Jensen's inequality formula
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berrymucho
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In all generality Jensen's inequality says that for a convex (concave) function f, the expectation of the function of a random variable X (RV) is greater (smaller) than the function of the expectation of the same RV: E{f(X)} ≥ f(E{X}). Jensen's inequality would typically be reflected in the convexity term when looking at the Taylor approximation of the price of a non-linear instrument (e.g. bond, option). Every time there's a combination of a RV, a non-linear function of this RV, and the expectation operator of either (mean values)... Jensen's inequality needs to be kept in mind...
Thanks, that's helpful. So how do we calculate it for the exam? Is it as a described above? We are given this formula with E(1/1+r)>1/E(1+r)=1/1+E(r) but not sure how to apply it...
berrymucho
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To be honest I don't think there's much to "apply" directly without a given context. This is more a reminder to not automatically assume that E{f(X)}=f(E{X}), but instead E{f(X)}~f(a)+f '(a)*E{x-a} + 1/2*f ''(a)*E{(x-a)^2}, where f ' and f '' refer to the first and second derivatives of function f. Note that if f is convex then f ''>0 (i.e. convexity term non null), which gives you Jensen's inequality E{f(X)}≥f(E{X}).
The trick I think is that in practice, it is not always obvious to tell what quantity is a (realization of a) random variable versus a constant: for instance, say the mean of random variable X (population) is µ, a constant. However, the sample mean estimator of the same quantity is itself a random variable... depending on what you do with this variable, Jensen's inequality will apply, and you'd have to ask yourself "is there any convexity I need to worry about?". Hope this helps.
The trick I think is that in practice, it is not always obvious to tell what quantity is a (realization of a) random variable versus a constant: for instance, say the mean of random variable X (population) is µ, a constant. However, the sample mean estimator of the same quantity is itself a random variable... depending on what you do with this variable, Jensen's inequality will apply, and you'd have to ask yourself "is there any convexity I need to worry about?". Hope this helps.
Hi @bpdulog and @berrymucho I am catching up on some forum posts that I didn't get to in the pre exam flurry and, with respect to Jensen's Inequality wanted to share this XLS (screenshot below) here at https://www.dropbox.com/s/oifbfo90iaw6qvo/0529-jensens.xlsx?dl=0
This replicate Tuckman's Chapter 8 example (volatility and convexity) and tries to simplify the implication of convexity (aka, Jensen's inequality). It's a two-year rate tree such that the expected rate in the first and second year is 10.0% but with volatility (8 or 12%; then 6 or 10 or 14%). The "effect" is that--instead of a price (of a 2-year bond) being what we might expect as $100/1.10^2 = $82.64--the price is higher at $82.67 which a corresponding yield of 9.8181% (instead of 10.0% as we might expect). I hope that's helpful and that the exam went well for you, thanks!
This replicate Tuckman's Chapter 8 example (volatility and convexity) and tries to simplify the implication of convexity (aka, Jensen's inequality). It's a two-year rate tree such that the expected rate in the first and second year is 10.0% but with volatility (8 or 12%; then 6 or 10 or 14%). The "effect" is that--instead of a price (of a 2-year bond) being what we might expect as $100/1.10^2 = $82.64--the price is higher at $82.67 which a corresponding yield of 9.8181% (instead of 10.0% as we might expect). I hope that's helpful and that the exam went well for you, thanks!
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Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]), It's pretty hard for me!!
I was going through Jensen's Inequality and wanted to see the differences by creating a hypothetical scenario for a Zero Coupon Bond. While doing so in Excel, I noticed that the convexity adjustment changes as the maturity of the bond changes in a non linear fashion, as below:
Could you please have a look at this and help me understand why this is so? I thought that as maturity increased the bond's term structure became more and more convex but the above signifies that it does become more convex to a certain maturity but flattens out (relatively) afterwards. (Attaching a spreadsheet I created - Could you please have a look?)
Could you please have a look at this and help me understand why this is so? I thought that as maturity increased the bond's term structure became more and more convex but the above signifies that it does become more convex to a certain maturity but flattens out (relatively) afterwards. (Attaching a spreadsheet I created - Could you please have a look?)
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