hi this is the first of several videos where we're going to look at symbolic examples it's I think important to be able to symbolically represent values of annuities and understand the symbols first and then we'll work on numeric values for those symbols and later videos later in the course but for now let's look at an example where we're symbolically representing the value of an annuity for this particular annuity we've got 17 payments the payments are for I see their monthly payments and I could describe that valuation date as what three months before the first payment now I don't have a symbol one symbol that represents the value of the annuity at that valuation date so I'm gonna use an intermediate valuation date I'll put a dotted line where that intermediate valuation date is and I'm gonna choose there because I have a symbol to represent the value of that annuity at that valuation date at the dotted line at 17 payments of for one period before the first of those payments of four and so four times a angle seventeen is what I use to value that annuity at that dotted line and then of course I need to take that value and discount at two periods to get back to the valuation date and I would do that by multiplying by V squared so I could symbolically represent the value of this annuity by four times a angle seventeen times a V squared I wanted to introduce this because there's some actual notation that you're expected to know on the exam for this sort of this type of annuity that this type of annuity is called a deferred annuity I'll get to that the symbol for that that we represent so that a angle seventeen times V squared can be represented by the symbol in red there and that's read as a two deferred that vertical bars read as deferred to deferred a angle seventeen what's in red then is the in words the present value of a two-month deferred 17 month basic annuity immediately its two-month deferred because I'm thinking of it as an annuity immediate as immediate the first payment is at the end of the first period so the begin date would be at the at where the dotted line is and of course that's two months after the original validate valuation date so that's why I call it a two month deferred this Lots a two month deferred 17 month basic annuity immediate okay there's other ways that I could get an symbolic representation of the value of the annuity there for instance if I would have chose the intermediate valuation date at the time of the first payment which makes perfect sense to me to do because I could use the symbol four times in a double dot angle 17 as the symbol that represents the value of the annuity at the dotted line shown now and of course I would need to discount that for three months to get back to the original valuation date and I would do that by multiplying by V cubed likewise with the previous notation there's deferred notation for this too so the set 8 double dot angle 17 times V to the third I could write with this symbol the symbol in red would be the symbol in red would be red 3 deferred a double dot angles 17 and now I'm thinking of this as a 3-month deferred 17 month basic annuity due it's three month deferred because I'm thinking of it as an annuity due with that annuity do the first payment as at the beginning of the first period and so the start date would be where the dotted line is and of course that's three months after the original valuates valuation date so that's why it's a three month deferred annuity do okay so that was the validate that I used here those were two the last two expressions in are perfectly acceptable expressions there's lots of different ways to do this and you could even do some odd thing you might just be an odd person and say hey I'm going to use a valuation date here at the time of the last payment that's actually I'm ok with that because the value of the annuity at that time is a four times an S angle 17 we have a symbol to represent the value there now you're gonna you're gonna have to figure out how many periods to discount that back to to get to the original valuation date and if you take off your shoes and socks do whatever you have to do count and you'll see that there are 19 periods that you're going to have to discount that back in order to get to the original valuation date so the value of the annuity would be a 4 times s angle 17 times a V to the 19th okay so let's look at the expressions that we have then we have this four times angle 17 times B squared that's that's one way we could value the annuity that's one expression that we could use to value the annuity we also had a four times a double dot angle seventeen times V cubed I wanted to show you in this slide the reason that those two are going to give you the same answer is because remember the a angle 17 is an a double dot angle 17 times V so in the first expression that we have the four times a angles 17 times V squared if for a angle 17 you would substitute in a double dot angle 17 times V then group that V with the V squared yell and end up with the second expression that we have there and then finally we we did this one again that's not natural to me but that might be natural to you four times an S angle 17 times a V to the nineteenth again I want to convince you that you're going to get the same numeric value when we do eventually get to calculate numeric values for these things and I want to show you that by looking at the relationship between an angle 17 and an S angle 17 remember those valuation dates are 17 periods apart so I'd have to discount the S angle 17 17 periods I would do so by multiplying by V to the 17th to get an angle 17 so in the expression the top expression in red if for the angle 17 if you would substitute in that S angle 17 times B to the 17th group the V to the 17th with the V squared you'll get the V to the 19th in the bottom expression in red so those are all equivalent expressions in the sense that you're going to get the same numeric value a regardless of which one you use okay but to me the more natural choices would be one of these two and and they're kind of equally equally natural for me okay let's look at another example in this example let's symbolically represent the value of the perpetuity shown at the valuation date so I can describe this since the payments are monthly I'm sorry since the timeline is monthly the payments or every three months are quarterly so I could describe this as a quart quarterly payments of seven forever valued what two months before the first payment that's how I could describe this picture now the payments are quarterly so I know I'm gonna have