The value of differential equation to an actuary?
Actuary : A person who makes calculations relating to insurance. NOTE: Not just life assurance, as some people seem to think. Differential Equation: A mathematical means of calculating the consequence of change, especially applicable to situations where the change is non-linear. So... An actuary might use a differential equation to calculate an appropriate premium in a situation where one of the risks changes with time.
Gompertz Differential Equation?
You have: dy/dt = y*(a - b ln(y)) As suggested, let z = ln(y), then y = exp(z) and dy/dt = exp(z)*dz/dt With this substitution, the equation becomes: exp(z)*dz/dt = exp(z)*(a - b*z) dz/dt = -b*(z - a/b) dz/(z - a/b) = - b dt Integrate: ln(z - a/b) - ln(zo - a/b) = -b*t where zo is the value of z = ln(y) at t = 0 ln((z - a/b)/(zo - a/b)) = -b*t z = a/b + (zo - a/b)*exp(-b*t) Back substitute for z: ln(y) = a/b + (ln(yo) - a/b)*exp(-b*t) where yo = y(0) y(t) = exp[a/b + (ln(y0) - a/b)*exp(-b*t)] A) here, a = 1, b = 4, and yo = 8, so: y(t) = exp[1/4 + (ln(8) - 1/4)*exp(-4t)] As t -> ∞, exp(-4t) -> 0, so lim y(t -> ∞) = exp(1/4)
What is the application of differential equations in our every day life?
A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variables.Unlike the elementary mathematics concepts of addition, subtraction, division, multiplication, percentage etc, which are used on a day to day basis, differential equations are not generally used/observed in our every day life.Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time.Having said that, I have compiled a list of applications of differential equations. I knew some of them and some of them were googled.One of the most basic examples of differential equations is the Malthusian Law of population growth. dp/dt = rp shows how the population (p) changes with respect to time. The constant r will change depending on the species. Malthus used this law to predict how a species would grow over time.Some other uses of differential equations include:In medicine for modelling cancer growth or the spread of diseaseIn engineering for describing the movement of electricityIn chemistry for modelling chemical reactions and to computer radioactive half lifeIn economics to find optimum investment strategiesIn physics to describe the motion of waves, pendulums or chaotic systems. It is also used in physics with Newton's Second Law of Motion and the Law of Cooling.In Hooke's Law for modeling the motion of a spring or in representing models for population growth and money flow/circulation.As you can see from the above examples, unless you are a physicist or a chemist or a biologist or an actuary or an electrical/electronics engineer, chances are that you might not get a chance to use differential equations.If you are interested in knowing the maths behind these concepts then do check out Applications of Differential Equations.
3-variable differential estimation anyone?
Here, f(x,y,z) = 2xy + 2xz + 2yz. We let (x,y,z) = (80, 60, 50) with Δx = Δy = Δz = 0.2. Taking partial derivatives, f_x = 2y + 2z f_y = 2x + 2z f_z = 2x + 2y So, f_x (80, 60, 50) = 220 f_y (80, 60, 50) = 260 f_z (80, 60, 50) = 280 Hence, the total differential dz at (80, 60, 50) is given by dz = 220 dx + 260 dy + 280 dz. ==> Δz ≈ 220 Δx + 260 Δy + 280 Δz = (220 + 260 + 280) * 0.2 = 152. So, the maximum error is approximately 152 sq cm. I hope this helps!
Is Advanced Calculus essential for an actuary? If so, how does it benefit in our studies/works?
If by Advanced Calculus you mean “Real Analysis” then I'm going to say the answer is no. Having knowledge of it might help, but as an actuary you won't need to do epsilon-delta proofs.As an actuary here is the following math you should know:CalculusCalc 1Calc 2Calc 3Linear AlgebraProbabilityFinancial MathOrdinary Differential EquationsMAYBE PDEs (I did an actuary shadow/intern day back home and some people said they regularly use black scholes, though you might just be able to pick this up on the job). The background knowledge obtained from this course wouldn't hurt though. I doubt you would even use ODEs much at all, just knowing how to solve them when they come up will be useful.StatisticsTime SeriesApplied RegressionBayesian StatsNow on the general day to day basis, you will likely not be using much of the above math. Instead, you’ll be using SAS, SQL, Excel/VBA, and R to do the vast majority of your work. You should just know the above math so on the rare occasion you'll need to do it, you'll be able to know how to do it or be able to quickly pick it up again.Read around on various forums and websites to get a better idea. To get the best answer though, you should consult an actual actuary. They'll be able to provide you with the best answer. I listed the above information from talking to a few actuaries that visited my school’s Math/CS club and surfing the web.I hope this helps you![edit 4/24/17]I should clarify that you'll definitely be using calculus, linear algebra, and probability every single day in your job, just most of the time it is going to be done by a program or a computer. You absolutely need to know that stuff, but unless you're fact checking something by hand, it'll normally be done via some kind of software because technology makes everybody’s lives easier and you can get more work done! This is not to say you'll have to not know how to do this stuff. You won't always do everything by a computer, so you should absolutely know it. I'm sorry if that wasn't clearly addressed in my original answer.
What's the annual effective discount rate?
I'm actually working on this same problem right now... ran into this by googling the question. Sigh! Well, let's see if I can do it on my own and end up helping you, too!.... part a) She wants $4000 now. So she will owe a(t) = (1 + i)^t = (1 - d)^(-t) = 4000(1 - .035)^(-6) = 4953.31687. So about $4,953.32. My book says I'm correct. part b) My book is asking "what is the annual effective INTEREST rate of Latisha's loan?" Is that what you meant? If not, sorry. If so, then... d = i / (1 + i) which is .035 = i / (1 + i). Solve for i and you should get i = 3.62694%. (That's what my book says the answer is). Sorry there's not a whole lot explanation. Hope I helped anyway!!! :)
What is some good initial literature for actuarial science and applied mathematics?
I have got some good, but complicated books, depending on where you want to go:There is a good book on derivatives and the models for valuation. Hull - options, futures and other derivatives. This provides a good introduction as well.There are a couple of amazing mathematical books on life insurance and risk by Rob Kaas (Faculty of Economics and Business) , to get you into the mathematics of insurance and riskAnd if you like insurance, try googling for copula's and extreme value theory - I am not up to date with the books, but that is an absolutely interesting field of mathematics.A bit dated, but good book on simulation (it uses a language that no-one uses anymore), but has a good overview of techniques and distributions you need for simulation.Simulation modeling and analysis (4th edition), A.M. Law en W.D. Kelton, McGraw-Hill International Series, ISBN 97890071255196Econometrics is an interesting field of applied mathematics in general (if you like economy) - look for regression, macro economic models, micro economic models, and if you can, go to an university library (again, not up to date with the latest books).Remember, these are good books to actually learn the subject from, not to get an introduction into the field. You already need to understand some of the mathematics. Or maybe find someone to explain more. But you will get a great overview of what is possible.