C7 - Quantitative Methods (10-20 Qs) SC Flashcards

Question

Estimate the mean and population standard deviation of the portfolio size from the data given below. Portfolio Size (£'000) / No of Clients 20 and under 40 / 50 40 and under 80 / 160 80 and under 120 / 126 120 and under 200 / 35 200 and under 400 / 29 A) Mean = £95,000, SD = £66,652 B) Mean = £66,652, SD = £95,000 C) Mean = £95,000, SD = £44,425 D) Mean = £44,425, SD = £95,000

Answer 1

**A - Mean = £95,000, SD = £66,652** Computing the GEOMETRIC MEAN: **Step 1:** Compute mid point of each interval = £30, £60, £100 £160, and £300 (figures in £'000) **Step 2:** Multiply mid-point by the frequency and sum = (£30 x 50) + (£60 x 160) + (£100 x 126) + (£160 x 35) + (£300 x 29) = 38,000 **Step 3:** Take solution from Step 2 and divide by the total number observations (in this case, clients) = 38,000/(50+160+126+35+29) = 38,000/400 = £95 adding back in the £'000 = £95,000 Computing the POPULATION STANDARD DEVIATION: **Step 1:** Take each mid-point, deduct the mean and square, then multiple by the frequency and sum the results = (((£30 - £95)^2) x 50) + (((£60 - £95)^2) x 160) + (((£100 - £95)^2) x 126) + (((£160 - £95)^2) x 35) + (((£300 - £95)^2) x 29) = 1,777,000 **Step 2:** Take the solution form Step 1 and divide by the number of observations to get variance = £1,777,000/400 = 4,442.5 **Step 3:** Take the square root of the vairance to arrive at SD = √4,442.5 *(OR (4,442.5)^(1/2) )* = £66.652 adding back in the £'000 = £66,652 Note both geometric mean and SD can be computed using the calculators stats function...DATA > CLEAR MEMORY > STATS > LIST 1 ENTER MIDPOINTS > SELECT ONE FREQ > SELECT LIST 2 ENTER FREQUENCIES > SWITCH TO STATS MODE (2ND DATA 1) > 1-VAR > SELECT LIST 1 FOR DATA > SELECT LIST 2 FOR FREQUENCY > CALC > ANSWERS: n = VARIABLES / x = GEOMETRIC MEAN / Sx = SAMPLE SD / σx = POP SD *Unless stated sample SD is wanted, always use the population SD for answer*

Answer 2

**D - Semi-Log Graph** When we wish to visually display the rate of change in a variable over time we plot the natural logarithm of the variable over time.

Answer 3

**C - 1 (show percentage change from base) and 4 (All components in an arithmetic index have the same impact on index value)** Indexes show percentage change from the base, NOT actual change. The base value of the index can be anything but it is normally either 100 or 1,000.

Answer 4

**D - 5.54%** (1.1 x 1.08 x 1.2 x 0.95 x 1.06 x 0.75 x 1.3 x 1.1)^(1/8) - 1 = 0.0554 i.e. 5.54% *While the arithmetic mean is based on adding and dividing values, the geometric mean multiplies and finds the root of values. Even though the geometric mean is a less common measure of central tendency, it's more accurate than the arithmetic mean for percentage change and positively skewed data.*

Answer 5

**A - The MODE will be greater than the MEDIAN and MEAN** A negatively skewed distribution is caused by the presence of large negative values of variable. With negative skew MEAN

Answer 6

**B - They always reflect absolute changes since the base year** Indexes are designed to show percentage change in a vairable overtime NOT absolute chnages

Answer 7

**A - 65.99** To get answer on calculator: **Step 1:** Clear memory... DATA > DATA > 4 **Step 2:** Enter midpoints in List 1 (SO £25, £75, £125, £175, and £225) **Step 3:** Enter frequencies into List 2 (SO 78, 97, 43, 50, 32) **Step 4:** Switch to STATS MODE.... 2ND > DATA > 1(1-VAR) **Step 5:** Select List 1 for DATA and press enter **Step 6:** Select List 2 for FRQ and press enter **Step 7:** Select CALC by hitting enter when it is flashing: **Step 8:** Answers... 1:n = Variables (300), 2:x = Geometric Mean (101.83), 3: Sx = Sample SD (66.1024761) and **4:σx = POP SD (65.99221335)**

Answer 8

**B - £1,712** -£5,000 + (£13,500/(1.15)^5) = £1,711.89

Answer 9

**C - Mean = 4.8, Median = 4.4** The MEAN is the sum of all the returns divided by the number of returns (SUM OF RETURN)/11 = 4.8 The MEDIAN us the central value after putting hem all in numercial order, so with the 11 numbers in numerical order the 6th number (4.4) is the median.

