Relationship between bond prices and interest rates

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Relationship between bond prices and interest rates

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 vs. interest rates.

So to start off I'll just start with a fairly simply bond, one that does pay a coupon.

And we'll just talk a little bit about what you would 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 from the US government.

Let's say it's a bond for $1000, let's say it has a 2 year maturity 2 year maturity - and let's say it has a 10% coupon 10% coupon paid semi-annually so this 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 twenty-four months in the future.

Half-way is twelve months, then this is eighteen months and this right here is six months.

And we went over a little of this in the "Introduction to Bond" video but it's a 10% coupon paid semi-annually, so it will pay us 10% of the par value per year, but it's going to break it up into 2 six-month payments.

So 10% of 1000 is $100.

So they are going to give us $50 every 6 months.

They are going to give us half of our 10% coupon every 6 months.

So we're going to get $50 here $50 here these are going to be our coupon payments - $50 there and finally at two years we'll get $50, and we'll also get the par value of our bond!

We'll also get $1000.

So we'll get $1000 plus $50, 24-months from today.

Now, the day that this, let's say that it is today we are talking about, the bond is issued.

And you look at that and say, you know 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 a 10% coupon is just about perfect.

So you say, I think I will pay $1000 for it.

So the price of that bond the price of the bond right when it gets issued, or on day 0 if you will, you would be willing to pay $1000 for it.

cause you say, you look I'm getting roughly 10% a year, 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, obviously things don't, interest rates don't move that quickly... but the moment after you buy that bond, or let's be a little bit more realistic 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 rates go up.

And let's say that they go up in such a way, now that they go up for this type of a company, for this type of a risk, you could go out on the market and get 15% coupon.

For this type of risk you would now expect, you would now expect a 15% interest rate.

Interest rate. Obviously for something less risky you would expect less interest.

And now for a company just like Company A you would now expect a 15% interest rates.

So interest rates have gone up.

Now let's say you need cash, and you come to me, you say, "Hey Sal, are you willing to buy this certificate off me?

I need some cash; I need some liquidity. I can't wait for the 2 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 $1000, because this bond is only giving me 10%, I'm expecting 15%.

So I want to pay something less than $1000, that after I do all the fancy math in my spreadsheet, it'll come out to be 15%.

So I'm going to 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, that if interest rates go up, someone willing to buy that bond, they'll say: "Gee, that's not the 15% coupon I can get on the open market, I'm going to pay less than $1000 for this bond."

So the price, the price will go down.

Or, you can essentially say, the bond would be trading at a discount to par.

Bond would trade a discount!

At a discount. to par.

Now, let's say the opposite happens.

Let's say interest rates go down.

Let's say we're in a situation where interest rates interest rates go down.

So now, for this type of risk - like 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 I had to just go to companies issuing their bonds, I would have to pay $1000, or roughly $1000, for a bond that only gives me a 5% coupon.

Roughly, give or take, I'm not being precise with this math, I just want to give you the gist of it.

So I would pay $1000 for something giving a 5% coupon now, this thing is giving a 10% coupons, 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 a 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... 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.

I'm going to 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 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 face value - the par value - is $1000 2 years from today.

There's no coupons.

So if I were to draw a payout diagram, it would just look like this.

This is today, this is one years, this is two years.

You just get $1000.

Now let's say that one day one, interest rates for a company like Company A this is Company A's bonds.

So this is starting off, day 1.

Day 1. Let's say people's expectations for this type of bond, is they want 10% per year, 10% 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 $1000 in 2 years.

So the way to think about it is, let's P in this. I'm going to do a little math now, and hopefully it won't be too bad.

So let's say P is the price that someone would be willing to pay for a bond.

So whatever price that is, if you compound it by 10% for 2 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 2 years, if I'm getting 10% on my initial payment, or the innial amount that I'm paying for my bond.

So this should be equal to the $1000.

So let me just be very clear here: Let's P is 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 2 years, that should be equal to $1000.

