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# Relationship between bond prices and interest rates

AP.MACRO:
MEA‑3 (EU)
,
MEA‑3.A (LO)
,
MEA‑3.A.3 (EK)

## Video transcript

what I want to do in this video is to give a not too mathy explanation of why bond prices why bond prices move in the opposite direction as interest rates so bond prices versus interest rates so to start off I'll just start with a fairly simple bond one that does pay a coupon and we'll just talk a little bit about what you'd be willing to pay for that bond if interest rates moved up or down so let's start with a bond from some company so let me just write this down this could be company company a it doesn't just have to be from a company it could be from a municipality or it could be from the US government and let's say it's a bond for $1,000 let's say it has a two year maturity two year maturity and let's say that it has a 10% coupon 10% coupon paid semi-annually so this is semi-annual annual payments so if we just draw the diagram for this obviously I ran out of space on the actual bond certificate but let's draw a diagram of the payments for this bond so this is today let me do it in a different color that's today let me draw a little timeline right here this is two years in the future when the bond matures so that is 24 months in the future halfway is 12 months then this is 18 months and this right here is 6 months now we went over a little bit of this in the introduction to Bond video but it's a 10% coupon paid semi-annually so it'll pay us ten percent of the par value per year but it's gonna break it up into two six month payments so 10% of a thousand is$100 so they're gonna give us $50 every six months they're getting give us half of our 10% coupon every six months so we're gonna get$50 here $50 here if these are going to be our coupon payments$50 there and and finally at two years we'll get $50 and we'll also get the par value of our bond will also get a thousand dollars so we'll get a thousand plus$50 24 months from today now the day that this let's say this is today that we're talking about the bond is issued and you look at that you say you know what for a company like Company A for this risk profile given where interest rates are right now I think a 10% coupon is just about perfect so 10% coupon is just about perfect you say you know what I think I will pay $1,000 for it so the price of the bond the price of that bond right when it gets issued or on day zero if you will you'd be willing to pay$1,000 for it can you say look you know I'm getting roughly 10% a year and then I get my money back 10 percents a good interest rate for that level of risk now let's say that the the moment after you buy that bond just to make things a little bit obviously things don't interest rates don't move this quickly but let's say the moment after you buy that bond or maybe to be a little bit more realistic let's say the very next day interest rates go up if interest rates go up let me do this in a new color so let's say that interest interest rates go up and let's say they go up in such a way that now that they've moved up for this type of a company for this type of risk you could go out in the market and get 15% a coupon so let's say for this type of risk you would now expect you would now expect a 15% interest rate interest rate obviously for something less risk you'd expect less interest in that but for a company just like company you would now expect a 15% interest rate so interest rates have gone up now let's say you need cash and you come to me and you say hey Sal can you know are you willing to buy this certificate off of me I need some cash I need some liquidity I can't wait for the two years for me to get my money back how much are you willing to pay for this bond well I'll say you know what I'm gonna pay you less than $1,000 because this bond is only give 10% I'm expecting 15% so I want to pay something less than$1,000 that after I do all of the fancy math in my spreadsheet it'll come out to be 15% so I'm gonna pay so the price so in this situation the price will go down and I'll actually do the math with a simpler bond than one that pays coupons right after this but I just want to give you the intuitive sense if interest rates go up someone willing to buy that bond they'll say gee this only gives a 10% coupon that's not the 15% coupon I can get on the open market I'm gonna pay less than $1,000 for this bond so the price price will go down or you could use essentially say that the bond would be trading at a discount to par bond would trade at a discount at a discount to par now let's say the opposite happens let's say that interest rates go down let's say that we're in a situation where interest rates interest rates go down so now for this type of risk like a company a people expect 5% people expect 5% rate so how much could you sell this bond for well if you weren't there and if I had to just go to companies issuing their bonds I would have to pay$1,000 or roughly $1,000 for a bond that only gives me a 5% coupon roughly give or take I'm not being precise with the math I really just want to give you the gist of it so I would pay a thousand dollars for something and giving a 5% coupon now this thing is giving me a 10% coupon so it's clearly better so now the price would go up so now I would pay more than par or you would say that this bond is trading at a premium a premium to par premium to par so at least in the gut sense when interest rates went up people expect more from the bond this bond isn't giving more so the price will