1. Quantitative Methods Himanshu Shah shahhr21@gmail.com 9970836701

3. Discounted Cash Flow Applications

4. Statistical Concepts and market return

5. Probability Concept

6. Common Probability Distribution

7. Sampling and Estimation

8. Hypothesis Testing

10. Concept of Time Value of Money helps in arriving at comparable value of different rupee amounts at different points of time into equivalent values of particular time.

11. Investment decisions are generally taken on cash flow analysis – TVM is pre-condition

12. For TVM analysis, it is better to draw time line first denoting at what time how much cash flows have occurred

17. 0 1 2 3 Annuities An annuity due is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately. Ordinary Annuity Timeline i% A A A 10

18. Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT PV FV Ordinary Annuity vs. Annuity Due 11

20. Discounted Cash flow Applications

22. How much value is created from undertaking an investment?

23. The first step is to estimate the expected future cash flows

24. The second step is to estimate the required return for projects of this risk level.

25. The third step is to find the present value of the cash flows

27. A positive NPV means that the project is expected to add value to the firm and will therefore increase the wealth of the owners.

29. It is based entirely on the estimated cash flows and is independent of interest rates found elsewhere

30. It is the interest rate at which PV of cash flows equates with PV cash outflows. Alternatively, at IRR, NPV of the project is zero.

32. IRR can be misleading when mutually exclusively projects are compared as this method assumes that cash flows are reinvested at IRR only which may not be possible in every situation

34. Holding period return is independent of the passage of time.When comparing investments, the periods should all be of the same length. Ending Beginning value value Income _ Holding period = return + Beginning value

36. Reflects compounding of returns

37. It is the rate at which Re.1 is compounded over a specified period of time

39. Money weighted Return The Money-Weighted Rate of Return (MWRR)…same as internal rate of return of portfolio. Initial deposit is considered as inflows and all withdrawals and ending value is considered as outflows. Not good for comparing different fund managers as MWRR over 0-T is also, generally, a function of the amount and timing of net new money – which is not in the control of the fund manager.

41. Returns on T-bills are quoted on a “bank discount basis”:

42. Here, discount is calculated from par value and not market value of investment

44. Considers compounding of interest

46. Convert holding period yield into effective annual yield

55. It is sum all observations divided by number of observations

56. Arithmetic mean is unique

57. Sum of deviation from its mean is always zero.

58. A.M = X1+X2+…….+Xn n = sum of all observations Number of observations

60. Geometric mean is used while calculating investment results for multiple periods or measuring compound growth rate

61. G.M. of n observation is defined as ‘n’th root of the product of n observation

62. G.M. is always less than or equal to A.M.

64. In such cases the usual mean is not good representative of data. Therefore we are obtaining weighted mean by assigning weights to each item according to their importance.

65. WM = sum(wx) sum(w)

67. It divides whole data into equal portion. In other words 50% observations will be smaller than the median and 50% will be larger than it.

69. Unimodal: distribution with one most frequently occurring value

71. Rank ordering of data: Quartiles, Quintiles, Twentiles, Deciles, Percentiles

72. 2nd quartile is the median

73. Position of a percentile in an array with n entries sorted in ascending order

75. It is the difference between the maximum and minimum values in a dataset

76. Mathematically, Range = Maximum value – Minimum value

77. Mean Absolute Deviation

78. It is the average of the data’s absolute deviations from the mean.

79. Mathematically MAD = ∑|xi – x(bar)| / n

80. Variance

81. It is the average of the population’s squared deviations from the mean.

83. It is simply the square root of the population variance.

84. Mathematically,= square root [∑(xi – m)^2/ N ]

85. All of the above are population parameters, however in investment management quite often we need to work with subset or sample of the population

87. For k > 1, the proportion of observations within k standard deviations of the mean is at least (1 – 1/k^2)

89. Allows for comparison of disparate distributions (because it would be in relation to the mean of the respective distributions)

91. It measures the extra return an investor earns (over the risk free return) for the added risk taken

92. Sharpe Measure =

94. Characteristics of a normal distribution:

95. Mean , mode and Median are equal

96. Completely described by 2 parameters

97. Mean and Variance

98. 68% within 1 standard deviation, 95% within 2 standard deviations and 99% within +- 3 standard deviations

101. Area under curve gives the total probability

103. Skewness occurs because the arithmetic mean of the population is not equal to its median since the mean is influenced by some extreme outliers

104. Degree of skewness of a distribution can be measured using the coefficient of skewness, Sk

105. If Sk is positive the distribution is positively skewed

107. Usually: mode < median < meanExample: Gamblers tend to like returns that are positively skewed because of the likelihood (however small) of a very large return