So if you do the math here, you get P times 1.10 squared is equal to 1000 or P is equal to 1000 divided by 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 $1000 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.

Let's get the calculator out.

So if we have 1000 divided by 1.1 squared.

That's equal to $826. and, well I'll just round down $826.

So this is $826.

So if you were to pay $826 for this bond and in 2 years, that company would give you back $1000, you will have essentially go a 10% 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 2 years out, and 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 second 1.

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, interest, people expect more Interest goes up.

The next expectation is to have a 15% return on a loan to a company like Company A.

So now what's the price we would be willing to pay?

We'll use the same formula.

The Price is going to be equal to $1000 divided by, instead of discounting it by 10%, we're going to discount it by 15% over 2 years.

So 1 plus 15% compounded over 2 years Bring out the calculator.

Bring out the calculator, and I think you have a sense, if you have a larger number now in the denominator, so the price is going to go down.

So let's actually calculate the math: $1000 divided by 1.15 squared is equal to $756 dollars, give or take a little bit.

So now the price has gone down.

The price is now $756.

This is how much someone will willing to pay in order for them to get a 15% return, and get a $1000 in 2 years, or get a $1000 in 2 years and essentially for it to by a 15% return.

Now just to finish out the argument, let's look at if interest rates go down.

So let's say Interest, the expected interest rate on this type of risk, goes down.

Now let's say it's now 5%.

What is someone willing to pay for this $0 bond?

Well the price, if you compound it 2 years, by 1.05, that should equal 1000.

Or the price is equal 1000.

Divided by 2 years of compounding at 5%.

Get the calculator out again.

We get $1000 divided by 1.05 squared is equal to $907 dollars.

So all of a sudden we're willing to pay, the price is now $907.

So you see mathematically, when interest rates go up, the price of the bond went from $826 to $756.

The price went down.

When interest rates went down, the price went up.

I think it makes sense: the more you expect, the higher the return you expect, the less you are willing to pay for that bond.

Anyway, hopefully you found that helpful.

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Relationship between bond prices and interest rates발음듣기

다른 번역문 등록

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.발음듣기

다른 번역문 등록

So bond prices vs. interest rates.발음듣기

다른 번역문 등록

So to start off I'll just start with a fairly simply bond, one that does pay a coupon.발음듣기

다른 번역문 등록

And we'll just talk a little bit about what you would be willing to pay for that bond if interest rates moved up or down.발음듣기

다른 번역문 등록

So let's start with a bond from some company.발음듣기

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So let me just write this down.발음듣기

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This could be Company A It doesn't just have to be from a company - it could be from a municipality, or from the US government.발음듣기

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Let's say it's a bond for $1000, let's say it has a 2 year maturity 2 year maturity - and let's say it has a 10% coupon 10% coupon paid semi-annually so this semi-annual annual payments.발음듣기

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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.발음듣기

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So this is today, let me do it in a different color, that's today, let me draw a little timeline right here.발음듣기

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This is two years in the future, when the bond matures.발음듣기

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So that is twenty-four months in the future.발음듣기

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Half-way is twelve months, then this is eighteen months and this right here is six months.발음듣기

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And we went over a little of this in the "Introduction to Bond" video but it's a 10% coupon paid semi-annually, so it will pay us 10% of the par value per year, but it's going to break it up into 2 six-month payments.발음듣기

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So 10% of 1000 is $100.발음듣기

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So they are going to give us $50 every 6 months.발음듣기

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They are going to give us half of our 10% coupon every 6 months.발음듣기

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So we're going to get $50 here $50 here these are going to be our coupon payments - $50 there and finally at two years we'll get $50, and we'll also get the par value of our bond!발음듣기

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We'll also get $1000.발음듣기

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So we'll get $1000 plus $50, 24-months from today.발음듣기

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Now, the day that this, let's say that it is today we are talking about, the bond is issued.발음듣기

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And you look at that and say, you know 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.발음듣기

