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## AP®︎/College Macroeconomics

# Relationship between bond prices and interest rates

AP.MACRO:

MEA‑3 (EU)

, MEA‑3.A (LO)

, MEA‑3.A.3 (EK)

Bond prices and interest rates are inverseley related. Learn about the relationship between bond prices change when interest rates change in this video. Created by Sal Khan.

## Want to join the conversation?

- Mr. Khan said that if people expect interest rates to go up, they will be willing to pay less for a bond. This makes sense for bonds with coupons and zero coupons. He also said that if people expect interest rates to go down, they will be willing to pay more for the bond if the coupon is greater than the expectation. I dont understand this logic with a zero coupon bond. Why would people pay more if the interest rate dropped? His explanation is on12:58(20 votes)
- Hi Rbalafx,

LooQuanxiang's answer is good, but a bit unclear. I'll try to give a simpler answer, the way I understand it. Note also that my answer relates to zero-coupon bonds, which is what Sal is explaining about in his video.

First, the scenario for YOU:

==========

1. DAY 1: On the day that a bond certificate is issued, you go out and buy it. The certificate you have comes with:

- a par value of $1000

- a coupon rate of 10% per year

- a maturity period of 2 years

2. DAY 2: The next day, the interest rate in the market shoots up, all the way to 15%.

3. DAY 3: You decide that you don't want to hold onto the bond any more. Now, you want to resell it.

4. DAY 4: You find out that no one on the secondary market wants to buy your bond for the original price you bought it for, at $1000. Why? Because other people out there can easily buy a bond priced at $1000 that gives returns of up to 15%, which is higher than what your bond offers (only 10%). So now you have to sell your bond at a lower price, and here's how you'll set its price.

==========

Now, my explanation. While I do this, please keep in mind the following two concepts:

a) That you are the original purchaser of a bond, and anyone who buys the bond from you thereafter is a subsequent purchaser;

b) That a par value, for a subsequent purchaser, works as a "ceiling" price;

c) That this is a zero-coupon bond.

Now let's begin:

1. The**original**purchaser of a bond (that's YOU) usually gets his returns ON TOP of that bond's par value. (E.g. At a 10% coupon rate, you will eventually receive your $1000 principal, then $100 per year for every year you hold the bond.)

2. However, the**subsequent**purchaser of a bond (NOT YOU, this is another person) cannot hope to get (competitive) returns on top of the par value, the way you do. (Why? Because your bond only offers 10% returns -- this subsequent purchaser could instead just avoid you and get bigger returns by buying bonds with higher coupon rates of 15% in the open market.) So, if the subsequent purchaser still wants to purchase your bond from you while making returns (not losses), he MUST do so while simultaneously matching existing competitive market interest rates of 15%. To do this, he will offer to buy your bond at a lower price, so that he can work UP TO the bond's par value. (E.g. He buys your bonds at $756, then at 15% per annum, gets $122 per year for every year he holds the bond.)

3. Compare and contrast (1) and (2) above -- for the**original purchaser**, his returns come ON TOP of the bond's par value, while for the**subsequent purchaser**, his returns begin from a lower value UP TO the bond's par value. (Why, again you ask? Because the subsequent purchaser needs to do this in order to maintain returns similar to those he can get by instead purchasing bonds at prevailing market interest rates; otherwise, he will suffer losses when he buys your bonds. In the subsequent purchaser's simpler words: "Sure, instead of the bonds on the market, I can buy your bonds, no problem. But I still want my 15% returns like what the market is currently giving -- so if you really want me to buy your bonds, you will have to offer me a a lower beginning price. At 15% in returns, I'll just make my profits by working from there up to the original $1000 par value then." )

4. In summary, for a**subsequent purchaser**of a zero-coupon bond, its par value acts as a ceiling.

5. To illustrate my explanation more concretely (and be careful not to get confused here, but I'll clarify any possible confusions in Paragraph 6 further below, so don't worry):

a) If interest rates go up (e.g. from 10% to 15%), the price of the bond will be less than the par value of $1000, and GO DOWN: to $756.

- The logic: For the subsequent purchaser to achieve 15% returns (similar to competitive market rates), he must buy the bond at $756, and over the maturity period of the bond, collect interest returns of a higher 15% per annum ($122 per year) until he receives his full sum of the par value at $1000.

b) If interest rates go down (e.g. from 10% to 5%), the price of the bond will still be less than the par value of $1000, and GO DOWN: to $907.

