Recall that e x lim n 1 x n n we can approximate the effective return by the

Recall that e x lim n 1 x n n we can approximate the

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Recall that e x = lim n →∞ (1 + x n ) n , we can approximate the effective return by the formula 1 + r = e i ( ) , where i ( ) is known as continuously compound return . Therefore, continuously compounding with rate r means an effective return of e r - 1. When investing in stocks, when the price appreciate in the first day (i.e. profit), it is very unlikely that the investor will sell some of it to realise the profit (due to transaction costs, unless there is a dividend payment). Hence, we shall use “compounding”(in a sense of multiplication) in this case as the capital changes over time which captures profit/loss. (not the above formula as different period may have a different rate of return) As the stock price changes very frequently, we need to multiply the return for a large number of small time intervals when computing the effective annual return, i.e. 1 + r = (1 + r 1 )(1 + r 2 ) ... (1 + r n ) where r k is the effective return for the k -th period. In view of the formula above, it would be much convenient if it is of the form e ˜ r = e ˜ r 1 e ˜ r 2 ...e ˜ r n = exp(˜ r 1 + ˜ r 2 + ... + ˜ r n ) ⇐⇒ ˜ r = ˜ r 1 + ˜ r 2 + ... + ˜ r n which can be done by using the continuously compounding return instead of the effective return. It has an obvious advantage that the returns can be summed. For example, if the effective return is 10% annually, then the 2 year return effective is 21% instead of 20%, which would be difficult to visualise without calculators (when the situation is more complicated). However, if the continuously compounding return is used, it is simply summing some numbers, which is much easier. In this sense, it is quite “natural” 1 to use continuously compounding. In fact, when calculating Net Present Value (NPV) or Future Value (FV), all you care is 1 + i or v = 1 1+ i , not i itself. Note the continuously compounding return is sometimes called the log-return as it involves taking (the natural) log. Therefore, continuously compounding is an important concept. When we quote “continuously com- ponding interest rate r % annually” , it means that a $1 investment at inception will grow to e r/ 100 in one year. 3. Basic Probability In this section, we only consider random variables that take real values. 1 Also “natural” because it involves the number e , which is natural. 2
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3.1. Discrete and Continuous Random Variables Assume the random variable X is a variable which takes a random value. If the set of possible values are discrete (e.g. tossing a dice where the outcome is { 1 , 2 , 3 , 4 , 5 , 6 } , i.e. you can list it in some way), it is called discrete random variable. If the set of possible values is continuous (e.g. the time of death of a dog) then it is called continuous random variable. For discrete random variable, most of the time you need to sum off something, e.g. to calculate the expectation. For continuous random variable the sum is replaced by an integral, which is generally more difficult to evaluate.
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  • One '16
  • Jin Xia Zhu

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