Financial Mathematics for Actuarial Science, Lecture 1, Interest Measurement

hi I'm bill Kenny and this is my first relatively long lecture about financial mathematics for actuarial science on giving these lectures of Dudley University for consumption in my courses and well these lectures that I'm going to do will be hopefully about an hour and will cover a lot of content from a broad perspective I'm using the sixth edition of the mathematics of investment and credit by saying the Obermann for this class which you want to get you also will want to get a solutions manual for it and also often going to use the theory of interest by Stephen Kellison as a reference this is the second edition is about twelve years old there's probably newer editions right now again these are going to be a broad perspective on things an overview of the chapters I've tried to do all chapter one essentially in one one-hour lecture here if you want help with problem solving you should look down in the description of this video and I will have links to some problem-solving videos I've got lots of them that are typically 7 to 13 that's going over problems in Robins book so you need to get practice with those problems if you're really going to get this again these lectures are for a big perspective on events so lecture 1 you can see in the title layer is about financial quantities for representing growth in decay especially exponential growth and exponential decay so I'm going to talk about how do we represent these kinds of quantities especially in finance on again from a broad perspective I will mention real life practical realities from time to time but you're only going to get more of those from reading the book you're going to talk about the book talks about things like Treasury bills and bonds and that kind of thing it gets into those kinds of details I'm going to focus more than math in these lectures but there's a basic reality the most fundamental fact in finance is that money has what's called a time value also known as the time value money you may have heard this before thinking about putting money in your savings account you get interest your money growers so to speak but the idea of the time value of money is broader than that it arises from the basic reality that most people would prefer a dollar or a pound or a euro or whatever right now at this instance than a dollar in the future like one year from now right what are the causes of this well psychology / personality is that people's greed they want to have their money right now they can't control themselves they need to go buy something with it right away there is also something called opportunity cost meaning they could take this dollar today and in essence they could make money off their money they could also buy things with it that would help them start business that's called opportunity cost there's also inflation the value of your money goes down over time people typically think of that as the price of things that you buy go up over time and so you want to get the money right now so that you can buy what you need right away there's also this idea at least in America keeping up with the Joneses meaning your average next-door neighbor you see that they buy a nice they have a nice toilet or they go buy a boat or a nice car and you want to do the same thing you want to keep up with them you know one life to live etcetera there are lots of possible causes for this idea of the time value of money and what we're trying to do is we're trying to model this not just the savings accounts letting lots and lots of situations a blast from the past I hope you've seen formulas for simple interest in compound interest before maybe you've seen these formulas your simple interest high which is the amount of interest that you earn is P times R times TP being the principal R being the interest rate typically given as an annual rate may be 10 well maybe 2% for years Amelia's like that and he needed time maybe in years and the accumulated of future value of your money the original principle the deposit plus the interest which can be simplified to that expression and other is the formula for compound interest which looks like this take your principal and multiply by this expression one plus the interest rate divided by the number of compounding periods per year raid 2 and times T where again n is the number of compounding periods per year times T is the number of years ago by to get the future accumulated value in the account if you want to figure out the interest then use this form so maybe you've seen this before and this is certainly something we're going to talk about in this lecture and throughout this class in financial math actuarial science though we will mostly talk about compound interest we will actually use different notation for the most part than what you see here and when you're comparing simple interest versus compound interest look back at those formulas thinking of these things as functions of T it's really linear growth versus exponential growth that's a linear function of T this is an exponential function of T I hope you remember from algebra and or precalculus that linear growth has a constant rate of change or slope where an exponential growth if you're less like it is likely to know this has a constant relative rate of change for a given change in time delta T represents the change in time what do the graphs look like for linear growth the graph is a straight line and you see now if you ride the ramona's constant the change in the amount Delta a divided by the change in time delta T this quality of here stays constant no matter where you are on the curve take the riser to run give it any two distinct points on this straight line you look at the same slope that's also called the rate of change with exponential functions it's a little bit more subtle is the change in the amount divided by the starting amount think of a and T is being started on during a certain interval of time so T here looks like it's about 1.6 that's the starting moment in