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So a 10% coupon is just about perfect.발음듣기

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So you say, I think I will pay $1000 for it.발음듣기

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So the price of that bond the price of the bond right when it gets issued, or on day 0 if you will, you would be willing to pay $1000 for it.발음듣기

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cause you say, you look I'm getting roughly 10% a year, then I get my money back.발음듣기

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10% is a good interest rate for that level of risk.발음듣기

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Now, let's say that the moment after you buy that bond, just to make things, obviously things don't, interest rates don't move that quickly... but the moment after you buy that bond, or let's be a little bit more realistic the very next day, interest rates go up.발음듣기

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If interest rates go up let me do this in a new color - so let's say that interest rates go up.발음듣기

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And let's say that they go up in such a way, now that they go up for this type of a company, for this type of a risk, you could go out on the market and get 15% coupon.발음듣기

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For this type of risk you would now expect, you would now expect a 15% interest rate.발음듣기

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Interest rate. Obviously for something less risky you would expect less interest.발음듣기

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And now for a company just like Company A you would now expect a 15% interest rates.발음듣기

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So interest rates have gone up.발음듣기

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Now let's say you need cash, and you come to me, you say, "Hey Sal, are you willing to buy this certificate off me?발음듣기

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I need some cash; I need some liquidity. I can't wait for the 2 years for me to get my money back.발음듣기

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How much are you willing to pay for this bond?"발음듣기

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Well I'll say, "You know what, I'm going to pay you less than $1000, because this bond is only giving me 10%, I'm expecting 15%.발음듣기

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So I want to pay something less than $1000, that after I do all the fancy math in my spreadsheet, it'll come out to be 15%.발음듣기

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So I'm going to pay, so the price, so in this situation, the price will go down.발음듣기

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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, that if interest rates go up, someone willing to buy that bond, they'll say: "Gee, that's not the 15% coupon I can get on the open market, I'm going to pay less than $1000 for this bond."발음듣기

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So the price, the price will go down.발음듣기

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Or, you can essentially say, the bond would be trading at a discount to par.발음듣기

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Bond would trade a discount!발음듣기

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At a discount. to par.발음듣기

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Now, let's say the opposite happens.발음듣기

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Let's say interest rates go down.발음듣기

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Let's say we're in a situation where interest rates interest rates go down.발음듣기

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So now, for this type of risk - like Company A - people expect 5%.발음듣기

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People expect 5% rate.발음듣기

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So how much could you sell this bond for?발음듣기

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Well if you weren't there and I had to just go to companies issuing their bonds, I would have to pay $1000, or roughly $1000, for a bond that only gives me a 5% coupon.발음듣기

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Roughly, give or take, I'm not being precise with this math, I just want to give you the gist of it.발음듣기

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So I would pay $1000 for something giving a 5% coupon now, this thing is giving a 10% coupons, so it's clearly better!발음듣기

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So now the price would go up!발음듣기

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So now I would pay more than par.발음듣기

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Or you would say that this bond is trading at a premium a premium to par premium to par.발음듣기

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So at least in a 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.발음듣기

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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.발음듣기

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Now let's actually do it with an actual... 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.발음듣기

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And to do this, I'm going to do what's called a Zero Coupon Bond.발음듣기

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I'm going to show you a Zero-coupon Bond.발음듣기

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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.발음듣기

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So zero-coupon bond is literally a bond that just agrees to pay the holder of the bond the face value.발음듣기

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So let's say the face value, the face value - the par value - is $1000 2 years from today.발음듣기

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There's no coupons.발음듣기

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So if I were to draw a payout diagram, it would just look like this.발음듣기

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This is today, this is one years, this is two years.발음듣기

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You just get $1000.발음듣기

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Now let's say that one day one, interest rates for a company like Company A this is Company A's bonds.발음듣기

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So this is starting off, day 1.발음듣기

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Day 1. Let's say people's expectations for this type of bond, is they want 10% per year, 10% per year, per year interest.발음듣기