- The logic: At this point, the coupon rates of other bonds on the market are lower than yours (theirs is 5%, your bond is 10%) -- i.e. your bond gives higher returns, so people deem your bond more attractive than the market and may want it more. However, if you price your bond too expensively, no one will want to buy it either. So, to compete against existing interest rates on the market, you have to match prevailing market rates, and sell your bond at a lower price, but not at too low a price. Switching perspectives, here's how it works for the subsequent purchaser: he will buy your bond at a higher sum ($907), then collect lower interest returns of only 5% per annum ($46.5 per year) until he receives the full sum of the par value at $1000.

6. Now, to explain one confusing aspect of (5) (a) and (b):

a) Note that in (5) (a) and (b) above, despite interest rates moving up and down, the moving prices of a bond COMPARED TO THE PAR VALUE still go in one direction:**down**. Why? Because, as I've said before, the par value acts as a ceiling. For a subsequent purchaser of the bond, the price of that bond cannot be**higher**than the par value. (Otherwise, you would be buying the bond at a loss, which defeats the whole point of buying a bond in the first place.)

b) HOWEVER, when interest rates move up and down, the moving prices of a bond COMPARED TO ITSELF will work inversely: they go both**up and down**. Thus, when interest rates go up, bond prices go down (e.g. to $705), and when interest rates go down, bond prices go up (e.g. to $907). The reason for this has already been explained above; to compete against interest rates on the market.

Always remember, in any case, that the reason why a bond's price moves is due to the need to remain competitive against the interest rates of other bonds already existing on the open market.

I hope this explanation helps out a bit. (If I'm mistaken, someone out there please correct me!)(105 votes)

- When talking about zero coupon bonds you use compound interest. When discussing bonds with a par value and scheduled/coupon interest payments compound interest is not used. Why the difference? Why doesn't a 2 year zero coupon bond at 10% sell for $800?(14 votes)
- The 10% coupon, 2 year, $1000 par bond sold for $1000 in a world where the fair interest rate was 10% APR, compounding every 6 months. If you just keep the cash from the coupons in your wallet, you get $1200, and you're not compounding your gains. However, that's
**your choice**to not compound. If, on the other hand, you took the coupon payments and invested them immediately in other assets that**also**got 10% APR (also compounded 6-monthly), then you would be able to compound your gains.

If you reinvested to compound your interest: after 2 years you would get 1000 + 50 + 50*(1.05) + 50*(1.05)^2 + 50*(1.05)^3 = 1215.51. The last coupon can't be reinvested at all before bond maturity, but the second-to-last one can be reinvested 1 time, the third-to-last one can be reinvested 2 times, etc.

Note, if you re-factor all of the terms of the equation, this is identical to 1000*(1.05)^4 = 1215.51, which is the original $1000 at 10% APR compounded every 6 months over the 2 year maturity term. So if you**choose**to compound your gains, it's identical.

(But since you have more flexibility with the coupons, your risk is lower than with a zero-coupon bond, and so the market trade value might be higher for the coupon bond, giving it a lower effective yield. But that gets into a different discussion of risk/reward valuation of maturity periods, which Sal discusses in the "Annual Interest Varying with Debt Maturity" video.)(6 votes)

- Hi Sal, thanks for the video!

Who or what controls this interest rate? Is it suply and demand?

Also is there any relationship between the interest rate in the bond market to the overnight interest rate that is set by the Federal Reserve bank?

Thanks!(7 votes)- Yes it's basically regulated by supply and demand. The company might issue bonds originally for a certain percentage rate that it feels comfortable with and see if anyone takes to them before adjusting to their results; or they might consult with a financial adviser who will tell them that similar companies issued bonds at a given rate and people bought them at that rate, so that is their target rate.(6 votes)

- Sal, how do interest rates change in the first place? What does central bank do to influence interest rates? I am confused because I can't find the link between interest rates and the yield on bonds. Yields pertain to bonds and interest rate is just a general term. Please assist..(4 votes)
- Interest Rate (the one which Sal mentions as going up or down) is the benchmark rate (In US, it is the Fed rate). This is the rate at which Federal reserve is willing to lend money to the banks. Depending on a complex set of factors, the Fed changes these rates.