time that we're thinking about here we're letting T increase from about 1.6 up to 6 so it increases by 4.4 that's Delta T Delta a the change in a which is this distance right there divided by AMT which is this is Sue's right here also this is a cinch right there is constant for a given delta T it was giving Delta here being about four point four if I move this back and forth for that given delta T that ratio this distance divided by this one will stay constant I actually have a Mathematica animation for this let's see here and exit sorry about this there we go mathematic is a program that I use a lot to make animations and this illustrates how the slope stays constant for linear drove I can change the starting location I can also change delta T and no matter what happens with this particular line the triangle is similar no matter what and you have the same slope whereas the exponential growth again the ratio the system's of that distance will stay constant I'm not going to take the time to check that in my calculator you might want to pause the video and try to estimate the distances for that particular starting value of a which is now about 2.7 or so and they'll change a and you can Charlie calculating the ratio of this divided by that for this new starting value of XK which is the starting value of T right there and see that you get about the same thing change delta T and you just different ratios but for any six delta T the ratio for different days different starting you really are probably shouldn't called it a different starting times stays the same version selfie okay so that's one of your vs. exponential growth what's about actuarial occasion for all this and compounded your solutions this terminology is important to the future or accumulated value at a time T of a payment or the closet of 11.1 euro one-pound whatever at time zero is given by this expression right here and actually there are times where we don't even write it like you see here let me just go ahead and write it like that what do these symbols represent well first of all there is a one here at the positive one that's in front of this that we go about reading since when you multiply by one it doesn't change it i n would be the interest rate compounded n times per year so often times n might be four if it's quarterly interest rate and might be twelve if it's compounded monthly and in the number of compounding periods per year let's just pretend time as and years again that's the same thing up there and T is the amount of time in years don't get confused we're not doing complex analysis here on the imaginary unit that's not square root of negative one no imaginary version here I stands for the interest rate this is often called the nominal rate so I wrote I 12 and by the rating and amount of power you're either I'm not reading I to the 12 power it's indicating that this is a nominal interest rate compounded 12 times per year in for example it was 4% for zero for that mean that that is the nominal in name only interest rate compound in 12 times a year which means the corresponding monthly interest rate would be that thing divided by 12 and that would be point three repeating or the percent percent per month that's how much your money would grow every month if if a 4% nominal rate per year compounded monthly it grows by that percentage every month which actually we will see makes it grow more than four percent of the entire year that's the so-called magic of compound injures something else you'd be mentioned about this n times T and as the number of compounding periods per year say he has a number of years also can be thought it was just representing the number of times your compound interest over those two years so for example if n is 12 and key is to n times K would be 24 you come from interest 24 times over two years if the payment of time zero is it something other than one called a sub zero then capital NT is often used for the future value of the situation and you just multiply literally in T by a 2-0 if that future value of that is the positive that's something other than 1 but we often work with the little a of T we often think about amounts of 1 instead of arbitrary amounts in a lot of the problems that we do here's a technicality they can often be ignored they're not always as stated the equations in the previous slide are only true when n times T is a whole number and in reality the graph of a of T is piecewise constant the blue horizontal lines here make up the true graph of a of T you're only getting interest if it's compounded say monthly at the end of every month that's when you get the jumps in the amount we're not depositing any more than just the first amount if it was quarterly once every three months you get the jumps at time 3 6 9 12 etc and lungs and if time were mirrors you get those jumps at time 1/4 1/2 3/4 and 1 etcetera well this technicality can often be ignored and we will often ignore did not worry about it but there are some situations where you have to pay attention to here is an essential and I really do mean essentially this is really important point of view for problem solving if the creation of timelines number lines where you mark off a lots of money and times given that amount of money a valued at a time t1 and some other time t2 that's in the future say we can promote or push a forwarding I literally think in terms of fiscal pushing like I kicked that anomaly and push it forward from time one to an equivalent value of time to by multiplying by a ratio hey and multiply it by this ratio little a of t2 divided by little a of T 1 and you can visualize it on a number line like that you're going to over line your time line yours time T log nears time t2 you've got this amount a at time t1 what is future or accumulated value other this interest regime where common interest or maybe something else where complication is going on what is that future value take and multiply it by this ratio now often times t1 is taken to be zero and a of zero little a of 0 is 1 and therefore this fraction