 Hey guys, it's MJ the student-actory and in this video we're going to be looking at Immunization so this is part 2 of chapter 14 for course CT1 Now immunization It's a bit weird. It's kind of confusing the first time you look at it and At first glance it doesn't make much sense at all But what you're going to find is that it is very mathematical and it actually is very easy If you draw your timelines So what we're going to do is before we get into all the mathematics You can see we're going to be looking at stuff like convexity, which has got the double derivative we're going to be looking at volatility and Duration mean term and all these weird and wonderful things before we dive into that Let's just take a step back. So what I've got here. It's a program called affinity designer and what I'm going to do is I'm going to go back to beginning I'm going to go back to the beginning of the course because it's important that we understand the basics so that we understand Immunization and the basics of the course is it's very simple. It's if we have a cash flow So the length of this represents the magnitude. So this could be like say a hundred dollars If we have a cash flow in the future So you can see this is time zero and this is time future If we have a cash flow in the future, how much is it worth in the beginning and What we've learned throughout this course is that We take this amount and it becomes a little bit smaller and This amount here is equal to that amount there. So this amount these two amounts are exactly the same When you take the time value of money into consideration The difference between them, so if I had to draw let's say with a different color Oh, no, we keep that color there, but I want to make a little value there this red part here is what the interest component is and What makes the subject interesting? excuse the pun is that We could have this situation Or we could have a situation like this So these two colorful blocks here Either one of them could be that value at time zero where red represents the interest that will be earned and blue represents the present value The amount of this red value Depends on what the interest rate is now how people actually get to the interest rate that is a very philosophical Discussion and in my opinion, it's kind of just made up by the central government Depending on what economic motors they want to chase. So that's why this value here This red value can change and that's what immunization seeks to protect your portfolio against Immunization was developed by an actually called Reddington British Actory and he noticed that Asset portfolios were sensitive to changes in interest rates because if you have a payment You know in the future and the interest rate is very high Then that amount that you need in beginning is very low but if interest rates are low, then you're going to need a much bigger value here and Vice versa the amount of money you invest now depending on the interest rate will affect the amount of money You actually receive at the end So in order to combat this interest rate sensitivity that is in your portfolio Reddington devised three rules that should be implemented to protect your portfolio against these changes And you want to do this because the fact that interest rates can just change Can mean your company or your business or even yourself could lose lots of money So it's a good idea to immunize your portfolio So with the basics done and understanding that money has a time value Money's worth more worth more in the present than it is in the future because of interest Let's dive back into the mathematics. So Let's come back to to my mathematics and before we get into the The you know the nitty-gritty of it. Let's look at one of the three conditions that Reddington wants for immunization He says that the present value of the assets must equal the present value of the liabilities Now straight away when you think about it You could actually relax this so that the present value of the assets should be greater than or equal to the present value of the Liabilities, but present value of assets equal present value of liabilities It's a fair statement to make as you'll you want to chase the cheapest way to immunize Your portfolio Because if your present value of your assets was much much bigger than your present value of the liabilities You wouldn't really have to worry, but they'd say your assets and your liabilities are the same What you want is the volatility of your assets and your liabilities to be the same and you want the Convexity of your assets to be greater than the convexity of your liabilities Now these are the three conditions you need to hold so that you can be protected against changes in the interest rate Now this isn't perfect. I mean straight away. We can see there are five big problems with Reddington's Theory it requires constant rebalancing of your asset portfolio. So that's going to be expensive Sometimes cash flows are uncertain. So it's really difficult to do the maths the assets that we're going to see that we might need may not even exist and It only really gives you protection against a small chunk of interest rate change And it assumes that the the yield curve is flat and we saw in the previous video that that isn't the case so straight away we know that this isn't a perfect system, but it is very elegant and We're going to go look into the mathematics and how you can do these exam questions I will end off with an example just so that we we all end on the same page But first of all present value of assets equal present value of liabilities You've been doing that the whole course. So I'm not going to go into that That is like the equation of value that was chapter 12 chapter 13 all of those ones What we're going to do now is just look at the mathematics behind point 2 and point 3 because it is a little bit tricky The first time you come around it so first thing we look