108. Negative Skewed Distribution Negative: tails to the left Usually: mean < median < mode Example Typically investors would not prefer negatively skewed instruments since it means that there is a likelihood (however small) of a very large loss

110. Kurtosis is an important consideration in risk management

111. Most portfolio returns exhibit some level of skewness and Kurtosis

113. Fatter tails

114. More likelihood of an observed value to be either close to the mean or far from the mean

115. Can be perceived to be of increased risk due to the fatter tail

120. Random Variable – uncertain or number

121. Outcome – observed value of random variable

122. Mutually exclusive events – events that cannot happen together

123. Mutually exhaustive events – includes all possible outcomes

124. Basic Properties

125. Sum of probabilities of all possible outcomes is 1.

128. Probability of an event regardless of the past/future occurrence of other events. Ex. Probability of Economic recession without considering probability of change sin interest rate and inflation

129. Conditional Probability :

132. P(A and B) = P(A)*P(B)

133. Dependent Events : If the above two conditions are not satisfied , events are dependent eventsExample: Economic recession and decrease in interest rates are dependent events.

135. Conditional Expected values are calculated using conditional probability

136. Forecasts are made using the expected value for a stock’s return, earning etc.

138. A negative covariance between X and Y means that when X is above its mean its is likely that Y is below its mean value.

139. If the covariance of the two random variables is zero then on average the values of the two variables are unrelated.Cov( Ri, Rj) = E { [Ri– E(Ri) ] [Rj– E(Rj) ] }

141. Correlation takes on values between –1 and +1.

142. A correlation of 0 means there is no straight-line (linear) relationship between the two variables.

143. Increasingly positive (negative) correlations indicate an increasingly strong positive (negative) linear relationship between the variables.

144. When the correlation equals 1 (-1) there is a perfect positive (negative) linear relationship between the two variables.Corr ( Ri, Rj) = [ Cov ( Ri, Rj) ] / [(Ri) (Rj) ]

148. P(R|A) is probability that fruit is red and round given it is an Apple

149. P(R) is the probability that a fruit is red and round (prior probability), regardless of whether it is an apple or not

150. P(A) is the probability that a fruit is an apple (prior probability) regardless of whether or not it is round and red

151. Bayes theorem allows us to infer the posterior probability P(A|R) if we know P(A), P(R) and P(R|A)

152. P(A|R) = [P(R|A)*P(A)]/P(R) Updated probability = Varied applications in business decision making including evaluation of mutual fund performance

154. Only one item to be selected from each group and two or more groups are present. Use multiplication rule.

155. When n items are to be arranged and none of them belong to any group. Use ‘n’ Factorial.

156. Each element of entire group must be a assigned a place, label in one of the three or more sub groups of predetermined size. Use the labeling formula.

157. When one needs to choose or select from two groups of predetermines size. Use Combination formula

159. n-factorial , n! = n x (n-1) x (n-2) x (n-3) x … x 2 x 1.

160. Labeling :

161. There are n items and each can receive one of k different labels

162. The total number of ways labels can be assigned :( n!) / [ n1 + n2 + … + nk ]

163. Combination :

164. Total number of ways of selecting r items from a set of n items when the order of selection does not matter

165. nCR = n! / [(n-r)!r!]

166. Permutation :

167. Total number of ways of arranging r items from a set of n items when the order of selection does matter

170. The number of possible outcomes can be counted and probability of each possible outcome is measurable and positive

171. P(x) = 0 when x cannot occur

172. P(x) > 0 when x can occur

173. Continuous Random Variable :The number of possible outcomes are infinite even within a range of values P(x) = 0 , even when x can occur. We can define probability P(x1 <= X <= X2 )

175. Specifies the probability of a random variable being equal to a specific value

176. Two key property of a probability function :

177. 0 <= p(x) <= 1 , probability of an event will always be between zero and 1

179. Function used to generate probability that outcome of a continuous distribution lie within a particular range of outcomes

180. For a continuous distribution, it is equivalent of a probability function for a discrete distribution

181. Cumulative density function (cdf):

182. It defines probability that a random variable takes on a value less or equal to a specific value.

183. It can be expressed as F(x) = P (X<=x)

187. Trials are independent

188. Each trial yields one of two outcomes: "Success" or "Failure“

190. Expected value of X = E(x) = np

191. Variance of X = np( 1-p)The function below calculates the probability of x successes in n trials when the probability of success is p.