to use a quarterly effective interest rate somewhere in the problem so let me just denote that by Q once again I don't have a symbol a single symbol to represent the value of this perpetuity at the valuation date so I have to take some intermediate valuation date let me take the intermediate value equation date at the time of the first payment because the symbol that I would use to represent the value of the perpetuity there would be a seven times in a double dot angled infinity now I'm thinking of this as taking that number and I need to discount it for two months so in my mind I'm thinking I'm gonna be multiplying this by a V squared because I'm discounting for two months but that's months and so now I need to introduce another notation for the or another symbol for the monthly effective interest rate let me use an M for that now that I have two interest rates I need to be careful about I have a Q&M I got to be careful about what the a double dot angle infinity is with respect to those payments are quarterly so this is R if the the symbol there is with respect to the quarterly effective interest rate and now I'm going to discount that for two periods by multiplying by a V squared but the V squared is with respect to the M interest rate now remember the V when you have periodic effective interest rates like an eye given an eye the V is a 1 over a 1 plus I that's the periodic discount factor so using an M I would have just a V with respect to M would be a 1 over a 1 plus M that would be the monthly discount factor and that's what I would be using in this particular picture okay so now let's go back to the to the intermediate valuation date and I've got a seven times in a double dot England as the value of this perpetuity at the first pay at the time of the first payment and I said we're going to discount for two months now we could be thinking of the two months it's just two thirds of a quarter and if I do that then the V to the M with respect to remember V is 1 over 1 plus I so V with respect to M will be V over 1 plus M that will be the monthly discount factor if I change the M to a queue then a V with respect to Q would be the quarterly discount factor that would be 1 over a 1 plus Q and now at this point I'm not I'm discounting for again two months but the two months are two thirds of a quarter and so I would multiply by a V to the two-thirds power in order to get the expression at the valuation date the value of this into this perpetuity at evaluation day so this is a perfectly acceptable expression also notice that this is kind of a cleaner expression in the sense of I'm only using one interest rate the Q and so I don't really need to put the decorations and the extra symbols of the Q and and and so forth on like I did when I had two interest rates floating around so on this slide I have the that the a double dot symbol is with respect to Q and the V squared symbol is with respect to M but when only using the Q now I don't really need those extra decorations of course you could put them there but they're not needed they make it look a little bit more complicated than it really is in my opinion okay so that's a that's another way that I could write or that's a way I could write the value of the annuity by only using with only using the Q quarterly effective interest rate let's go back there's one more one more kind of mistake I want you to be careful of so I want to point that out to you a common mistake I see students may we're trying to value this perpetuity at the valuation date and students will want to use the symbol seven times an a angle infinity and sometimes they might think that's giving me a value that one month before the first payment of seven they think of that as one period before the first payment of seven but you got to be careful the seven the payments are seven or quarterly this is not right the payments of seven or quarterly so the a symbol there is with respect to a quarterly effective interest rate which means it's giving me a value seven times a angle infinity has given me a value one quarter before the first payment of seven the payments of seven are quarterly so a angle infinity has given me the value one quarterly period before so that would be where I have it shown and now I need to accumulate that for one period to get to the valuation date I can accumulate in four period as month so I can think of this as well I'm gonna accumulate it for one month well if I'm going to think of it that way that I'm accumulating it for one month then I need to introduce the symbol the monthly effective interest rate and then accumulate one month by multiplying by a one plus M now remember the V value is with respect to M as 1 over a 1 plus M and so I could actually even use V notation here too and say I'm accumulating the 1 plus M then would be the reciprocal of V I could use negative exponents of V in other words to accumulate when I'm taking a V to the positive power I'm discounting when I take V to a negative power I leukemia I'm accumulating so this would be times of e to the minus 1 notice all the extra decorations that I have though again I've got the the Q on the a angle infinity symbol I have the subscript of ill on the V that's all necessary to describe what I'm doing now on the other hand if I want to again use the seven times a angle infinity symbol to represent the value of the annuity one period before one month before the first payment I could think of accumulating one month as accumulating one third of a quarter and so accumulating one third of a quarter would mean I would multiply by a 1 plus Q to the 1/3 power and again with V being a 1 over 1 plus Q I could use a V notation here and say well that's a V to the minus one-third and notice I don't have to use extra symbols because I've only got one one interest rate in the problem there so there's no confusion on what interest rate is using is being used with the a symbol and what interest rate is being used with a V symbol it's just one one interest rate they're the quarterly effective interest rate so both of these would be acceptable expressions the seven times a little infinity times V to the minus one-third or seven times a double dot angle infinity times of e to the two thirds both are acceptable expressions for the value of the annuity R this case the value of the perpetuity at the valuation date okay so these are good examples they're going to show up again later on the course where we're looking for numeric values of these types of annuities and perpetuate ease so we'll do some more symbolic examples in the next video I'll see you then [Music]