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» Q&A: What's the swap value 90 days into its life...

» Q&A: When will an investor receive a margin call...

» Q&A: which order is most likely to be executed sooner...

» Q&A: Which Rho is greater for...

» Q&A: why would investors want to buy ADRs?

» Q&A: Your porfolio has the following three assets...

### Q&A: What's the swap value 90 days into its life...

...given the following 3-mon LIBOR term structure? The first floating payment was set at the 180-day rate of 9%.

Term (days) | Rate |

90 | 9.125% |

270 | 10% |

450 | 10.375% |

630 | 10.625% |

*Answer*

- The new LIBOR terms to used are 6m-90=3m, 9m, 15m, and 21m.

Term (days) Spot LIBOR Rate Discount Factor 90 3m LIBOR = L(90, 90) = 9.125% B(90, 180) = 1 / (1+L(90, 90)*(90/360)) = 1 / (1+9.125%*(180/360)) = 0.9777 270 9m LIBOR = L(90, 270) = 10% B(90, 360) = 1 / (1+L(90, 270)*(270/360)) = 1 / (1+10%*(360/360)) = 0.9302 450 15m LIBOR = L(90, 450) = 10.375% B(90, 540) = 1 / (1+L(90, 450)*(450/360)) = 1 / (1+10.375%*(540/360)) = 0.8852 630 21m LIBOR = L(90, 630) = 10.625% B(90, 720) = 1 / (1+L(90, 630)*(630/360)) = 1 / (1+10.625%*(360/360)) = 0.8432

- PV(fixed) = (180/360) x 0.0975 x (0.09777 + 0.09302 + 0.8852 + 0.8432) + 1 x 0.08432 = 1.02046963.
- PV(floating) = (0.045 + 1) x (0.9777) = 1.0216965
- the first floating payment was set at the 180-day rate of 9%. For a $1 notional principal, the payment would be 0.09 x (180/360) = 0.045
- The value of the floating payments, is based on discounting the next floating payment of 0.045 and the market value of the floating-rate bond on the next payment date (which is $1)

- PV(Swap) = (1.0216965 - 1.0204693) * $20 mil = $24,537.

*Category: C++ Quant > Finance > Derivatives > Swaps*

### Q&A: When will an investor receive a margin call...

...if he holds a long position in 10 oat futures contracts, with a required initial margin of $5 per contract, a maintenance margin of $2, and the following end-of-the-day prices?

Day 0 | Day 1 | Day 2 |

$100 | $101 | $96 |

*Answer*

- initial margin = $5 * 10, Maintenance margin = $2 * 10
- Day 1: Margin balance = $50 + (101-100) * 10
- Day 2: $60 + (96 - 101) x 10 = $10. Since the margin balance is below the maintenance margin, he receives a margin call: must restore his balance up to the initial margin. Variation margin = initial margin - margin balance = 50 - 10 = $40.

*Bonus Points*

- A long futures position suffers losses when futures prices fall.
- Will get the call when 5-2 = 100-x, solve for x = $97.

*Category: C++ Quant > Finance > Derivatives*

### Q&A: which order is most likely to be executed sooner...

...: limit buy, limit sell, market, stop or day?

*Answer* : Market order is executed immediate at the prevailing price.

- Limit order: A condition placed on a transaction executed through a stockbroker to assure that securities will be sold only if a specified minimum price is received, or purchased only if the price to be paid is no more than a given maximum.
- Stop order: A mechanism for locking in gains or limiting losses on securities transactions. The investor is not assured of paying or receiving a particular price but rather agrees to accept the price prevailing when the broker is able to execute the order after prices have
*reached*some predetermined figure - Day order: automatically cancelled at the end of the day

*Bonus Points*

- A stop-limit order is different from a simple stop order in that once the stock price reaches the preset stop price the order is converted into a limit order.
- A short sale is the sale of stock that you do not own with the intent of purchasing it back later at a lower price.
- When an investor buys or sells a security, he must deliver the cash or certificate to the broker within 3 days (settlement).

*Category: C++ Quant > Finance > Financial Markets*

### Q&A: Which Rho is greater for...

...the following options on a stock selling for $65, with 5% risk-free rate?

- Call option A: Exercise price X = $70, Days to option expiration = 15.
- Call option B: Exercise price X = $70, Days to option expiration = 120.

*Answer* : Option B. Rhoc (for call options) and Rhop (for put options) change as time passes, with both tending toward zero as expiration approaches.

*Bonus Points*

- The sensitivity of the option price to the risk-free rate is called the Rho.
- For a call option, Rho is always positive.
- For a put option, Rho is always negative.

- Large changes in the interest rate have relatively little impact on the option prices. (ie, the price of a European option on an asset is not very sensitive to the risk-free rate.)

*Category: C++ Quant > Finance > Derivatives > Options*

### Q&A: why would investors want to buy ADRs?

*Answer*: When a U.S. bank buys foreign stock shares, puts them in a trust and resells trust units, the shares become known as ADR (American Depository Receipts). ADRs allow foreign shares to be traded in the United States much like any other security. ie. reduced administration and duty costs on each transaction that would otherwise be levied.

*Bonus Points*

- ADR represents the ownership interest in a foreign company's common stock. ADRs are denominated in U.S. dollars, with the underlying security held by a U.S. financial institution overseas.
- ADRs do not eliminate the currency/fx and political risks (ie. is the government stable?)
- The process is as follows: The shares of the foreign company are purchased and put in trust in a foreign branch of a New York bank. The bank, in turn, receives and can issue depository receipts to the American stockholders of the foreign firm.

*Category: C++ Quant > Finance > Financial Markets*

### Q&A: Your porfolio has the following three assets...

Asset | X | Y | Z |

Expected returns | 5% | 10% | 15% |

standard deviations | 10% | 20% | 35% |

You views any porfolio return below a level of 3% as unacceptable. Find the asset that minimizes the probability that the portfolio will fall below 3% annual return, and what is the probability?

*Answer* : make use of Roy's Safety-First criterion.

- Compute the SFRatio for each asset: SFRatio = (R(e)-R(l)) / stderr
- The probability of shortfall will be N(-SFRatio) = 1 - N(SFRatio). Or in Excel, NORMSDIST(-SFRatio)

Asset X Y Z SFRation (5%-3%)/10% = 0.2 (10% - 3%)/20% = 0.35 (15% - 3%)/35% = 0.34 N(-SFRation) NORMSDIST(-0.2)=0.4207 NORMSDIST(-0.35)=0.3632 NORMSDIST(-0.35)=0.3669

- Y<Z<X: Thus, asset Y minimizes the probability that the return will fall short of 3%, with a probability of approximately 36%.

*Bonus Points*

- Safety-first rules are concerned with shortfall risk. According to Roy's safety-first criterion, an optimal portfolio minimizes the probability that the porfolio return (
**NOT**the returns on all the securities in the portfolio) fall below the threshold level. - If returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio).
- SFRatio gives the distance (from the mean return to the shortfall level) in units of standard deviation: it measures the excess return over the threshold level per unit of risk.
- The SFRatio becomes the Sharpe ratio if we substitute the risk free rate for the threshold return R(l): The safety-ratio focuses on the excess return over the threshold return, while the Sharpe ratio focuses on the excess return over the risk-free rate.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability*