Relationship between bond prices and interest rates | Finance & Capital Markets | Khan Academy
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Relationship between bond prices and interest rates | Finance & Capital Markets | Khan Academy

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.

About Ralph Robinson

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100 thoughts on “Relationship between bond prices and interest rates | Finance & Capital Markets | Khan Academy

  1. Thanks can you do a slightly more advanced one in context of the current situation. 10yr is trading at a premium (assume that also include the risk premium) and generally went up on the recent $10bn taper announcement. ย Since taper would raise rates, prices would fall, but then the fed promise low rates as well.. all quite confusing.

  2. can someone quickly explain, at the very end of the video, why would you want to pay a lower amount for a higher expected return? surely a higher expected return is a good thing? thus you'd pay more for the privilege?

    Thanks in advance!

  3. At the very end he says "The higher return you expect the less your willing to pay" … that doesn't make sense to me.

  4. @ fleshcookie
    interest rates fluctuate towards equilibrium by market pressures, just like any other market value on a good or service.

  5. sorry, something I just don't get quite grapple with – why when interest rates go up to say from 10% to 15% for a bond, wouldn't the issuer want to pay more so that he'll receive more money from the interest rate in the end (or when the bond matures)? I'm still a little new to this so forgive me

  6. I have a doubt.. So is that mean in zero coupon bond interest is not paid in between years its directly been paid on maturity date(interest + par value)?…and while buying the bond how the rate of bond is fixed its the same way showed in above video? and other thing is instead of buying bond we can keep put this money in fixed deposit where we can get more interest rate and i can take that money whenever i wanted…..?
    reply will be appreciated

  7. coupon rate and interest rate are different things right ? and interest rate u mean as in the interest u get when u save money in a bank ? or is there another interest rate for bonds specifically ? dumb questions but pls someone answer

  8. Thank you for the explanation, I had a hard time trying to understand this relation !
    I just have a question on your calculus when you talk about a zero coupon : if the interest become ten percent, and reasoning by the logic you show at the beginning : The first year I get 10% of 1000 that 100, the second year I also get 10% of 1000 that 100 + the 1000 of the bond.
    Finally I got : 1000+100+100=1200=1000 + 1000*i + 1000*i = 1000 (1 + 2*i) and not 1000*((1+i)^2) : because the money of the first interest is not reinvest or is it and that would explain why we have 1000*((1+i)^2).
    Looking Forward to your response ! Thanks

  9. Quick question. So if the interest rates go down and the demand for bonds increase, do the prices of bonds increase? And if they do, doesn't the increase in these prices discourage buyers to go back to the interest rates savings?

  10. But what if I don't want to sell my bond and I'm willing to wait to the end of the maturity. How would the interest rate effect me ?

  11. The 10% coupon is SEMI ANNUAL payment. So instead of $50 paid per 6 months, it should be $100 per 6 months right? I'm confused right now…

  12. If its 10% interest of a $1000 bond, why is it compounded? It's not as if the interest from the first year is going to be added to the value of the bond, can someone explain to me? Would really appreciate it.

  13. So if I go buy a bond, am I getting the old one or the new one? I always hear this concept described in terms or old bonds vs. new bonds.

  14. Okay i need a little help .. ( the video at 8:30 mins) i don't understand why he is adding the 1.10 .. i get the .10 as this is the interest rate but the part i don't understand is where is the 1 coming from ?? thanks a bunch

  15. Very well explained, what could take someone hours in the library to understand, you do that in 13 minutes, thank you very much.

  16. do the interest rates stay consistent to the yield at the date of investment or does it fluctuate with the market?

  17. Bonjour,
    Pourquoi le prix de l'obligation est il infรฉrieur au prix d'achat ร  taux d'intรฉrรชt รฉgal ?

  18. So the price I am willing to pay will continue to decline inversely related to the increasing interest rate percentage quoted on the bond.

    But why?

    If an investor is selling me a bond with x amount interest rate, but new bond interest rates increased in value after the investor purchased the underlying bond

    Then, it is in my best interest to buy the underlying bound at a discounted price paralleling the value of the higher interest rates, so that I can benefit in transferring that 15 percent value in the form of a discount, saving me money when I buy the bond.

  19. Thanks so much for the video. Is there any connection between bond yields and interest rate setting behavior of reserve banks?

  20. From Abuย Hurayrah :ย The Prophet, , said: "Riba has seventy segments, the least serious being equivalent to a man committing adultery with his own mother." (Ibn Majah)

  21. I thought the 10% interest is only on the initial principal invested (ie $1000) and does not compound! This math only makes sense if the interest compounds!!!! ….and I don't think it usually does with bonds (but I'm no expert)

  22. Wow! Amazing!! I'm almost done studying for financial mathematics for actuaries and not once did the #1 recommended study manual (asm manual) mention what's really the difference between a given yield rate and the rate of the coupon payments.

  23. What a brilliant way of explaining the present value of a bond using discounted cash flows, without even introducing it as an NPV of a bond. This inductive reasoning is brilliant.

    PV of Zero Coupon Bonds= Bond's Price/(1+R) where R= Coupon Payments
    PV of Normal Bond= Bond's Coupon payments/ (1+R)^t +Bond's Principal/ (1+R)^t
    t= number of years

    Every time the interest rate, or discount rates, increase, the price of a bond, or its present value decreases.

    We also have to distinguish between:

    Current Yield= Annual Coupon Payments/Current Market Price of a Bond
    Yield to Maturity= [Cash-Flows + (F+P/Number of years to Maturity) / (F+P/2)
    where F: Face Value of a bond
    P= Market Price of a bond

    When the credit issuers are solvent and there's no or every small change of default, then there bonds' yields are lower while their prices are higher. This is because they will most likely pay their debt obligations.

    That's the reasons economically-weak countries like Greece, Spain, Portugal and Argentina offer very high yields on their bonds.

  24. This are just the interest rates on bond right ? we are not considering the interest rates offered by bank on deposits & how they would affect the bond prices.

  25. Thank you very much. This video helped me to get the understanding of Keynes' Speculative demand theory of money.

  26. im confused. why are we comparing yield to maturity / interest rates to COUPON rates? and that is how the prices changes? doesn't make sense