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So given that, how much would they be willing to pay for something that's going to pay them back a $1000 in 2 years.발음듣기

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So the way to think about it is, let's P in this. I'm going to do a little math now, and hopefully it won't be too bad.발음듣기

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So let's say P is the price that someone would be willing to pay for a bond.발음듣기

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So whatever price that is, if you compound it by 10% for 2 years so I do 1.10, that's 1 plus 10%.발음듣기

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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.발음듣기

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This essentially is how much I should get after 2 years, if I'm getting 10% on my initial payment, or the innial amount that I'm paying for my bond.발음듣기

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So this should be equal to the $1000.발음듣기

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So let me just be very clear here: Let's P is someone who expects 10% per year, for this type of risk, would be willing to pay for this bond.발음듣기

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So when you compound their payment by 10% for 2 years, that should be equal to $1000.발음듣기

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So if you do the math here, you get P times 1.10 squared is equal to 1000 or P is equal to 1000 divided by 1.1 squared.발음듣기

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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 $1000 in two years, discounted by 10%.발음듣기

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This is 1.10 or 1 plus 10%.발음듣기

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So what is this number right here?발음듣기

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Let's get a calculator out.발음듣기

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Let's get the calculator out.발음듣기

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So if we have 1000 divided by 1.1 squared.발음듣기

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That's equal to $826. and, well I'll just round down $826.발음듣기

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So this is $826.발음듣기

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So if you were to pay $826 for this bond and in 2 years, that company would give you back $1000, you will have essentially go a 10% compounded interest rate on your money.발음듣기

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Now, what happens if the interest rate goes up, let's say the very next day?발음듣기

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And I'm not going to be very specific, I'm going to assume it's always 2 years out, and you know it's one day less, but that's not going to change the math dramatically.발음듣기

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Let's say it's the very next second that interest rates were to go up.발음듣기

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So second 1.발음듣기

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So it doesn't affect our math in any dramatic way.발음듣기

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So let's say interest rates go up, so now all of a sudden, interest, people expect more Interest goes up.발음듣기

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The next expectation is to have a 15% return on a loan to a company like Company A.발음듣기

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So now what's the price we would be willing to pay?발음듣기

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We'll use the same formula.발음듣기

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The Price is going to be equal to $1000 divided by, instead of discounting it by 10%, we're going to discount it by 15% over 2 years.발음듣기

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So 1 plus 15% compounded over 2 years Bring out the calculator.발음듣기

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Bring out the calculator, and I think you have a sense, if you have a larger number now in the denominator, so the price is going to go down.발음듣기

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So let's actually calculate the math: $1000 divided by 1.15 squared is equal to $756 dollars, give or take a little bit.발음듣기

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So now the price has gone down.발음듣기

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The price is now $756.발음듣기

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This is how much someone will willing to pay in order for them to get a 15% return, and get a $1000 in 2 years, or get a $1000 in 2 years and essentially for it to by a 15% return.발음듣기

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Now just to finish out the argument, let's look at if interest rates go down.발음듣기

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So let's say Interest, the expected interest rate on this type of risk, goes down.발음듣기

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Now let's say it's now 5%.발음듣기

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What is someone willing to pay for this $0 bond?발음듣기

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Well the price, if you compound it 2 years, by 1.05, that should equal 1000.발음듣기

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Or the price is equal 1000.발음듣기

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Divided by 2 years of compounding at 5%.발음듣기

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Get the calculator out again.발음듣기

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We get $1000 divided by 1.05 squared is equal to $907 dollars.발음듣기

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So all of a sudden we're willing to pay, the price is now $907.발음듣기

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So you see mathematically, when interest rates go up, the price of the bond went from $826 to $756.발음듣기

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The price went down.발음듣기

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When interest rates went down, the price went up.발음듣기

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I think it makes sense: the more you expect, the higher the return you expect, the less you are willing to pay for that bond.발음듣기

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Anyway, hopefully you found that helpful.발음듣기

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