1.01.2005

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» Q&A: What is the yield to maturity for...
» Q&A: List at least 2 difficulties that specialists have with block trades
» Q&A: List 3 common usage of market indexes.
» Q&A: At what price will an investor get a margin call?
» Q&A: A researcher wishes to estimate the mean SAT score and...
» Q&A: Suppose there is a price-weighted market index...
» Q&A: A researcher wishes to estimate the mean reading speed of...
» Q&A: In an equal-weighted market index...
» Q&A: What's the appropriate alternative hypothesis if...
» Q&A: List some of the market index differences...

Q&A: What is the yield to maturity for...

...a corporate bond with a face value of $1,000 matures in 3 years, has a 9% coupon paid at the end of each year, and a market price of $950?

Answer : With Excel Rate function, FV=$1000, Nper=3, PMT=$1000x9%, PV=-$950, Solve for rate: 11.04% .

Category: C++ Quant > Finance > Debt > Valuation

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Q&A: List at least 2 difficulties that specialists have with block trades

Answer : the "three Cs" with block trading (block transactions of over 10,000 shares)
  • Capital: no enough capital needed to acquire blocks of 10,000 shares
  • Commitment: unwilling to commit the capital because of large risks involved
  • Contacts: prohibited from soliciting interest in shares from other institutional traders due to Rule 113

Category: C++ Quant > Finance > Financial Markets

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Q&A: List 3 common usage of market indexes.

Answer
  • As benchmarks for portfolio performance (on a risk-adjusted basis )
  • To create index funds (i.e to track the performance of the specific market series over time.)
  • To help "technicians" predict future market movement. ie. as inputs for technical analysis.
  • To examine factors that affect aggregate market movements.
  • As a proxy for the market portfolio to calculate the systematic risk of an asset.

Bonus Points

  • Key factors to keep in mind when computing an index
    • the size of the sample: the larger, the better - but eventually the costs of taking a larger sample will outweigh the benefits.
    • the breadth: must represent the total population.
    • the source: must be taken from each different segment of the population
    • the weight given to each member in the sample: Price-Weighted, Value-Weighted, or others.
    • the computational procedure

Category: C++ Quant > Finance > Financial Markets

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Q&A: At what price will an investor get a margin call?

...suppose the investor pays $6,000 toward the purchase of 100 shares at $100 per share), borrowing the remaining from the broker. The maintenance margin is set to be 30%.

Answer

  • The initial percentage margin = 60%.
  • If the price of the stock falls to $57.14: the percentage margin becomes = (100 *57.14-4,000) / (100*57.14) = 29.9%. The investor will get a margin call.
  • Leverage factor = 1/margin% = (100%/60%) = 1.7

Bonus Points

  • Investors who purchases stocks on margin borrow from their brokers, and leave purchased stocks with the brokerage firm as collateral
    • The interest rate charged by the broker is typically 1.5% above the rate charged by the bank making the loan. aka. the call money rate.
    • The Federal Reserve sets the margin requirement (ie. the minimum proportion of total transactions value that must be paid in cash).
  • The Leverage effect: margin transactions enable investors to earn returns higher than those possible with a cash purchase. Also expose to larger possible gains and losses
  • Return on margin transaction = (change in investor's equity - interest - commission) / initial investor's equity.
  • Percentage margin: the ratio of the net worth, or "equity value" of the account to the market value of the securities.
  • Maintenance margin: the required proportion of your equity to the total value of the stock. The minimum maintenance margin specified by the Federal Reserve is 25 percent.
  • If the percentage margin falls below the maintenance margin, the broker issues a margin call requiring the investor to add new cash or securities to the margin account. If the investor fails to provide the required funds in time, the broker will sell the collateral stock to pay off the loan.

Category: C++ Quant > Finance > Financial Markets

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Q&A: A researcher wishes to estimate the mean SAT score and...

...compute a 95% confidence interval from a random sample of 10 scores: 320, 380, 400, 420, 500, 520, 600, 660, 720, and 780. Assume that the standard deviation of SAT verbal scores in a school system is known to be 100.

Also, how large should n be so that a 95% confidence interval for m has a margin of error of 0.0001?

