing device, not a forecast.
Merksatz: Real-world probabilities embed a risk premium; option pricing uses q and ignores them.
5. Mutale Copper Mines produces copper of a grade that differs slightly from the grade deliverable under exchange-traded copper futures. To hedge next year's output, it sells these futures, which are cleared through a central counterparty and margined daily. Which risk is Mutale's hedge least likely to expose it to?
A. Basis risk, because Mutale's copper and the futures' grade may not move together B. Liquidity risk, because rising copper prices would trigger variation margin calls C. Significant counterparty credit risk, because the other side could default
Lösung und Erklärung Richtige Antwort: C Significant counterparty credit risk, because the other side could default
Once Mutale's trades are cleared, the central counterparty becomes the buyer to Mutale as seller, and daily variation margin settles gains and losses before they can build up. Even if the trader who originally bought the futures defaults, the clearinghouse still stands behind the contract. Counterparty credit risk is therefore minimal compared with an uncleared over-the-counter forward.
Warum die anderen Optionen falsch sind
A: Mutale does face basis risk: because its copper grade differs from the contract grade, the value of its output and the futures price can diverge, so the hedge may not offset its losses exactly. B: Mutale does face liquidity risk: as a short futures holder it must pay variation margin in cash when copper prices rise, even though its unsold copper gains value at the same time. Kernregel: Centrally cleared, daily-margined futures carry little counterparty credit risk but can still create basis risk and liquidity risk.
Merksatz: A cleared futures hedge removes most counterparty risk, not basis risk or margin-call liquidity risk.
6. A bullion dealer at Hesperon Metals quotes the same one-year gold forward price to a highly risk-averse jewelry maker and to a risk-tolerant hedge fund. Storage costs are negligible. Why do the two clients' attitudes toward risk not affect the quoted forward price?
A. The dealer can hedge by buying gold with risk-free borrowing, so the price reflects that hedge's cost. B. The forward price equals the market's forecast of the gold price in one year, which both clients share. C. Gold earns no return above the risk-free rate, so no risk premium could ever enter its forward price.
Lösung und Erklärung Richtige Antwort: A The dealer can hedge by buying gold with risk-free borrowing, so the price reflects that hedge's cost.
If the dealer sells gold forward, it can borrow at the risk-free rate, buy gold today and deliver it at expiration. That hedged position is riskless, so the forward price must equal the spot price grown at the risk-free rate; any other quote would open an arbitrage. Since the price comes from the cost of this hedge, it is the same for every client, whatever its attitude toward risk.
Warum die anderen Optionen falsch sind
B: A no-arbitrage forward price is not a forecast of the future spot price; it is the spot price carried forward at the risk-free rate, whatever investors expect gold to be worth. C: The result does not depend on gold having no risk premium; forwards on risky shares with large risk premiums are priced the same way, because the hedge removes risk regardless of the asset's expected return. Kernregel: Because a derivative can be hedged with the underlying and risk-free borrowing or lending, its no-arbitrage price depends on the cost of that hedge, not on anyone's risk preferences.
Merksatz: Hedgeable means risk-neutral pricing: derivative prices come from replication costs, not from investors' risk attitudes.
7. Kahurangi Dairy Cooperative hedges the price of its milk powder exports with exchange-traded futures cleared through a central counterparty, instead of an uncleared over-the-counter forward with a trading house. Compared with the forward, the futures position is least likely to:
A. be tailored to the exact quantity and delivery date of Kahurangi's shipments B. require Kahurangi to post initial margin and settle gains and losses daily C. reduce Kahurangi's exposure to a default by the party on the other side of its trade
Lösung und Erklärung Richtige Antwort: A be tailored to the exact quantity and delivery date of Kahurangi's shipments
Futures are standardized: the exchange sets the contract size, quality and delivery months, so Kahurangi must use the nearest standard terms rather than its exact shipment amounts and dates. Customization is a feature of the over-the-counter forward it chose not to use, which is why the futures position is least likely to be tailored.
Warum die anderen Optionen falsch sind
B: Futures do require this: the clearinghouse collects initial margin and settles gains and losses through variation margin every day. C: Futures do provide this: the central counterparty becomes the counterparty to both sides and, with daily margining, sharply reduces default risk compared with an uncleared forward. Kernregel: Exchange-traded derivatives have standardized terms, daily margining and central clearing, whereas OTC derivatives can be customized but, if uncleared, carry bilateral counterparty risk.
Merksatz: Futures trade flexibility for safety: standardized terms in exchange for clearing and daily margin.