simplifies to little a of T divided by will a of 0 which would be little a and T divided by 1 which would just be teens we take our amount at time zero and find its future value by multiplying by little AAT so it often simplified that but in general you want to think of it as being multiplied by this ratio at least if T 1 is not 0 and then suppose if you were thinking about a situation where this was not 1 so that is the typical way to think about the delay and then you want to think about that restore to what happens in the case of compound interest the ratio simplifies by properties of exponents right will a of t2 divided by little day of the team law before using compound interest let's keep it simple we'll take n to be 1 here it's where we get 1 plus I to the T to power divided by 1 plus I to the 1 power and you subtract those exponents like you see here are property of exponents and so you really can't just do a multiplication complete upshot of that on the other hand if you were doing simple interests which actually can tell you the truth we rarely think about in a lot of the problems with chapter 1 then it becomes that and it doesn't really simple language ok so that would be what happens if you're doing compound interest or some filters but again they're on business these are the main models simple interest and compound interest so we will consider more complicated situations including in this lecture we will look at a more complicated situation um so interest can become how-to in various ways you could compound quarterly you could compound monthly you could compound semi-annually which would be twice a year or annually once a year you could compound daily so n would be 365 or maybe 360 or keep it a nice round number on you can compound with different frequencies and by varying the interest rate the corresponding the nominal rate you can get the equivalencies here and this equation is not showing up very well I use a Mac and we're using a PC here and not showing up I see fraction bars in here general equation relating : rates that are compounded with different frequencies so you've got I add that is the nominal interest rate when you compound incomes per year and you've got I am which is the nominal interest rate when you compound M times per year if you're going to get the same accumulated or future value stay after one year these two things you need to equal and so you could also solve this equation for one of the variables in terms of the other like this your fraction here to think about that that's not too hard to see I'm going to solve for I am I need to raise both sides to the 1 over N power to get rid of the end there to make it a 1 giving me an elbow brain power they're all evidently we were having difficulty seeing here there should be a horizontal log in there we raise both sides of the loan grand powers you get a demo version here we then subtract one from both sides to get a Vegas over there and then multiply both sides by hand the definition for that will end here here and here showing us what they did solve for I am in terms of I am and then oftentimes you might want to use this kind of equation one hand is one to give us what's called an effective annual rate of hey there's the line it's shown up this time an effective annual rate given I am the nominal rate compounded M times per years with n equal to one you can simplify this according to this one and just for your information you will see this as you work on the problems the effective annual rate when when you compound more than once a year is going to be more than the nominal rate again that's because you're compounding more than once a year your money will grow by more than the nominal rate called nominal rate because it's a name only it's not the effective rate it's not measuring in reality as far as what you actually in an intercept there's also something called continuous compounding maybe you've heard of this before it involves the number e about 2.71828 let IV the effective annual interest rate and choose the number Delta so that e to the Delta is 1 plus I in other words Delta the natural logarithm of 1 plus i when you do such a thing when you find such a number Delta it's called the continuous interest rate often times outside of actuarial science at least we're going to call something a little fancier sounding we're going to call the force of interest and we will actually see there are some situations where it's not a constant it can vary but here it's going to be a constant he was compounding the compound interest on here's a side note that is something worth knowing sometimes the following living to be derived from l'hopital's rule if you know some calculus which you really should if you're in this class you shouldn't have calculus to at least in especially I'll just tell you you should know a lot about geometric series if you don't you want to review that especially before today's chapter to all the bases for calling Delta continuous interest rate this woman's back you can think of this expression is essentially giving you the future value of one after one year when you the novel interest rate is Delta and compounding M times per year and then let n go to infinity let the number of compounding periods go to infinity into the limit you can prove this as e to the Delta which again is one plus I I would be interest rates one plus I have a name as well you can call that the annual growth factor and again Delta is the continuous interest rate or more typically you want to call it the force of interest some more notes about continuous compounding note that is a of T is common and your sis is one behind the keys a little hard to see up there is common interest then Delta it can be paused on another way he found as a prime of T divided by K of T think about that this is an exponential function we differentiated with respect T you might remember that it is the base here is not easy take the natural log of the base before multiplying by the exponential function we need to the derivatives we're dividing by the