at is Discounted mean term or otherwise known as DMT although I don't recommend that you Google DMT because you will find a psychedelic drug and not this mathematical concept So don't Google DMT Google discounted mean term if you want more information on it Okay, and what it is basically saying it is the average time of the cash flows Waited by their present value. So if I was to just show that graphically Let's just get rid of these things over here if I had Just one cash flow and let's say this at time five then the discounted mean term would be five but if I had one at five and one at Seven years then the discounted mean term would be six years five seven middle six But where this gets a little bit more sophisticated is it takes in the weight of the cash flow So if this cash flow here was double the amount so time seven It's double the amount of time five then the discounted mean term would be more like Six and a half or six point seven five or something like that It would lean more towards the heavily weighted cash flows now This is all easy in this case, but what you're gonna see with bonds is you have This let me actually draw the the normal cash flow pattern of a bond And you'll see why you actually have to use mathematics to figure this thing out Just bear with me as I copy and paste Okay, so this is how a normal bond looks you have your coupons and then you have your redemption amount So your discounted mean term needs to factor in that the redemption amount is normally much heavier Or actually draw to scale It'll be something like that So you can see the discounted mean term would lie somewhere Somewhere over there, but you don't have to guess That's why you use mathematics so that you don't have to guess so let's go back to the mathematics So what you're doing is you're taking the cash Flow value you're discounting it at multiplying it by its time and then you're just dividing it to get that weight And this gives you the DMT I think that symbol is called tau and it is connected to volatility by the following formula I'm not going to go too heavy into the mathematics and why it gets that but Please feel free and actually do think about why that formula makes sense. It'll be a good example for you Then I mean there is a quick example You can pause and just read through it The tricky part here is that coupons are paid half yearly So your DMT will give you a value in half years and so the final step you need to do is just convert to 4.1 What you will notice by looking at the maths here is that by taking By multiplying here the The time by the cash flow and because time is increasing, you know, you've got time 1 time 2 time 3 time 4 You are going to be getting an increasing annuity Function to deal with and I know well I remember I used to struggle a little bit with this because it is a little bit harder to handle mathematically But do a lot of practice and it will become easy. And so that's how you can get the DMT the volatility modified duration, it's just a little bit more sophisticated and You can also calculate it another way so you can calculate it either using the DMT and then using this formula here dividing by the interest or You can just take the present value Take the derivative of it divided by the present value and you'll know you'll start seeing that this is very similar to calculus you're using derivatives you're looking at different curves and all this type of stuff and What the the idea of immunization is or what we're going to see happens is you want to balance your Cash flows with another type of cash flow So you'll have your asset cash flow and your liability cash flow and Immunization that you balance them will give you that protection against interest rate and I'll show you guys that On the timeline a little bit later on but feel free to pause Go through this example. I don't want to go through it because otherwise this video will be too long And I think you guys can all read I will do the final the final example altogether Finally, there is this thing known as convexity It measures the spread of the payments so the more spread out your cash flow is the higher the convexity and I mean this if you think about it logically Let's actually just draw it on a little timeline Let's say This is my cash flow over here The blue cash flow and then let's make another one or color should we make it orange? If we have these two cash flows, okay, so let's pretend they're two different assets The orange one has got a lower convexity To the blue ones okay because blue ones are much more spread out than orange on orange one is just situated on one point Which means that the orange one is much more sensitive to interest rate changes in the blue ones Because if we change the interest rate with the blue ones This guy will be affected a lot, but these guys are not so much which member interest rate has that type of curve It's like an exponential Curve so this won't be affected that much by change in interest rate This will be a little bit more a little bit more a little bit more this one will be a little bit more But this guy will be affected much greater and because he's also being affected He gets counterbalanced by this one. So that's where convexity comes in. I hope I explained it well You'll see the first time you're looking at immunization. It is weird It is it is weird It is a little bit spacey and stuff like that But the more examples you do you'll finally click and it'll all make sense as it is an easy part of the course In the sense that you just need to be strong mathematically You know you just take know how to take second derivatives and stuff like that So yeah, that's what the convexity does again. This is more calculus So when people like, oh, how does calculus help me later on in life? Well, if you want to become an actually you need to know it So, yeah, convexity is the second derivative volatility. We saw was the very first derivative