193. Characteristics :

194. For all a <= x1 < x2 <= b, all values of x between a & b

195. P( X < a or X > b ) = 0 , probability outside range is zero

197. Can be an approximation of Binomial distribution for count and proportion when n is large

198. Characteristics :

199. Completely defined by mean and variance

200. Skewness = 0, ie. Symmetrical about mean

201. Kutrosis = 3 Excess kutosis for any distribution is measured relative to 3, ie of Normal distribution

202. Combination of normally distributed random variables is also normally distributed

204. Symmetrical

205. Total area under curve = 1Also Note: area within + or - 2 standard deviations is 95.4% 2) Area within + or - 1 standard deviation is 68.3% 99.7% Wide ranging applications in Modern Portfolio Theory and Risk Management

208. Hence they allow us to draw more meaningful conclusions on top of a point estimate

209. A Confidence intervals is a interval for which we can say with a certain level of probability, called the degree of confidence, that it will contain the parameter intended to estimate.

210. Three factors are relevant:

211. Standard deviation

212. Sample size

213. Precision levelPortfolio returns

215. s is used to estimate the standard deviation

217. We should use 1.96 for 95% confidence

218. 1.645 is used for 90% confidence

220. 95% confidence Interval X(bar) – 1.96s to X(bar) + 1.96s

221. 99% confidence Interval X(bar) – 2.58s to X(bar) + 2.58sProbability E(x) -  -2  2   68% 95%

223. The minimum acceptable level is threshold level

225. RL = threshold level of return

227. Lognormal distribution is skewed to the right

228. Lognormal distribution is bounded by zero from left

229. Application : Pricing assets which never take negative valuesNormal Distribution Lognormal Distribution

231. Procedure :

232. It is based on the repeated generation of one or more risk factors that affect security values, in order to generate distribution of security values

233. Random values of Risk factors are generated using the computer based on the parameters of the probability distribution these risk factors follow

235. Simulation used is a statistical method rather than analytical

237. It is based on actual changes in risk factors over some period of time

238. Simulation involves randomly selecting one of the past changes in risk factors and then calculating the value of security/ asset based on the changes in risk factors

239. Advantage

240. It uses the actual distribution of changes in risk factors which does not have to be estimated

241. Limitation

242. Past changes in risk factors may not be good indication of future

246. From each group , a random sample is taken

248. It includes observations over a period of time at specific and equally spaced time intervals

249. Ex. Stock returns of Reliance Industries from January 2009 to December 2009

250. Cross sectional data

251. It includes sample of observation taken at a single point in time

253. Types :

254. When standard deviation is known

255.  x(bar) = / ( sq. root (n) )

256. When standard deviation is unknown but estimated

257. s x(bar) = s x(bar) / ( sq. root (n) ), where ‘s’ is sample standard deviation

259. They are single values used to estimate population parameters

260. Desirable properties of an estimator :

261. Unbiased parameter

262. Gives the expected value of the estimator which is equal to the parameter being estimated

263. Efficient

264. If the variance of the sampling distribution is smaller than all other unbiased estimators of the parameter being estimated

265. Consistent

267. Properties

268. Symmetrical

270. As the degrees of freedom (sample size) gets larger, the shape of the t-distribution more closely approaches a standard normal distribution.

271. When compare to the normal distribution, the t-distribution is flatter with more area under the tails, as the degrees of freedom, df, for the t-distribution increases, its shape approaches that of the normal distribution.

272. Test Statistic Criteria

274. It pertains to excluding samples that may be critical in your analysis.

275. Survivorship bias

276. It is present if companies are excluded from the analysis because of problems with data availability.

277. Classic example is Mutual Funds which are closed are not taken into consideration while calculating performance

278. Look-ahead bias

279. It exists if the models uses data not available to market participants at the time the market participants act in the model.

280. Time-period bias

282. Estimation: Confidence intervals around a point estimate; post event

283. Hypothesis testing: assessment of the statistical significance of a hypothesis (or statement) about a population; defined a priori

284. Estimation vs. Hypothesis testing

285. Given a sample (post event) we can define confidence intervals to state our expectation that a population parameter falls within a certain range given the observations of the sample

287. Identify the appropriate test statistic

288. Specify the level of significance

289. Balance risk of false positive and false negative errors

290. Formulate the decision rule regarding the acceptance or rejection of the hypothesis

291. Collect sample and calculate the sample statistics

292. Accept or reject the hypothesis (statistical decision)

294. Null Hypothesis: it is the hypothesis to be tested

295. Alternative hypothesis: is the hypothesis to be accepted if the null hypothesis is rejected

296. If you believe the new strategy is better then the null hypothesis could be stated as - New strategy = Old strategy

297. Depending on what we are trying to accomplish we may state the hypothesis in 3 different ways:

298. H(0): Mean return = 6%; then alternative is H(A)” Mean return not equal to 6%

299. H(0): Mean return >= 6%; then alternative is Mean return < 6%

300. H(0): Mean return <= 6%; then alternative is Mean return > 6%

301. Note: The first example above calls for a two tailed test, i.e. null is rejected if the data suggests that the mean return is either smaller than 6% or larger than 6%