Answer : When constructing confidence intervals, the z-distribution is used when the population standard deviation is known.

  • In Excel, m =AVERAGE(320, 380, 400, 420, 500, 520, 600, 660, 720, 780) = 530
  • Standard error of the mean = 100 / 10^0.5 = 31.62
  • z = 1.96 (from the z-table)
    • It turns out that one must go 1.96 standard deviations from the mean in both directions to contain .95 of the scores.
  • Confidence interval = Point Estimate +/- Margin of Error = Point Estimate +/- Reliability Factor * Standard Error
    • Lower limit = 530 - 1.96 * 31.62 = 468.02
    • Upper limit = 530 + 1.96 * 31.62 = 591.98
  • The research can be 95% certain that the mean SAT in the school system is between 468 and 592.
  • To have a margin of error of 0.1: Reliability Factor * Standard Error = z * stdev /n^0.5 = 0.1
    • n = (z * stdev / 0.1)^2 = (1.96 * 100 / 0.1) ^ 2 = 3841600

Bonus Points

  • When use z-distribution to construct confidence intervals, one assumes
    • normal distribution
    • the population standard deviation is known.
    • Scores are sampled randomly and are independent
  • The value of z for the 95% confidence interval is the number of standard deviations one must go from the mean (in both directions) to contain 95% of the scores
  • If a larger sample size had been used, the range of scores would have been smaller.
  • The computation of the 99% confidence interval is exactly the same except that a different z value is used. The 99% confidence interval is even wider than the 95% confidence interval.

Category: C++ Quant > Finance > Quantitative Analysis

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Q&A: Suppose there is a price-weighted market index...

composed of two stocks, with stock A selling for $100, and stock B $25.
  • What's the price of the index?
  • What's the rate of return if stock A increases by 10%, and B increases by 20%? What if stock A decreases by 5%, stock B increases by 20%?
  • What if stock A splits two for one? If the divisor stays unchanged?

Answer

  • the price index = (100 + 25) / 2 = 62.5, with the divisor being 2
  • RoR = 70 - 62.5) / 62.5 = 12%
    • the price index = ( 100*(1+10%) + 25*(1+20%) ) / 2 = 70
    • if stock A decrease by $5 and B increase by 5$, RoR = 0 since they cancel out.
  • The divisor must be reduced to a value that leaves the average unaffected by the split (otherwise the average would fall, indicating incorrectly a fall in the general level of market prices)
    • divisor = (50 + 25) / 62.5 = 1.2 * If the divisor stays unchanged: stock A = 100/2 = 50. price index = (50+25)/2 = 37.5. Thus, a simple split will cause a price-weighted index to go down unless the divisor is reflected.

Bonus Points

  • Price-weighted market index is an arithmetic average of current prices.
    • Dow Jones Industrial Average is the best-know price-weighted series: computed by adding the prices of the 30 companies of NYSE and dividing by a "divisor".
    • Nikkei-Dow Jones Average: 225 stocks on the First Section of the Tokyo Stock Exchange.
  • High-priced stocks have greater impact on the index than low-priced stocks, as the scheme assumes that an investor purchases an equal number of shares for each stock in the index.
  • Since the weight of each firm in the index is proportional to the share price, Successful firms consistently lose weight within the index as they tend to split their stocks more often. Over time, low growth, small firms with high prices will dominate the index.

Category: C++ Quant > Finance > Financial Markets

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Q&A: A researcher wishes to estimate the mean reading speed of...

...high-school graduates and compute the 95% confidence interval from a random sample of 6 graduates: 200 words per minute, 240, 300, 410, 450, and 600.

Answer : When constructing confidence intervals, the t-distribution is used when the population standard deviation is unknown.

  • In Excel, m = AVERAGE(200, 240, 300, 410, 450, 600) = 366.6667
  • sample standard deviation = variance^0.5, where variance = sum( (x(0)-m)^2 + ... + (x(n)-m)^2 )/n 60.9736
    • In Excel, s = STDEV(200, 240, 300, 410, 450, 600) = 149.3541652
  • Standard error of the mean = s / (N^0.5) = 149.3541652 / (6^0.5) = 60.97
  • df = 6-1 = 5
  • t = 2.571 (from the t-table)
  • Confidence interval = Point Estimate +/- Margin of Error = Point Estimate +/- Reliability Factor * Standard Error
    • Lower Limit = 366.67 - 2.571*60.87 = 209.904
    • Upper limit = 366.67 + 2.571*60.87 = 523.430
  • the researcher can be 95% sure that the mean reading speed of high-school graduates is between 209.904 and 523.430.