go down likewise if interest rates go down this bond is giving more than what people's expectations are so people are willing to pay more for that bond now let's actually do it with an actual if actually do the math to figure out the actual price that someone a rational person would be willing to pay for a bond given what happens to interest rates and to do this I'm gonna do what's called a zero coupon bond I'm gonna show you a zero coupon bond and actually the math is much simpler on this because you don't have to do it for all of the different coupons you just have to look at the final payment so a zero coupon bond is literally a bond that just agrees to pay the holder of the bond the face value so let's say the face value the par value is$1,000 two years from today two years from today there's no coupon so if I were to draw a payout diagram it would just look like this this is today this is one year this is two years you just get \$1,000 now let's say on day one interest rates for a company like Company A this is Company A's bonds people so this is starting off so day one day one let's say people's expectations for this type of bond is they want 10 percent 10 percent per year per year interest so given that how much would they be willing to pay for something that's going to pay them back a thousand dollars in two years so the way to think about it is let's pee in this I'm gonna do a little bit of math now but hopefully we'll be too bad so let's say P is the price that someone is willing to pay for a bond so whatever price that is if you compound it by 10 percent for two years so I do 1.10 that's one plus ten percent so after one year if I compound it by ten percent will be P times this and then after another year I'll out multiply it by one point one o again this essentially is how much I should get after two years if I'm getting 10 percent on my initial payment or the initial amount that I'm paying for my bond so this should be equal to this should be equal to the thousand dollars so let me just be very clear here let's P is what someone who expects 10% per year for this type of risk would be willing to pay for this bond so when you compound their payment by 10 percent for two years that should be equal to a thousand dollars so if you do the math here where you get you get P times 1.1 squared is equal to 1,000 or P is equal to one thousand divided by 1.1 squared another way to think about it is the price that someone would willing to be willing to pay if they expect a 10% return it's the present value of one thousand dollars in two years discounted by ten percent this is one point 100 or 1 plus 10% so what is this number right here I think it's gonna calculator out the calculator out and so if we have 1,000 divided by 1.1 squared that's equal to 826 dollars and well I'll just round down 826 dollars so this is 826 dollars so if you were to pay 826 dollars today for this bond and in two years that company would give you back a thousand dollars you will have essentially have gotten a 10 percent annual compounded interest rate on your money now what happens if the interest rate goes up let's say the very next day and I'm not gonna be very specific I'm gonna assume it's always two years out you know it's one day less but that's not going to change the math dramatically let's say it's the very next second that interest rates were to go up so let's say second one so it doesn't affect our math in any dramatic way so let's say interest rates go up so now all of a sudden so interest people expect more interest goes up the new expectation is to have a 15% return on and it on a on a loan to a company like company a so now what's the price we're willing to pay well use the same formula the price is going to be equal to thousand dollars divided by instead of discounting it by 10% we're gonna discount it by 15% over two years so one plus 15% compounded over two years bring out the calculator bring out the calculator I think you have a sense we have a larger number now in the denominator so the price is gonna go down so let's actually calculate the math one thousand dollars divided by one point one five squared is equal to seven hundred and fifty six dollars give or take a little bit so now the price has gone down the price is now seven hundred and fifty six dollars this is how much someone's willing to pay in order for them to get a fifteen percent return and get a thousand dollars two years or get a thousand dollars in two years and essentially for it to be a fifteen percent return now just to finish up the argument what happens if interest rates go down so if let's say interest the expected interest rate on this type of risk goes down and let's say it's now five percent what is someone willing to pay for this zero coupon bond well the price is if you compound it two years by 1.05 that should be equal to thousand or the price is equal to a thousand divided by two years of compounding at five percent and get the calculator out again we get one thousand dollars divided by 1.05 squared is equal to nine hundred and seven dollars so all of a sudden we're willing to pay price is now nine hundred and seven dollars so you see mathematically when interest rates went up the price of the bond went from eight hundred twenty six to seven hundred fifty six the price went down when interest rates went down the price went up and I think it makes sense the more you expect the higher return you expect the less you're willing to pay for that bond anyway hopefully you found that helpful
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