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Actuarial Exam 2/FM Prep: Given a Force of Interest, Find the Interest Earned Over a Time Period - YouTube
Channel: Bill Kinney
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hi in this video I'm gonna be solving a problem聽
one point 6.5 s in the mathematics of investment聽聽
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in credit by Samuel Abraham in the 6th edition is聽
this book we will be finding the interest earned聽聽
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in a time period when we are given a force of聽
interest so here we go Ernie makes two deposits聽聽
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a deposit of 100 times zero and deposit of X at聽
time three the fund grows add a force of interest聽聽
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equal to this function here it's not a constant聽
force of interest constant forces of interest to聽聽
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give you exponential growth this is positive聽
increasing fairly rapidly this is definitely聽聽
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going to give you faster than exponential growth聽
the amount of interest earned from time three two聽聽
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times six is X so X is showing up in two ways in聽
this problem both as a deposit at time three and聽聽
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the amount of interest earned over all from time聽
three two times six the goal is to calculate X we聽聽
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haven't made a number line in a while so let's聽
go ahead and do that here let's make a number聽聽
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line three relevant times time zero time three and聽
times six you've got a deposit of 100 at time zero聽聽
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and of X at time three what does it mean to grow聽
according to a force of interest I'm just going to聽聽
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tell you it's something you should look up if you聽
forgotten for example it means this one hundred聽聽
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deposit at time zero will grow to one hundred聽
times e to the integral of the force of interest聽聽
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from time 0 to time three integral from zero to聽
three of Delta sub T the force of interest it聽聽
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also means at times six that initial one hundred聽
will grow to one hundred times e to the integral聽聽
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of the force of interest from time zero to ten聽
six that's what it means to have that force of聽聽
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interest that's how that money grows what about X聽
X is also growing according to the same force of聽聽
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interest but it's starting at time three rather聽
than times zero and therefore you multiply X by聽聽
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e to the integral not from zero to six but instead聽
from three is six it's a three down there if it's聽聽
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difficult to see integral from 3 six of the force聽
of interest so once I find these quantities these聽聽
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integrals and multiply these by a hundred I can聽
find the total interest room from time 3/2 times聽聽
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6 by taking this quantity minus this one and聽
adding this quantity minus this one set that聽聽
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equal to X then solve the resulting equation for聽
x let's do the integrals first so what about the聽聽
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integral from 0 to 3 here's our force of interest聽
T squared over 100 antiderivative is going to be聽聽
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T cubed over 300 that needs to get evaluated聽
from 0 to 3 it's going to give you 27 over 300聽聽
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minus 0 which is the same as 9 over 100 or point聽
0 9 the integral from 0 to 6 think about it it's聽聽
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going to give you the same antiderivative from 0聽
to 6 you're going to get 6 cubed over 300 minus聽聽
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0 that'll be 216 over 300 which equals 72 over聽
100 which can be reduced further but you don't聽聽
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have to be let's just write that as 0.72 the聽
integral from 3 to 6 is going to give you 216聽聽
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over 300 minus 27 over 300 or if you prefer聽
0.72 minus point zero 9 it equals point 6 3聽聽
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I need to exponentiate those things I've got e to聽
those integrals it seems a different color we use聽聽
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our calculator here so e to the integral from 0聽
to 3 is e to the points type in point zero nine聽聽
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then my e function is above the ln i've got to聽
use the second function key e to the point zero聽聽
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nine is about one point zero nine four one聽
seven four two eight four and I think I'll聽聽
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use all my decimals this time just playing聽
it safe I also need e to the point seven two
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five point seven two second gives you two point聽
zero 5 4 4 3 3 2 1 1 I also need a to the point聽聽
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six 3 that will be one point eight seven seven six聽
one zero five seven nine and now I'm ready to set聽聽
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up my equation think back up here this minus this聽
what's that going to be that's going to be this聽聽
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was e to that integral which was this quantity聽
here one point eight seven seven six one zero聽聽
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five seven nine times X minus X that difference聽
will be the interest earned on the deposit of X聽聽
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at time three two times six add on to that the聽
interest earned by the initial 100 deposit from聽聽
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time three two times six this minus this that's聽
gonna be two point let's see I have to multiply by聽聽
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hundred and forgot about that two hundred and five聽
point four four three three two one one - looking聽聽
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here multiply that by 100 one hundred and nine聽
point four one seven four two eight four I'm聽聽
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sure I could get away with fewer decimals but聽
I've already started this many salt to continue聽聽
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set this equal to X because you're told that the聽
amount of interest earned from time three two聽聽
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times six is X this is our equation to solve for聽
x two oh five point five four three three two one聽聽
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one minus one hundred nine point four one seven聽
four two eight four four one seven four two eight聽聽
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four yo okay give Siri nine ninety six point zero聽
two this is going to be point eight seven seven聽聽
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six one zero five seven nine X subtract that from聽
both sides you're going to get ninety six point聽聽
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zero two five eight nine two seven equals x minus聽
that times X let's just go ahead and store this 96聽聽
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take one minus twenty eight seven seven six one聽
oh five seven nine gives you a point one two two聽聽
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three eight nine sorry about that four-two-one X聽
now divide both sides by that so I'm going to take聽聽
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its reciprocal and multiply what I have stored聽
in register zero there we go finally there's the聽聽
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answer X is seven eighty four point five nine and聽
that is correct you know in the video with that
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