function itself the month of I the teen camp loudly get natural log of n plus I which is Delta in other words Delta uses this value now what that means is Delta measures the relative the constant relative rate of change for an exponential growth function here a prime of T is the rate of growth that's the derivative instantaneous rate of growth at any moment in time if I divided by km T itself is called the instantaneous relative rate of growth I think the rate of growth divided by how much you start with just like I mentioned before with the exponential growth you have a constant arm confer give that as an annual rate of growth but it's also a constant instantaneous rate of growth as exemplified by this equation therefore the case of compound interest Delta represents the content instantaneous relative rate of growth like I've just said in other cases not a common interest then this is not necessarily constant this comedy right there a prime divided by a and we can write it as a function of time as traditional for actuaries through use often times subscripts of the variable they often use super strips and pre superscripts and subscripts for variables as well as people who are confusing but here that fell script is the variable for this thing and we have function of T it's a kind of T divided by a of T in general which is not necessarily constant in general so again for compound interest it is can we do anything with this Delphis of T beyond continuous compounding yeah you can actually reverse the modeling process and start with the given force of interest function a little hard to see there that the Delta s of T that keeps going on top of my column there related to the amount function by the same equation that I showed you and interestingly enough this ratio this relative rate of change can be thought of as this derivative think about that use the chain rule of this function take the derivative of natural log of a of T since with revenues of with respect to X say of natural log of X is 1 over X if you take the derivative of natural log of a of T you're going to get 1 over a to T times the derivative of the inside function with the chain rule the inside function today of T so if derivative will be a prime of T we'll get this exact same thing the relative rate of growth and this equation can then be integrated to try to find the formula for a of T another way you can think about this is you can think about this a differential equation that can be solved for a of T dat / a will Delta's of T another way to think about that and you can integrate after separating variables to find a of T but let's just think about it this way integrate both sides from say t1 to t2 and you get this kind of equation when you integrate a trivet instance you're going to get the function itself right this should be familiar we're entering D DT of this thing we're going to get the function itself evaluated from t1 to t2 so I plug in t2 and the function and subtract what I get when I plug in T 1 on the right hand side you just get this integral integral to Delta CP function you can solve this for a sub T a of T 2 in terms of a of T 1 and delta T by combining these logarithms we can write this as the log of a of T 2 divided by M T 1 then you can exponentiate take e to both sides multiply both sides by a of T 1 you're going to disagree so there's a general equation relating on telling you what the amount function is at time t2 based on knowing will be a known function at a time t1 + knowing the force of interest most commonly this is used with t1 equal to 0 and if you take t1 to be a 0 you take a or 0 B 1 it's the one then natural log of 1 is 0 you can solve for the natural log of a of T and then again ultimately exponentiate get a AP in simplest form like that I got a new variable here tau that's just done done to emphasize that in this function the true variable is this T that's in the upper limit of the integral so I use a different variable there sometimes people are a little sloppy at least owe you Devon Irv and I like to try to use different variable towers degree versioning team here's an oddball example where we can apply this formula suppose elements of T the force of interest as a function of time is point one plus point zero five times the cosine of two pi t you think about that for a second it's going to be an oscillating function periodic function with amplitude point zero five and average value or midline of point one we will graph this in a minute few minutes and it's got a period of one right this is going to go through a cycle of two pi units as T goes from zero to one this might not be such a bad model if you're thinking about just sort of general interest rates varying periodically over a giving you that's possible it could happen at least approximately the goal here is to find a formula for an T and find the value of for example a of 1.25 I'm also going to find what the effective growth or interest rate over this one point two five year period is and what's the equivalent of effective annual interest rate in this example what does the formulas use the formula for the previous slide and there we go there was the delta T function with the talents of T we need to integrate that with respect to tau and then plug in the limits of integration so we get point 1 tau excuse me plus over the point zero five divided by two pi times sine of two pi K tau when you integrate that evaluated from zero to T when you plug in 0 ie in both situations sine of 0 and 0 u 0 with one towel and how equal zero you only get something nicer when you plug in the deep most spots and this forty pi comes from the point zero five divided by two pi so that's what the function aft and the function T is and you can evaluate it at a different value if you like for today of 11.25 it turns out to be approximately one point one four to two meaning that over that one year in three months the money you buy off fourteen point twenty two percent what's the equivalent effective annual interest rate I'll let you figure it out I'll let you