STAD70 Statistics & Finance II<\/p>\n
Assignment 1<\/p>\n
Due date: Feb 11, 2021 (by 11:59pm). Late submissions will be penalized.<\/p>\n
Include your codes in your submission.<\/p>\n
1. (Normal mixture)<\/p>\n
(a) Recall that the moment generating function (mgf) of the standard normal distribution N(0, 1) is<\/p>\n
E[etX ] = e 1 2 t2,<\/p>\n
where X \u223c N(0, 1) and t \u2208 R. Use this result to show that the kurtosis of N(\u00b5,\u03c32) is 3 for any \u00b5 \u2208 R and \u03c3 > 0. Hint: You may differentiate (repeatedly) under the integral sign, e.g.<\/p>\n
d<\/p>\n
dt E[etX ] = E<\/p>\n
[ d<\/p>\n
dt etX ] .<\/p>\n
Then evaluate at t = 0.<\/p>\n
(b) One way of modelling fat-tailed distributions is to use normal mixtures. Given two normal distributions N(\u00b51,\u03c3<\/p>\n
2 1) and N(\u00b52,\u03c3<\/p>\n
2 2), a normal mixture (with two compo-<\/p>\n
nents) has density<\/p>\n
p(x) = (1 \u2212\u03bb)\u03c6(x; \u00b51,\u03c321) + \u03bb\u03c6(x; \u00b52,\u03c3 2 2),<\/p>\n
where \u03c6(x; \u00b5,\u03c32) is the density of N(\u00b5,\u03c32) and 0 \u2264 \u03bb \u2264 1. Assume \u00b51 = \u00b52 = \u00b5. Compute the kurtosis of the distribution in terms of the parameters.<\/p>\n
(c) Let \u00b51 = \u00b52 = 0, \u03c31 = 1, \u03c32 = 4, and \u03bb = 0.05. Simulate n = 50000 samples from the corresponding normal mixture. Plot the density estimate p\u0302(x) together with the theoretical density p(x).<\/p>\n
(d) For the distribution p in (c), compute (numerically) the 5% quantile, and compare it with that of the normal distribution whose standard deviation matches that of p.<\/p>\n
2. (Relative entropy and excess growth rate) Given two positive probability vectors p = (p1, . . . ,pn) and q = (q1, . . . ,qn) (i.e., pi,qi > 0 and<\/p>\n
\u2211 pi =<\/p>\n
\u2211 qi = 1), define the relative<\/p>\n
entropy H(p||q) by<\/p>\n
H(p||q) = n\u2211<\/p>\n
i=1<\/p>\n
pi log pi qi .<\/p>\n
(a) Using Jensen\u2019s inequality, show that H(p||q) \u2265 0 and equality holds if and only if p = q. Hint: Think of H(p||q) = \u2212Ep<\/p>\n
[ log<\/p>\n
q(X) p(X)<\/p>\n
] where X \u223c p.<\/p>\n
1<\/p>\n
(b) Suppose there are n assets. At time t, t = 0, 1, . . ., let Xi(t) > 0 be the value of asset i. Consider a buy-and-hold portfolio whose value at time t is<\/p>\n
V1(t) = X1(t) + \u00b7 \u00b7 \u00b7 + Xn(t) X1(0) + \u00b7 \u00b7 \u00b7 + Xn(0)<\/p>\n
.<\/p>\n
Let V2(t) be the rebalanced portfolio with weights w = (w1, . . . ,wn), where w is a positive probability vector. That is, we have<\/p>\n
V2(t)<\/p>\n
V2(t\u2212 1) =<\/p>\n
n\u2211 i=1<\/p>\n
wi Xi(t)<\/p>\n
Xi(t\u2212 1) .<\/p>\n
Let mi(t) = Xi(t)<\/p>\n
X1(t)+\u00b7\u00b7\u00b7+Xn(t) be the capitalization weight of asset i. Show that for each<\/p>\n
t we have<\/p>\n
log V2(t)<\/p>\n
V1(t) = H(w||m(0)) \u2212H(w||m(t)) + \u0393\u2217w(t), (0.1)<\/p>\n
where \u0393\u2217w(t) is the cumulative excess growth rate of the rebalanced portfolio up to time t.<\/p>\n
(c) Consider the stocks Ford (F), JPMorgan (JPM), IBM (IBM) and Coca-Cola (COKE). Consider monthly stock prices from Jan 1, 1990 to Dec 31, 2021. Normalize the prices so that the beginning value Xi(0) is 1 and m(0) = (<\/p>\n
1 4 , 1 4 , 1 4 , 1 4 ). Let w =<\/p>\n
(0.1, 0.2, 0.3, 0.4) (in the same order). Compute the performance of the two portfolios and illustrate the decomposition (0.1) with a figure (similar to Figure 2.6 in the notes). Comment on the results you get.<\/p>\n
3. Consider the 4 stocks in Problem 2 (c). Now, consider the sample period Jan 1, 2000 to Dec 31, 2021.<\/p>\n
(a) Compute the sample skewness and sample kurtosis for the log returns at daily, weekly and monthly frequencies (similar to Figure 1). For each asset-frequency pair, perform the Jarque-Bera test and report the p-values. Also compute the autocorrelation (lag 1) for the log return and absolute value of the log return (similar to Table 2). Comment on the results.<\/p>\n