Answer 10

**C - £597.03** This calculation must be done with semi-annual time periods. Three (3) years equates to six (6) semi-annual periods. The semi-annual interest rate is 6%/2 = 3% Future value = £500 x (1.03)^6 = £597.0261

Answer 11

**C - 1.60%** Monthly rate = 1.21^(1/12) - 1 = 0.016 SO 1.6%

Answer 12

**D - 172.50** The price index has increased by 15% from 1996 to 1997. Basing this on the old index price of 150 at 1996, the 1997 level would be 150 x 1.15 = 172.5

Answer 13

**D - £1,500** The calculation must be done with quarterly time periods. Six (6) years equated to 24 quarterly periods. Present Value (PV) = £2,713/(1.025)^24 = £1,499.95. £1,500 is the closest match to this figure.

Answer 14

**A - Market Capitalisation** The FTSE 100 and FTSE 250 are market value weighted arithemetic indexes. FTSE100 being the 100 largest companies and FTSE250 being the NEXT 250 largest companies.

Answer 15

**C - 3.2 and 2.0** **GEOMETRIC MEAN:** Geometric mean can be calculated by taking the seventh roof of ^7√(0.91 x 1.12 x 0.81 x 1.22 1.28 x 0.95 x 1.02) - 1 = 0.032 so **3.2** For TI-30XS calculator: (0.91 x 1.12 x 0.81 x 1.22 1.28 x 0.95 x 1.02) ^(1/7) - 1 = 0.0554 i.e. 5.54% **MEDIAN:** -19%, -9%, -5%, **2%,** 12%, 22%, and 28%

Answer 16

**A - The arithemetic mean of the middle two numbers** Median = middle observation when data set is ranked from lowest to highest. With an odd number of observations there is only one middle obserbations, so this is the median. With even numbers of observations, there are two middle observations. The median is calculated by taking a simple mean average of the two middle values.

Answer 17

**B - 17.57%** We need to find the discount rate that gives an overall NPV of zero. Trial and error is the only approach for this exam. NPV = -£25,076 *Year 1* - £2,000/ (1+R) *Year 2* + £20,000/(1+R)^2 *Year 2* + £20,000/(1+R)^3 *Year 3* By trying each answer, you will find 17.57 is the discount rate that gets you closest to NPV of zero. Start with one of the two middle values offered and you will only need to do a max of 2 calculations.

Answer 18

**D - 27.1** £40,000 = £500,000 x (1-R)^8 Rearranged: R = **1 -** ((£40,000/£500,000)^(1/8)) = 0.2707 SO 27.1% The 1 - at the beginning is key!

Answer 19

**C - 107 and 48.7** To get answer on calculator: **Step 1:** Clear memory... DATA > DATA > 4 **Step 2:** Enter midpoints in List 1 (SO 25p, 75p, 125p and 175p) **Step 3:** Enter frequencies into List 2 (SO 80, 120, 200, and 100) **Step 4:** Switch to STATS MODE.... 2ND > DATA > 1(1-VAR) **Step 5:** Select List 1 for DATA and press enter **Step 6:** Select List 2 for FRQ and press enter **Step 7:** Select CALC by hitting enter when it is flashing: **Step 8:** Answers... 1:n = Variables (500), **2:x = Geometric Mean (107)**, **3: Sx = Sample SD (48.7930479)**and 4:σx = POP SD (48.74423043)

Answer 20

**A - £30,637** **Step 1:** Compute the PV of the £600 annual deductions PV = £600 x annuity factor Annuity factor = 1/R(1-(1/(1+R)^Years)) = 1/0.08(1-(1/(1.08)^8)) = 5.7466 PV= £600 x 5.7466 = £3,448 **Step 2:** Compute net amount at at dates NPV = £20,000 - £3,448 = £16,552 **Step 3:** Compound the net amount for 8 periods to get the terminal value £16,552 x (1.08)^8 = **£30,637**

Answer 21

**C - 1 (Total returns on a share consists of price changes and dividends received) and 3 (The major bond market indices are total return indices)** Most stock market indexes focus on share prices (price return) and not total return (which includes any cash flow distributions). Most bond indexes reflect total returns over a period (coupons, price changes, and reinvestment income).