check this on your own it would be about eleven point two two percent maybe I should couple chest the only like the attached on here okay that's that's pretty close yeah okay it's about eleven thirty two percent I'll leave that to you try to check it out on your own okay you can graph these things there's a graph of the force of interest as a function of time here's a graph the u-bahn function it's kind of like exponential growth but with some wobble limit you understand the relationship between these well Delta sub G is the most of means a prime MT divided by a of T at any moment in time it's supposed to give you the instantaneous relative rate of return any moment times that equals four there would be the slope of the tangent line there divided by the height of the function at that point that would be the value of Delta sub T at T equals four it looks like it goes to a maximum there in people's for a tangent line is close to a maximum in slope and dividing it by the amount this is close to a maximum here actually these slopes do have to be overall getting higher and higher if the relative rate of growth is going to stay periodic like this the same maximum minimum over time because you keep dividing my bigger numbers with relative rate growth hello to your us we're on to the second page of the lecture which I think this is shorter than the first page we can talk about future or a cumulative value but we can also talk about something called present value what about it's motivated by this question what about P should be deposited now as a principal to accumulate to a half of two years you go back to the original compound interest formula you can solve it for P right the original formula I will cover in the more deeply okay original one gold I have looked like that we can solve that equation for P by multiplying both sides by 1 plus R over N to the negative NT power I guess I'm using I in here if you multiply both sides by 1 plus I R and the negative and keep hours you will get exactly this equation okay so that would need a bad answer my question how much you should deposit now it's accumulated to a after two years thinking in terms of an accumulation function little a of T with little a of zero taken to be 1 what should be deposited now at time zero to grow to one after two years it should be multiplied by the reciprocal one should be multiplied by the reciprocal of a dirty I should say go to st. it I should save one that future value should be multiplied by you're at the word x it should be okay over the amount is going to be the reciprocal of a of T okay that's the amount of deposit right now careful this is not an inverse function okay you should you should avoid writing something like this though I have actually seen this in some books I think you should avoid it it's not an inverse function that is too reminiscent of inverse function I would strongly strongly advised to avoid confusion that you write it either like this with a negative one power or as a fraction 1 over a of T that's how much you should pause it now to a single 8 to 1 and years think about if you got multiplied by a of T to find its future value tears and the 80s are canceled leaving with the one on the in the case of compound interest where AM T is 1 plus I to the T we can all often use different rotations to this we can define something called the present value discount factor or just discount factor for short letting V equal the reciprocal of 1 plus I 1 over 1 plus I or if you prefer 1 plus I to the negative 1 power we more commonly will write this present value as just B to the T this is the present value at time 0 of a future payment of 1 at time T you kind of write in like this but if you let B be the reciprocal of 1 plus I you can also rate it like this some of the books I should tell you write need more like a new and tell you in truth I've heard people think of it both ways news a brief letter that I typically writing like that though in this program here didn't quite write it like that V is called the present that I just kind of factor some other make it look like it's a Greek letter nu again I've heard people different people called V and Colin new ok it's hard to tell the difference this particular PowerPoint made might look like that it's not a big deal you can make it look like that I more typically just going to think call it V and write it isn't over very V look like that and that's what I do in my problem-solving videos if you watch those this is a very common notation use on your to see it all the time in your problem solving finding this and using this present value discount or just discount factor for short well as an example if the effective interest rate say annual interest rate is 10 percent you can find the reciprocal of 1 point 1 it would be 1 divided by 1.1 and you get about point nine zero nine zero nine activates point nine zero repeating about you can think of that as percentages to think of this is about ninety point nine percent if your money grows by ten percent in a year you can think of the opposite way you can say the present value of an amount of one one year the future is nine point nine percent of what it is 0.9 one cents if you will at time zero and in both of these situations you can think of both of these as functions of time so the red graph is one point one to the T exponential growth the blue graph is its reciprocal is about point nine zero nine zero an activity looks like that that's exponential decay I also made a horizontal line here you can see head one for any given value of T like cables for you look at the outputs of these functions they have to multiply to one their multiplicative inverses they are reciprocals of each other about one point four looks like it was one around point seven something a bit okay those are will multiply to work you can think of present value in terms of timelines as well and that's also very important given some amount of money a value that some time t2 I can pull it back again I think physically I'm literally amount pulling that back to