Relationship between bond prices and interest rates Why bond prices move inversely to changes in interest rate
Relationship between bond prices and interest rates
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- [MUSIC PLAYING]
- What I want to do in this video is to give a not too
- mathy explanation of why bond prices move in the opposite
- direction as interest rates.
- So bond prices verses 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 A.
- It doesn't just have to be from a company, it could be
- from a municipality or it could be from the U.S.
- And let's say it's a bond for $1,000, let's say it has a
- two-year maturity, and let's say that it has a 10% coupon
- paid semi-annually.
- So, if we just draw the diagram for this-- obviously I
- ran out of space on the actual bond certificate-- let's draw
- a diagram of the payments for this bond.
- So this is today-- let me do it in a different color-- 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.
- And 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 10% of the par value per year, but it's going
- to break it up into two six-month payments.
- So 10% of $1,000 is $100, so they're going to give us $50
- every six months.
- So they're going to give us of half of our 10% coupon every
- six months.
- So we're going to get $50 here, $50 here, these are
- going to be are coupon payments, $50 there, and then
- finally at two years will get $50.
- And we'll also get the par value of our bond.
- We'll also got $1,000 plus $50 24 months from today.
- Now, let's say this is today that we're talking about, the
- bond is issued.
- And you look at that and 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 you say you know what, I think I will
- pay $1,000 for it.
- So the price of the bond right when it gets issued, or on day
- 0 if you will, you'd be willing to pay $1,000 for it,
- because you say, look you know I'm getting roughly 10% a year
- and then I get my money back.
- 10% is a good interest rate for that level of risk.
- Now, let's say that the moment after you buy that bond-- just
- to make things a little bit-- obviously 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.
- 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% coupons.
- So let's say for this type of risk you would now expect a
- 15% interest rate.
- Obviously for something less risky, you'd expect less
- interest. In fact, for a company just like Company A,
- 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, 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 going to pay you less than
- $1,000 because this bond is only giving me 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 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 going to pay less than $1,000 for this bond.
- So the price will go down.
- Or, you could essentially say that the bond would be trading
- 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 go down.
- So now, for this type of risk like Company A,
- 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 to 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 $1,000 for something giving a 5% coupon
- now, this thing's 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 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 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 going to do what's called
- 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.
- 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, for interest rates for a company
- like Company A-- this is Company A's bonds-- let's say
- people's expectations for this type of bond is they want 10%
- per year interest. So given that, how much would they be
- willing to pay for something that's going to pay them back
- $1,000 in two years?
- So the way to think about it is, and I'm going to do a
- little bit of math now but hopefully it won't 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% for two
- years, so I do 1.10, that's 1 plus 10%.
- So after one year, if I compound it by 10% it'll be p
- times this.
- And then after another year, I'll
- multiply it by 1.10 again.
- This essentially is how much I should get after two years if
- I'm getting 10% on my initial payment, or the initial amount
- that I'm paying for my bond.
- So this should be equal to the $1,000.
- So let me just be very clear here.
- 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% for two years,
- that should be equal to $1,000.
- So if you do the math here, when you get P times 1.1
- squared is equal to 1,000.
- Or P is equal to 1,000 divided by at 1.1 squared.
- Another way to think about it is the price that someone
- would be willing to pay if they expect a 10% return is
- the present value of $1,000 in two years discounted by 10%,
- this is 1.10 or 1 plus 10%.
- So what is this number right here?
- Let's get a calculator out.
- And so if we have 1,000 divided by 1.1 squared, that's
- equal to $826 and well I'll just round down, $826.
- So if you were to pay $826 today for this bond, and in
- two years that company would give you back $1,000, you will
- have essentially have gotten a 10% 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 going to be very specific, I'm going to 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 the very next second that interest
- rates were to go up.
- So 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 people expect more.
- The new expectation is to have a 15% return on a loan to a
- company like Company A, so now what's the price we are
- willing to pay?
- We use the same formula.
- The price is going to be equal to $1,000 divided by-- instead
- of discounting it by 10% we're going to discount it by 15%
- over two years.
- So 1 plus 15% compounded over two years.
- Bring out the calculator.
- And I think you have a sense, we have a larger number now in
- the denominator, so the price is going to go down.
- So let's actually calculate the math. $1,000 divided by
- 1.15 squared is equal to $756, give or take a little bit.
- So now the price has gone down.
- The price is now $756.
- This is how much someone is willing to pay in order for
- them to get a 15% return and get $1,000 in two years.
- Or get $1,000 thousand dollars in two years and essentially
- for it to be a 15% return.
- Now, just to finish up the argument, what happens if
- interest rates go down?
- So if let's say the expected interest rate on this type of
- risk goes down.
- Let's say it's now 5%.
- What is someone willing to pay for this zero-coupon bond?
- Well the price, if you compound it two years by 1.05,
- that should be equal to 1,000.
- Or the price is equal to 1,000 divided by two years of
- compounding at 5%.
- Get the calculator out again.
- We get $1,000 divided by 1.05 squared is equal to $907.
- So all of a sudden we're willing to pay,
- price is now $907.
- So you see mathematically, when interest rates went up,
- the price of the bond went from $826 to $756,
- 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.
- [MUSIC PLAYING]
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