so now that we have the basics of Immunization under our belt we understand that there are the limits and learn these five points I mean in the exam they will maybe ask you this and you can get to easy marks by just you know remembering these five points But I wouldn't encourage you guys also just to think about the points think about why may an asset not exist That's a nice thing to think about You know some exact assets just don't exist because the market is not sophisticated at the moment However, in the future those assets may exist I mean, it's nice to think about and you can really get lost in the thought Anyway, let's get to a typical immunization Exam type question. This one is quite easy What it's saying is that you have liabilities of 20,000 RAND due in 15 years time so in 20 in 15 years time you owe 20,000 RAND, okay now you need to set up your assets in such a way that you are Immunized or protected against changes in interest rate because remember if interest rates were to come down So if we had low interest rates it would mean that this liability is very expensive to us If interest rates are very high then this liability is very cheap to us And if you're not understanding that pause the video just think of that statement I've just said through and then click resume and we'll continue, but it's important to understand that but like I said that's The post post 13 chapters have been dealing with that idea So let's go back to our question. We owe 20,000 RAND in 15 years time Okay, and we're going to immunize it with two zero coupon bonds bond X and bond Y now bond X is going to be 10,000 RAND and At time 10 and now we need to figure out What what is the term needed for bond Y? So when when do we need to invest or what type of asset should we purchase in? order to immunize this position and and Sorry, I've still got a bit of a cough straight away Do you think Y is going to be greater than 15 less than 15? What do you think? Okay, it's good to make these these little guesses early on so that you can see If you're on sir at the end of the day if it's reasonable or not So back to the example. We've got a 20,000 RAND bond due in 15 years time We have a bond where people are going to pay us 10,000 back in 10 years time We can now set up another bond and we need to determine the term in order to give us immunization so What we can do straight away is we can figure out what is the present value of the assets equals the present value of the liabilities and we set up our equation of values our assets are 10,000 discounted by 10 10 years plus this value of Y and This must be equal to the bond discounted by 15 years So we know that the value of Y is going to be at time zero is going to be equal to 2165 bucks, but remember this is the value of Y at time zero Now what we're going to do is we're going to calculate the Discounted mean term of our assets and we're going to make them equal to our liabilities and What we do is we use that formula where we time the term by The cash flow amount and we're going to see that 10x because 10 is the duration of the first bond plus ny because n is the value It's the unknown that we're going to try to solve for and we're going to need to make it equal We're going to need to solve for n in this equation We know that when you add these values together, we're going to get something like this and We see that n is equal to 26 comma 7 years So like we see we made the guess here We had to put the value here Which you should have you should have realized because remember we want the discounted mean term to be balanced If this was a little bit shorter, we'd have to balance it by putting a value here It's like a little bit like a seesaw or some scales So hopefully you guys did realize that you had to put the value later We see that n is bigger than 15 so it makes reasonable sense and now straight away We can see that the spread of the assets is greater than the spread of the liabilities in the sense that the liabilities is a single amount and Our assets are two amounts, which means remember my blue and orange cash flows. It has the greater convexity So you can actually just say that in the exam you can get away if if you use general reasoning like I've just done there You can say the spread of the assets is Greater than the spread of the liabilities and therefore its convexity is greater and then you can conclude by saying Immunization is achieved so you didn't actually even have to go out and do that second calculation But that is because this was a very simple example So let's actually just end of this video by drawing this Thing here on our timeline. So Let's just delete all of these here. So, yeah, we have the 20,000 red bond over there and What I'm going to do is just raise it up here. So we've got this 20,000 red bond there and By setting up our one bond Over here, I should actually set put these ones on the top and orange one up below because those are assets and Those are our liabilities You can see I think this one's even like a little bit smaller and It's going the future there. So this is what it looks like graphically and if this was like say a seesaw scale Immunization kind of shows that the portfolio has been balanced and when it is balanced It is less sensitive to a change in interest rates and that you are that kind of is immunization and that ends off Chapter 14 of course CT1 Actual science. Thank you guys so much for watching. If you do have any questions, please let me know in the comment section I will do my best to to answer or explain any point that I maybe went About too quickly and if someone does ask a question and you know the answer Please feel free to to help them out and job to reply to that comment But yeah, we've got One more video to do for CT1 and then the course is done. I know the exam is very very soon So I am going to get on to making that video straight away But yeah, thanks for watching guys. Cheers