some time t1 and I'm demoting in that making it smaller typically to the times public value at time t1 by doing this multiplication it's the reciprocal the one that I had in the other slide you can visualize it like that that's the opposite kind of picture so before we had air revered we've put forward to time t2 here we have a at k92 at work yet in back in time to time t1 by doing this multiplication again in the typical situation we take T 1 to be 0 and T 2 DZ t and assuming little a of 0 is 1 you're effectively just a game divided by a of team up with the subsidy ok if you take T 1 to be 0 and T 2 to be T that's just its vision typically or the multiplication by in visa team and become about the cupcakes compound interest is this place is the same as these people again this comes up all the time in problem-solving so you really want to get used to this timely perspective very very helpful what's the present value in general for a varying force of interest function suppose the force of interest is Delta satine that interest aft tonight and one the present value discount function is essentially the same thing I have in this slide a while ago of a cup of the negative slander okay because of three simple the multiplicand reciprocal so it's the same formulas before except with the vigorously and that's what the formula would be in general and so we're going to go back to the oddball example now where the force of interest is periodic so once again tilde sub T is point one plus point zero five cosine 2 pi key we saw that the accumulation function was that and therefore the present value function is the same thing except with negative sign in the exponent the reciprocal of the previous one and this means for example that the present value at times you're all the pinna in at one time 1.25 is this particular example that's related to that other particular example and in fact don't that remember the one point effect at one point two five year rate was about fourteen point twenty two percent and yet this number is approximately the reciprocal of that now you should check out these calculations on your own fun way use your calculator to check all these calculations get back to the oddball example we can graph once again the force of interest and now I'm not graphing the accumulated value of the graph to the present value is a function time it's still a bit wobbly but it's exponential decay now instead of exponential growth um and as far as the relationship goes between these things um let's see certainly relate this to the original function of T into instantaneous relative rate of growth actually I didn't think about those coming the link to the instantaneous rate of growth of this function self-esteem folks go ahead and spend a little time thinking about this what would happen if I was finding the derivatives dysfunction a and T to the negative one power divided by the function self and shitty in the negative Romney is its rate of growth is negative and sold its relative gradient roads would be negative what would happen if I did this calculation I'm not remembering okay it looks just it happens I use the chain rule here I get negative 1 and T to the negative 2 times of a chain rule a prime of T divided by a of T to the negative 1 we can keep the negative 2 this is a times here you can keep the negative sign on the front we can say subtracting X buns negative 2 minus negative 1 is negative 1 meaning we can have an a of T in the Bob so we took it a negative quantity but it's the opposite of the instantaneous relative rate of growth of the accumulation function that's interesting and now I'm reverberating than that happened so that's kind of cool if you find the instantaneous rate of change of the relative orientation of functions the slope divided by the Alpha value it will be negative and in sum I but the absolute value is going to be the same as the relative instantaneous relative rate of growth through the accumulation function getting close the end here few more things to say um there's also something called a present value discount rate is cooperating that discount factor called de for a different effect of periodic interest rate and by the way when I say periodic here that doesn't mean the periodic function I mean is like an annual interest rate could be monthly as well we can define the equivalent effective periodic present value discount rate D by this equation D equals r1 plus I what does this represent this represents for an investment of one a payment of blood in time zero the percent growth in that investment relative to the future value rather than the present value instead of taking I divided by 1 which would be the amount of interest divided by the starting now we're taking I divided by 1 plus I which is the amount of interest divided by the ending amount right if you've got $100 and grows to 110 dollars that's 10% growth come out of a hundred and ten percent as far as this discount rate goes you take 10 and 110 should be 90 or so be the discount oh you might wonder why do this um let me just give you a little hint that you'll see in reading if you do the reading that this kind of rate it's often how interest is quoted especially with bonds whether they be government bonds or corporate bonds they say they are selling you say a thousand dollar bond meaning year to get a thousand dollars in the future they're selling ink you have a discount they say you pay less than a thousand to get this on that's sort of a historical origin of this idea but it's certainly a way to can measure interest there are some other relationships knowing worth knowing relating di and B D which again is I r1 plus I is also equal to one minus B in listen calculation verifies that the reciprocal of 1 plus I so you get 1 minus v1 minus this get a common denominator of 1 plus I in the numerator you will have the 1 plus I minus 1 the one who pants will leave you with an i should say Mindy and so therefore V is also 1 minus B D and D have to add one if V is at one nine zero repeating the bar here means repeating