Answer 22

**C - 5.4 and 2.29** To get answer on calculator: **Step 1:** Clear memory... DATA > DATA > 4 **Step 2:** Enter midpoints in List 1 (SO £2,000, £5,000, and £8,000) **Step 3:** Enter frequencies into List 2 (SO 7, 12, and 11) **Step 4:** Switch to STATS MODE.... 2ND > DATA > 1(1-VAR) **Step 5:** Select List 1 for DATA and press enter **Step 6:** Select List 2 for FRQ and press enter **Step 7:** Select CALC by hitting enter when it is flashing: **Step 8:** Answers... 1:n = Variables (30), **2:x = Geometric Mean (5.4)**, 3: Sx = Sample SD (2.328237457) and **4:σx = POP SD (2.289104628)**

Answer 23

**B - 27.38%** Geometric mean return = (1.20 x 1.25 x 1.30 x 1.35)^(1/4) = 1.273773 SO 27.38%

Answer 24

**B - It’s a measure of distribution NOT dominated by extreme values** *Range* - Highest to lowest value in a data set. *Median* - midpoint of the dataset when arranged in numerical order (low to high). *First Quartile* - will be 1/4 through the data set (this can be described as the 25th percentile). *Third Quartile* will be 3/4 of the way through the dataset (i.e. halfway between the highest value and the median). *Interquartile range* is computed as the 3rd quartile value to the 1st quartile value. By definition the interquartile range will be the middle 50% of the dataset.

Answer 25

**B - 114.24** (0.955 x 1.038 x 1.05 x 1.12 x 0.98) x 100 = 114.24

Answer 26

**A - Non-random sampling techniques** Non-random sampling increases the chance of ALL groups in a population being represented.

Answer 27

**C - Only 2 (Primary data are collected with a specific task in mind) is true** Population is ALL members of a well defined group and non-random sampling has judgements in selectings groups.

Answer 28

**C - 2% (mean) and 3% (median)** (93 + 98 + 103 + 107 + 109)/5 = 102 - 100 = 2% for mean When put in order, 3% is the middle number and therefore the median.

Answer 29

**D - 7%** √ = suqare root ^5 √ (1.2 x 0.95 x 1.02 x 1.09 x 1.11) - 1 = 0.0706 so 7% OR ((1.2 x 0.95 x 1.02 x 1.09 x 1.11)^0.2)-1 = 0.0706 so 7%

Answer 30

**C - ...the mode is less than the median is less than the mean** D is normal distribution (no skew). A and B are wrong as Median is ALWAYS in the middle. * Positive skew = MODE < MEDIAN < MEAN * Negative skew = MEAN < MEDIAN < MODE *Tip - being mean to someone is bad (negative), so negative starts the chain when referring to LESS than*

Answer 31

**B - The market model is a bivariate model** The market model relates to the return on a security to a constant and the return on the market *A bivariate model is a statistical model that examines the relationship between two variables. It can be used to identify patterns, make predictions, and understand how changes in one variable affect another.*

Answer 32

**A - FT30** The FT 30 (FT Ordinary Index or FTOI, not "FTSE 30") is a now rarely used index that is similar to the Dow Jones Industrial Average. As an index of stocks to represent the real trends on the market, the FT 30 has been superseded by the FTSE 100, which was introduced in 1984. The FT 30 index was calculated using the geometric mean.[3][4] As Rowley states, this had the effect of understating movements in the index compared to using the arithmetic mean; and there are circumstances where this is undesirable.

Answer 33

**C - Nikkei 225** Nikkei is an unweighted arithmetic index

Answer 34

**D - The prices of a representative sample of 30 large US blue chip stocks** The Dow Jones Industrial Average is a simple arithmetic aggregation of unweighted share prices of 30 major US industrial companies

Answer 35

**A - £181.21** Three years and 3 months = 39 months = £150 x (1.005)^39 = £182.2080905

Answer 36

**A - £1.17** 2 years = 8 quarters Compound: £100 x (1.02)^8 = £117.17 SImple: £100 + (8 x (£100 x 0.02) = £116 £117.17 - £116 = £1.17

Answer 37

**A - £260.25** £575 / ((1.0825)^10) = £260.25

Answer 38

**D - £181.67** 5 years = 60 months £100 x (1.01)^60 = £181.67

Answer 39

**A - Total interest paid in a year divided by the initial balance** Although APR involves annualising a measure of return usually quoted on a monthly, quarterly, or semi-annually basisn by compounding, an alternative calculation involves summing the interest paid throughout the year and dividing by the initial amount.

Answer 40

**A - £9.17** NPV = -£75 + (£100 / (1.09)^2) = £9.17 Make sure you sure the £75 as negative at the start.

Answer 41

**B 0.76** 1 / (1.07)^4 = 0.76

Answer 42

**D - The discount rate on an investment always equals the IRR** The IRR equates the PV of the cost of an investment with the PV of the cash flows coming from the investment.

Answer 43

**C - 5%** NPV = £0 = -£231.29 + (£100 / 1.05) + (£150 / (1.05^2)) Answer will answer 0.002517 which is the closest to 0

Answer 44

**D - Minimise the residual sum of square error** Linear regression calculate the value of coefficient by minimisign the residual sum of the squaref errors (Least squares method).