That is, if interest is compounded continuou sly at an a nnual rate of 9. Unless otherwise stated,when we refer to retu rns we mean net ret urns. The o n e month investment i n Micro so ft yielded a 5. This is completely arbitrary and isused only to simplify calculations.

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That is, if interest is compounded continuou sly at an a nnual rate of 9. Unless otherwise stated,when we refer to retu rns we mean net ret urns. The o n e month investment i n Micro so ft yielded a 5.

This is completely arbitrary and isused only to simplify calculations. The an-n u alization p rocess depends on the holding period of the in vestment and animplicit assumption about compounding. We illustrate with several exam-ples. In this case, no c ompounding is required to create an annu al return.

Next, con sider a one month inve stme nt in an a sset with r etu rn Rt. Whatis the annualized return on th is investment? Pretty good! Now, consider a t wo month investm ent with r et urn Rt 2. Example 8 In the se cond example, the two month r e turn, Rt 2 , on Mi-crosoft stock was To complicate matters, now suppose that our investmen t horizon is t woy ears. What is the annualreturn on this two y ear inves tm ent? In this case, the annual return is compounded tw ice to get the two yearreturn and t he relationsh ip is then so lved for the annu al re turn.

Properties of logarithms and exponentials are discussed in the ap-pendix to this chapter. Notice that rtis slightly smaller than Rt. Con tin uously compounded returns are ve ry similar to s imple returns aslong as the retur n is relatively small, wh ich it generally will be for mont hly ordaily returns.

For modeling and statistical purposes, however, it is mu ch mo reconve nie nt to use con tinu o usly compounded returns due to the additivityproperty of mu ltiperiod contin uously com pounded returns and unless notedotherwise from h ere on we will work w ith con tinuously compounded retu rn s.

Hence t he contin uously compounded two m onth return is just the sum of thet w o contin uously compounded one mon th returns.

Recall that with simplereturns the two month return is of a multiplica tive form geometric averag e. Example 12 Using the data from example 2, the continuously compoundedtwo month re turn on Microsoft stock can be compu ted in two equiv alent ways. The s econd way use s the sum of the two continuously compounded one monthreturns. The additivitity of continuously compounded return s to form multiperiodreturns is an importan t property for statistical modeling purposes.

J and Parramore, London, UK. Microsoft stock. That is, we buy 1 share of Microsoft stock today and plan to sell it next month. Then the return on this investment is a random variable since we do not know its value today with.

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## Intro to Computational Finance with R

Monte Carlo simulation descriptive statistics and data analysis estimation theory and hypothesis testing resampling methods e. Prerequisites Formally, the prerequisites are Econ and an introductory statistics course Econ or equivalent. Econ Econometric Theory is not a prerequisite. More realistically, the ideal prerequisites are a year of calculus through partial differentiation and constrained optimization using Lagrange multipliers , some familiarity with matrix algebra, a course in probability and statistics using calculus, intermediate microeconomics and an interest in financial economics Econ would be helpful.

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