then D would be point zero nine or even and two one it also turns out that I times V equal Dean's the product of the effective is say annual interest rate times the discount factor equals this discount rate and onto the easy verification sometimes useful to note that Y minus B equals I times B that's kind of strange like it equals it begin equation okay no psychology here go by minus B you can verify this equals five times B I can sometimes be useful in it another thing that tells you decide the fact that these are equal in that I will be bigger than me because I times B will be positive therefore I might be really positive and therefore I a bigger than B and another point if you solve the original equation for I is function D you get I is e divided by 1 minus D and D is this cattle rate is certainly a way to represent these functions there is a bunch of different ways of representing the accumulation function with I the effective annual interest rate it's wonderful out of the T if Delta the force of interest that's the same as either delta T if V is the present value disco factor that's the same as V is a negative T and D is the effective on discount rate that's the same as 1 minus T to the greater T and in terms of compounding more than once per year you can use these kinds expressions and DM we represent the discount rate compounded more than once per year and the present value is the reciprocal of all those things either negative 1 to be the t is either negative delta t is 1 minus t to the positive t is 1 plus size see and you have these kinds of relationships as well down there these are all different ways of representing the same functions and you'll see in different various problems that they are all used in one way or another and you want almost you want to know this in your gut and just be able to use these kind of equations whenever you need final topic in Craig done in less than an hour here again I'm covered and a broad perspective pretty much everything in Chapter one of this book the last topic is a little bit of a discussion about um there's a little bit of a discussion about inflation there we go final topic what is the real interest rate in the context of inflation money losing real valley how should real growth the measure making this video in 2017 interest on savings accounts to really pitiful right like 1/100 of 1% per year just it's just terrible compounded four times a year yeah I knew when you're you're lucky if you get a dollar of interest over the course of year I you can't keep up with inflation inflation is not too bad these days but it's certainly more than point zero one percent you know of inflation generally speaking the prices of goods is going up by maybe two or three percent per year meaning your buying power is going down over time so if you're investing at low interest rates you can't keep up with inflation ideally you want your interest that you're getting from your investments to be higher than inflation but if that's even if that's true you're you're not getting as much money as you as helpful the if are the annual inflation rate and I is the annual interest rate for any investment then the real rate of interest is given by this equation I so real is I might as hard / 1 + are all why is this makes sense and why is it important our final issues to deal with here let's take an example say you have a hundred let's go ahead and call them dollars a hundred dollars at the start of the year to invest and your money grows by a great interest rate an effective annual interest rate of 20% 0.2 you're going to have 120 dollars at the end of the year but maybe inflation is kind of bad - maybe inflation is 10 percent so that means from a practical level that most people think of prices going are going out on average by about 10 percent now hundred dollars is going to buy something at the beginning of the year that it's going to take 110 dollars to buy at the end of the year why is something at CP equals zero that takes 110 dollars to buy at the end of eternity the inflation rate R is 10 percent 0.1 so how much did your money really grow as far as buying college people might initially think if I - are 20% less 10 percent 10 percent that's not quite right because a hundred and twenty is not ten percent more than other than 110 is temporal ten more than 100 120 is not ten percent more than 110 you have to figure out how much more than 20 is as a percent of undertand to figure out the percentage you're buying our first body make 120 is on your knee 10/10 10/10 111 which is reserved and repeating of leaves on 9.09% is how much you're buying probably not 10% and that is the same thing you're going to get to do I minus R divided by 1 plus R point two minus point one is the exact same thing or about personal so you're buying power didn't go up by 10% it went up by about 9.09% a little bit less okay that's the intuition behind why you need to divide by one this is really important actually we won't be thinking about much as we work through problems and talk about ideas in this book but in the grand scheme of things in real life no pun intended this is real important no pun intended on but maybe I guess I what kind of anyway that's it for lecture one broad overview of chapter one again look at the problem-solving videos if you weren't able to give the problem solving that is the name of the game ultimately in passing actual real examples Thanks

Channel: Bill Kinney

Views: 58,458

Rating: 4.9556103 out of 5

Keywords: actuarial science lecture, actuarial mathematics, financial mathematics, actuarial science, financial math, effective interest rate, linear growth model, differential calculus real life applications, discount factor, compound interest, future value, accumulated value, present value, accumulation function, force of interest, relative rate of growth, continuous growth rate, linear growth vs exponential growth, accumulation function interest, integrate force of interest, actuary

Id: e9smoDy8gJQ

Length: 52min 11sec (3131 seconds)

Published: Mon May 22 2017