(b) Illustrate the aggregational Gaussianity property using one of the assets with (i) kernel density estimates (similar to Figure 3.4) and (ii) normal q-q plots. Comment on the results.<\/p>\n
4. Consider weekly log returns of the FTSE 100 index from Jan 1, 2000 to Dec 31, 2015. Leave the last 80 observations for testing; these observations are not used for fitting the model.<\/p>\n
(a) Fit an ARMA model to the data. Explain carefully how you arrive at your chosen model. Estimate the parameters and examine the residuals.<\/p>\n
(b) We examine the out-of-sample performance of the model. Suppose your fitted ARMA model in (a) is has order (p\u2217,q\u2217). For each time t of the testing period, use the data up to time t to fit an ARMA(p\u2217,q\u2217) model (note that here the order is fixed), and let<\/p>\n
2<\/p>\n
r\u0302t+1|t be the forecast of the log return rt+1 computed at time t. Compute the sum of squared errors \u2211<\/p>\n
t<\/p>\n
(r\u0302t+1|t \u2212rt+1) 2<\/p>\n
and compare it with \u2211<\/p>\n
t(r\u0302t+1\u2212r) 2, where r is the sample mean in the training period.<\/p>\n
Comment on the results.<\/p>\n
3<\/p>\n
—<\/p>\n
STAD70 Statistics and Finance II is a two-semester course in statistics and finance.<\/p>\n
The first assignment<\/p>\n
The deadline is February 11, 2021. (by 11:59pm). Late submissions will incur penalties.<\/p>\n
Incorporate your codes into your submission.<\/p>\n
1. Formalized paraphrase (Normal mixture)<\/p>\n
(a) Recall that the typical normal distribution’s moment generating function (mgf) is N(0, 1).<\/p>\n
E[etX] = e1 2 t2<\/p>\n
where X N(0, 1) and t R Use this conclusion to demonstrate that the kurtosis of N(,2) is 3 for every R and > 0. Hint: You can distinguish (repeatedly) beneath the integral sign, for example.<\/p>\n
d<\/p>\n
E[etX] dt = E<\/p>\n
[d] Formalized paraphrase<\/p>\n
[dt etX]<\/p>\n
Then, at t = 0, assess.<\/p>\n
(b) Using normal mixtures to model fat-tailed distributions is one method. N(1, 2) is given two normal distributions.<\/p>\n
N(2, 1) and N(2,<\/p>\n
2 2), a standard mixture (with two components),<\/p>\n
nents) possesses<\/p>\n","protected":false},"excerpt":{"rendered":"
STAD70 Statistics & Finance II Assignment 1 Due date: Feb 11, 2021 (by 11:59pm). Late submissions will be penalized. Include your codes in your submission.\u2026<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7650,7624,1387,7654,667,7649,7652],"tags":[613,7594,1394,7598,7595,7593,7597,7599,7601,7590,7596,7591,7592],"class_list":["post-92660","post","type-post","status-publish","format-standard","hentry","category-assignment-writing-help","category-dissertation-help","category-dissertation-writing-service","category-healthcare-assignment-help","category-nursing-essay","category-research-essay-help","category-write-my-essay-online","tag-best-dissertation-writing-services-uk","tag-cheap-custom-essay-writing-services","tag-custom-research-paper-writers","tag-custom-research-papers-writing-service","tag-essay-writer-websites","tag-essay-writing-service-quality-cheap","tag-free-research-paper-writing-service","tag-help-me-write-a-research-paper","tag-i-need-help-with-my-dissertation","tag-professional-dissertation-writing-service","tag-quick-essay-writing-service","tag-top-rated-dissertation-writing-services","tag-top-uk-dissertation-writing-services"],"_links":{"self":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts\/92660","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/comments?post=92660"}],"version-history":[{"count":0,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts\/92660\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/media?parent=92660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/categories?post=92660"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/tags?post=92660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}