8. Sina Aumua, treasurer of Falealupo Freight, enters a two-year interest rate swap with semiannual settlement in which Falealupo receives a fixed rate and pays MRR. Viewed as a portfolio of forward rate agreements (FRAs), Falealupo's swap is most likely equivalent to:
A. a series of long FRAs, one for each settlement date B. a series of short FRAs, one for each settlement date C. a single long FRA covering the full two years
Lösung und Erklärung Richtige Antwort: B a series of short FRAs, one for each settlement date
On each of the four semiannual settlement dates, Falealupo receives the fixed swap rate and pays the floating MRR on the same notional, with the payments netted. Each of these exchanges is exactly the cash flow of a short FRA, which receives the fixed rate and pays the reference rate. The whole swap is therefore a portfolio of short FRAs, each struck at the swap's fixed rate.
Warum die anderen Optionen falsch sind
A: Long FRAs pay fixed and receive floating, which would replicate a pay-fixed swap, the opposite of Falealupo's position. C: A swap settles every six months, so it needs one FRA per settlement period; one FRA covering two years would settle only once, and as a long it would also take the wrong side. Kernregel: An interest rate swap is a series of FRAs, one per settlement, each at the swap's fixed rate; a long FRA pays fixed and receives floating, so a receive-fixed swap is a series of short FRAs.
Merksatz: Receive-fixed swap = series of short FRAs; pay-fixed swap = series of long FRAs.
9. Shares of Valtrebbia Optics, which pay no dividends, trade at 58.70. A nine-month (0.75-year) European call option on the shares with an exercise price of 60.00 is priced at 4.35. The risk-free rate is 3.5% per year with annual compounding. Using put-call parity, the value of a European put on the shares with the same exercise price and expiration is closest to:
A. 4.12 B. 4.58 C. 5.65
Lösung und Erklärung Richtige Antwort: A 4.12
The present value of the 60.00 exercise price is 60.00/1.035^0.75 = 58.47. Rearranging parity, the put equals the call price minus the share price plus that present value: 4.35 − 58.70 + 58.47 = 4.12. At this price, a protective put and a fiduciary call cost the same, so no arbitrage is possible.
Rechenweg
PV(X) = 60.00 / 1.035^0.75 = 60.00 / 1.026137 = 58.47 p0 = c0 − S0 + PV(X) = 4.35 − 58.70 + 58.47 p0 = 4.12 BA-II-Plus-Tastenfolge
60 ÷ ( 1.035 y^x 0.75 ) + 4.35 − 58.7 = [4.12]
Warum die anderen Optionen falsch sind
B: 4.58 rearranges parity with the wrong signs, adding the share price and subtracting the present value of the exercise price (c0 + S0 − PV(X)). C: 5.65 uses the full exercise price of 60.00 instead of its present value; the exercise price is paid only at expiration. Intermediate displayed workings may be rounded; retain full precision until the final answer.
Kernregel: Put-call parity: S0 + p0 = c0 + X/(1 + r)^T, so p0 = c0 − S0 + X/(1 + r)^T.
Merksatz: Isolate the put as c0 − S0 + PV(X), and always discount the exercise price.
10. Faizan Rabbani buys a European put option on shares of Gulmitar Apricot Foods with an exercise price of 54.00, paying a premium of 2.85 per share. At expiration, the share price is 49.30. What is Rabbani's profit per share?
A. −1.85 B. 1.85 C. 4.70
Lösung und Erklärung Richtige Antwort: B 1.85
With the share at 49.30 at expiration, the put lets Rabbani sell for 54.00 a share worth 49.30, a payoff of 4.70. He paid 2.85 for the put, so his profit is 4.70 − 2.85 = 1.85 per share. His breakeven price was 54.00 − 2.85 = 51.15, and the share finished below it.
Rechenweg
Put payoff = max(0, 54.00 − 49.30) = 4.70 Profit = 4.70 − 2.85 = 1.85 BA-II-Plus-Tastenfolge
54 − 49.3 = − 2.85 = [1.85]
Warum die anderen Optionen falsch sind
A: −1.85 is the profit of the put writer, who received the premium but must buy the share at 54.00. C: 4.70 is the put's payoff at expiration; it ignores the 2.85 premium Rabbani paid. Intermediate displayed workings may be rounded; retain full precision until the final answer.
Kernregel: A put buyer's profit is the payoff, max(0, X − S_T), minus the premium paid.
Merksatz: Profit for an option buyer = payoff − premium; a long put profits only below X − premium.
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