Yield on bonds is basically the annual rate of return the bond holder gets. By definition, the rate of return would depend on how much you pay for it. For the first purchaser of a bond (who directly buys it from the issuing entity), yield is same as coupon rate. However, for secondary purchasers, yield depends on the price at which the bond was purchased. If it was purchased at a discount, then Yield > Coupon Rate. If the bond was purchased at a premium, Yield < Coupon Rate(4 votes)

- I've read through the comments and i'm not alone. Despite that i've read more than 10 articles and 5 half hour videos on bonds. Nobody has explained what is meant by "interest rate" and reasoning of going up/down. Which interest rate? The Fed? Or the Coupon? Or The newly issued bonds vs old? If its the Fed what is the connection between. Pretty please explain the connection and the terms its too much ambiguity behind the term "interest rate". Thank you!(3 votes)
- The interest rate on debt with similar maturity to the bond in question.(3 votes)

- why would a company go for stocks if you can make bonds?(2 votes)
- First and foremost there is the issue of repayment with a bond that doesn't exist with a stock. Issuing bonds is creating debt, and at some point debts must be repaid. So, if you issue bonds and your company never turns a profit, you still have to pay back those bonds. Stocks, on the other hand, guarantee nothing. So if you never make a profit, you never pay dividends to stockholders.

Another reason that a company might choose stocks (or, equity) instead of bonds (or, debt) is that the interest payment on debts is an operating cost. That means your breakeven point as a company is higher with debt than it is with equity.

These are not the only reasons, just a couple of reasons why a company might prefer equity to debt.(3 votes)

- If the interest rate is high, doesn't it affect all the other securities and not just bonds?(2 votes)
- Im still confused, if the interest rate is higher, wouldn't someone invest more to get more raw amount of money as interest? Why would the investment be lower? Wither higher interest, the more you invest, the greater your benefit?(2 votes)
- If you bought a $1000 bond that pays 5%, and the prevailing interest rate goes up to 10%, your 5% bond is worth less, not more. Why would someone give you $1000 to earn $50 each year when they could instead buy a "new" bond for $1000 and earn $100 each year? In order to make your bond attractive to them, you will have to sell it for less than $1000.(2 votes)

- (12:05) Could zero coupon bonds with a $1,000 face value be issued below par value? Does that happen often?(2 votes)
- They have to be issued below face value if they are zero coupon. The return to the investor comes from paying less than face and receiving face value at the end.(1 vote)

- Let's say you buy a 2010 bond for $200. If this 2-year bond was a zero-coupon bond, would the $200 you get back in 2 years be in 2010 dollars or 2012 dollars?(1 vote)
- dollars that you get in 2012 are 2012 dollars

If it's a zero coupon bond with face value of $200, you wouldn't pay $200 for it, you'd pay less than that. That's how zero coupon bonds give you your interest.(2 votes)

## Video transcript

Voiceover: What I want to do in this video is to give a not-too-math-y explanation of why bond prices move in the opposite direction
as interest rates, so bond prices versus interest rates. 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. Let's start with a bond from some company. 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. government. 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, 10% coupon paid semi-annually, so this is semi-annual payments. 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. 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 six months. 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 will pay us 10% of
the par value per year, but it's going to break it up
into two six-month payments. 10% of $1,000 is $100, so they're going to give
us $50 every six months. They're going to give
half of our 10% coupon every six months, so we're going to get $50 here, $50 here, these are going to be our coupon payments, $50 there, and then finally, at two
years, we'll get $50, and we'll also get the
par value of our bond, and we'll also get $1,000. We'll get $1,000 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 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 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, the price of that bond right when it gets issued
or on day zero, if you will, you'll be willing to pay $1,000 for it because you say, look, 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. 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% coupon. So let's say for this type of risk, you would now expect a 15% interest rate. Obviously for something less risky, you would expect less interest. For a company just like company, you would now expect a 15% interest rate. 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?" 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 will come out to be 15%, so I'm going to pay, so the price, so in this situation,
the price will go down. I'll actually do the
math with a simpler bond than one that pays
coupons right after this, but I just want to give
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 just 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 company A, people expect 5%. People expect 5% rate. So how much could you sell this bond for? If you were 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 $1,000 for
something 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. 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 getting
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 zero-coupon bond. 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 is 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, 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% 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? The way to think about
it is let's P in this ... I'm going to do a little bit of math now, but hopefully it won't be too bad. 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 one plus 10%, so after one year, if I compound it by 10%, it will 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. This should be equal to, this should be equal to the $1,000. 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. If you do the math here, you get P times 1.1 squared is equal to 1,000, or P is equal to 1,000
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 $1,000 in two years discounted by 10%. This is 1.10, or one plus 10%. So what is this number right here? Let's get a calculator out. Let's get the calculator out. If we have 1,000 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 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. 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. Let's say second one, so it doesn't affect our
math in any dramatic way. 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 a loan to a company like company A, so now what's the price
we're willing to pay? We'll 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 one plus 15% compounded over two years. We bring out the calculator. We 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. 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 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? Let's say interest, the expected interest rate on
this type of risk goes down, and let's say it's now 5%. What is someone willing to
pay for this zero-coupon bond? The price is, 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%. You 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. 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. 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.