303. Our decision on accepting or rejecting the hypothesis would be based on a comparison between the calculated test statistic and a specified set of possible values; these values are primarily a factor of the level of significance

304. There are 4 possible outcomes from a hypothesis test:

305. Accept hypothesis and it is true; correct decision

306. Reject hypothesis and it is false; correct decision

307. Reject hypothesis and it is true: type I error

311. The probability of a type II error is denoted by the Greek letter beta

312. The power of a test is the probability of correctly rejecting the null (when it is truly false)

314. It is difficult to determine beta so we typically only specify alpha

315. We need to specify alpha ahead of calculating the test statistic (else bias can creep in)

316. 3 conventional levels: 0.10, 0.05, 0.01

317. O.10 => some evidence that the null hypothesis is false

318. 0.05 => strong evidence

320. Test statistic is a quantity calculated from a sample whose value is the basis for deciding whether or not to reject the null hypothesis

321. Sample mean provides an estimate of the population mean

322. For a sample mean x(bar) calculated from a sample generated by a population with a standard deviation sigma, the standard error is given by:

323. (xbar) =  /squareroot(n); if we know sigma

324. If we do not know sigma then we need to use the sample standard deviation s to estimate it

325. S(xbar) = s/squareroot(n)

326. The test statistic then becomes [X(bar) - ]/S(xbar)Sample statistic – Value of population parameter under H(0) Test statistic = Standard error of the sample statistic

328. It is also an alternate way of hypothesis testing

329. Compare p value of the hypothesis test with the level of significance of the test

330. If the p value is larger than the level of significance the null hypothesis is accepted

331. If it is smaller than the level of significance the null hypotheses is not accepted

332. The p test gives the same result as the t test but it provides additional information that makes it more powerful

334. t test: used for testing arithmetic means of populations;

335. f test: primarily for testing population variances

336. chi-square test: for hypothesis testing of independence of events

338. We may also apply this test for small to moderate departures from the normal distribution

339. For large sample sizes (>=30 observations) we may use the Z statistic in place of the t statistic

340. The t distribution is a symmetrical distribution defined by a single parameter: degrees of freedom

341. Compared to the normal distribution the t distribution has fatter tails

342. As the sample size increases (DOF increases) the t distribution converges into a normal distribution

343. Z statistic = [X(bar) - ]/ [ /squareroot (n) ]

345. Two approaches:

346. Assumes that the population variances are equal (although unknown); samples are independent

347. If we are not able to assume the population variance are equal then we have an approximate t test

348. The null hypothesis may be formulated as (1 - 2) = 0 (or <=0 or >=0)With appropriate alternative hypothesis

350. Test concerning mean diffrences For data consisting of paired observations from samples generated by normally distributed populations with unknown variance the t-test is as follows: Applications include: Test whether the mean returns earned by two investment strategies were equal over a particular period of time In this case the since both strategies are likely to have common risk factors such as market return the samples are dependent; hence we need to apply this test since by calculating the standard error based on differences we are accounting for correlation between the observations Pairs of before and after observations such as dividend returns before and after a change in tax laws t = [d(bar) – (d0)]/sd(bar) With n-1 degrees of freedom and Sd(bar) = sd/(n)^(1/2) Sample variance Sd^2 = {∑[(di – dbar)^2]}/(n-1) Sample mean difference, dbar=(1/n) ∑di

352. Unlike normal and t distribution it is a asymmetrical distribution

353. Like the t-distribution it is a family of distributions based on the DF = (n-1)

354. Lower bound of 0 (unlike t distribution)

355. If sample is not random or underlying population is not normal then inferences can be wrongTest statistic, X^2 = [(n-1)(s^2)]/(0^2) With n-1 degrees of freedom and S^2 = {∑[(xi – xbar)^2]}/(n-1)

357. For two samples the first with n1 observations and sample variance s1^2 and the second with n2 observations and sample variance of s2^2

359. F = S1^2/S2^2

360. DF1 = (n1 – 1) and DF2 = (n2 – 1)

361. Test is sensitive to the validity of the underlying assumptions

362. Convention is to use the larger of the two ratios (alternate one being S2^2/S1^2)

363. This ensures that the F statistic is always greater than 1

364. Hence the rejection point for any formulation of hypothesis (for this distribution and convention) is a single value in the right hand side of the relevant F distribution