Bonus Points

  • When use t-distribution to construct confidence intervals, one assumes
    • normal distribution
    • the population standard deviation is unknown.
    • Scores are sampled randomly and are independent
  • T-distributions are spread out MORE than a normal distribution with u = 0, s = 1.
  • T-distributions are robust against nonnormality of the population except in the case of outliners or strong skewness.
    • sample size < 15: can use T-distribution only if the data are close to normal.
    • sample size >= 15: can use T-distribution except in the presence of outliners or strong skewness.
    • sample size >= 40: can always use T-distribution.
  • To shorten a given confidence interval, one should increase the sample size.

Category: C++ Quant > Finance > Quantitative Analysis

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Q&A: In an equal-weighted market index...

composed stock A and stock B, what's the rate of return if stock A increases by 10%, and B increases by 20%?

Answer

  • An equally weighted arithmetic average of these returns would be: (10% + 20%) / 2 = 15%. Or
  • for the geometric mean: 1 + r = [( 1 + 0.1) x (1 + 0.2)]^(1/2)
    • r = 1.1489. So the geometric average = 14.89% < the arithmetic average.

Bonus Points

  • In an unweighted/equal-weighted series, all stocks carry equal weight regardless of their price or market value. It assumes that equal dollar amounts are invested in each stock in the index at the beginning of the period.
    • Value Line Index: It is equally weighted geometric average of the performance of about 1,700 firms.
  • It is typically generated by taking the arithmetic or geometric mean of the percentage changes in the value for the stocks in the index.
    • Whenever there is variation in performance among the stocks in an index, the geometric average will be less than the arithmetic average (For this reason value Line index provides a downward-biased measure of the rate of return that would be earned by an investor purchasing an equally weighted portfolio of all the stocks in the index.)

Category: C++ Quant > Finance > Financial Markets

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Q&A: What's the appropriate alternative hypothesis if...

...an analyst hopes to show that the mean monthly return for stocks listed on the XYZ Stock Exchange is significantly different from 1%?

Answer : H(1): u != 1%. The alternative hypothesis corresponds to what the researchers are trying to show.

Bonus Points

  • Accepting the null hypothesis does not prove that it is true. It simply means that there is not sufficient evidence to reject it.
  • Rejecting the null hypothesis does prove the alternate hypothesis is true.
  • The null and alternative hypotheses account for all possible values of the population parameter.
  • 3 basic ways to formulate the null hypothesis: the null hypothesis will always contain "="
    • H0: u = u0, versus H1: u != u0: this hypothesis is two-tailed, which means that you are testing evidence that the actual parameter may be statistically greater or less than the hypothesized value.
    • H0: u<= u0, versus H1: u > u0: one-tailed; it tests whether there is evidence that the actual parameter is significantly greater than the hypothesized value.
    • H0: u >= u0, versus H1: u < u0: one-tailed; it tests whether there is evidence that the actual parameter is significantly less than the hypothesized value

Category: C++ Quant > Finance > Quantitative Analysis

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Q&A: List some of the market index differences...

...between bond market and stock market.

Answer :

  • Bond market indexes are typically computed
    • using value-weighting based on the outstanding amount of par value of the underlying issues
    • based on total return as opposed to capital gain
    • using prices quoted by traders.
  • Stock market indexes
    • use a variety of weighting methods,
    • usually measure only price change
    • and use actual trade prices.

Bonus Points

  • Most of the major international stock indices are computed using value-weighting.
  • One of the most comprehensive stock market index is Wilshire 5,000 Equity Index. Comprising 5,000 equity securities. Includes all New York Stock Exchange and American Stock Exchange issues and the most active over-the- counter issues. Represents the total dollar value of all 5,000 stocks.

Category: C++ Quant > Finance > Financial Markets

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