  27. ๋‚˜๋„ ๋ฏธ๊ตญ์—์„œ ์ˆ˜๋…„๊ฐ€ ์นธ์•„์นด๋ฐ๋ฏธ ์ด์šฉํ–ˆ๋˜ ์‚ฌ๋žŒ์œผ๋กœ์„œ ์ด ์นธ ์„ ์ƒ๋‹˜์‘ ์ง„์งœ ์กด๊ฒฝํ•œ๋‹ค. ๋ฏธ๊ตญ์€ ๋‹ฌ๋ผ… ์ด๋ถ„์€ ๊ณต์งœ ๋ฌด๋ฃŒ ๋กœ ์ด๋Ÿฐ๊ฑฐ ๊ฐˆํ‚ค๋Š”๋ฐ ๊ฑฐ์˜ ์ฐฝ์‹œ์ž์ž„… ๋ฏธ๊ตญ์ด์–ด์„œ ๊ฐ€๋Šฅ ํ•œ๊ตญ์—์„œ๋Š” ์ ˆ๋Œ€ ๋ถˆ๊ฐ€๋Šฅ… ๋Œ€ํ•™๊ต์œก๋งŒ ๋ด๋„ ๋ฏธ๊ตญ๋Œ€ํ•™ ๊ทผ์ฒ˜์— ํ•œ๊ตญ์ฒ˜๋Ÿผ ์ˆ ์ง‘ ๋ฒˆํ™”๊ฐ€ ์ž์ฒด๊ฐ€ ์ƒ๊ฒจ๋‚ ์ˆ˜ ์—†์Œ ๊ทธ๋ƒฅ ๋‹ฅ์น˜๊ณ  ์•„๋‹ฅํ•˜๊ณ  ๊ณต๋ถ€ … ๊ธฐ๋ณธ์ ์œผ๋กœ ์ „๋ฌธ๋Œ€ ๊ฐ•์‚ฌ๋ผ๋„ ํ•ด๋„ ๋ฏธ๊ตญ์—์„œ์˜ ๊ต์œก์—๋Œ€ํ•œ ์ž๋ถ€์‹ฌ์ด ์ฉ”๊ณ  .. ๋‚ด๋Œ€ํ•™์˜ ๊ฒฝ์šฐ๋„ ํ†ต๊ณ„๊ต์ˆ˜๊ฐ€ ์–˜๊ธฐํ–ˆ๋˜๊ฒŒ ์šฐ๋ฆฌ๊ฐ€ ๋ฏธ๊ตญ์—์„œ ํ‹ฐ์นญ ์Šคํ‚ฌ์ด 10์œ„๊ถŒ์ด๋‹ค ์˜ฌํ•ด๋Š” 14์œ„๋กœ ๋‚ด๋ ค๊ฐ”๋‹ค ์ด๋Ÿฌ๋ฉด์„œ ์ ์ˆ˜๋ฅผ ์•ˆ์ฃผ๋ ค๊ณ  ํ•˜์ง€๋งŒ ํ•™์ ์— ๊ด€๊ณ„์—†์ด ์–ด๋ ค์›Œ๋„ ์ด๋Ÿฐ ๊ต์ˆ˜๋“ค ๋งŒ๋‚˜ ์˜๋ฏธ์žˆ๋Š” ๊ต์œก ๋ฐ›๊ณ  ๊ณต๋ถ€ํ–ˆ๋‹ค๋Š”๊ฑฐ์— ์ž๋ถ€์‹ฌ์ด ๋“ ๋‹ค. ์ด ์นธ ์„ ์ƒ๋‹˜๋•Œ๋ฌธ์— ์ธ๋„์‚ฌ๋žŒ๋“ค์ด ๋”์šฑ ์นœ๊ทผํ•ด์ง€๊ณ  ์กฐ์•„์กŒ์—ˆ์ง• ๋˜‘๋˜‘ํ•ฉ๋‹ˆ๋‹ค ์ง„์งœ ์ˆ˜ํ•™ 25๊ตฌ๊ตฌ๋‹จ์ด์—ˆ๋˜๊ฐ€ 25์ง„๋ฒ•์ธ๊ฐ€ ๊นŒ์ง€ ์™ธ์šฐ๋Š” ๋‚˜๋ผ. ํ•œ๊ตญ์€ ์ง„์งœ ๋‹ฌ๋ผ์ ธ์•ผ๋ผ ์ž๊ธฐ ๋ฐฅ๊ทธ๋ฆ‡๋งŒ ์ฑ™๊ธฐ๋ฉด ๋”์ด์ƒ ๋ฏธ๋ž˜๊ฐ€ ์—†์Œ. ๊ณต๋ฌด์› ๋งˆ๋‹ˆ ๋ฝ‘์ง€ ๋ง๊ณ  ์ค‘์†Œ๊ธฐ์—… ๋งˆ๋‹ˆ ๋ฝ‘์•„์š”

  28. Could someone pls explain to me why it makes sense that "the higher return you expect, the LESS you're willing to pay for that bond"?

  29. so what's the point, then? if you were to get a $1000 in 2 years buying its any interest rate the price would be inverse so the net effect at the end of 2 years would be same? so whats the gain?

  30. 498K views and 2.2K likes? Why?

    This is one of the simplest ways this particular subject couldโ€™ve been explained.

  31. Can you go over warrants and convertible bonds? Or link it to me if you already have? I don't see it in the playlist ๐Ÿ˜ฒ

  32. Man……..u get simple interest on bonds,dont you ?… why u've used compounded method to calculate the value of a bond with some previous coupon rate ?

  33. Why do you multiply both 10% per year times P at the same time and not separately. So the 826 will be 800$ cause it is 10% per year. 10% of the 1000$ per year is 100$ each year, times 2 years is 200$,,,,,, 1000 -200 = 800. Or is the formula used to reduce the profit of the one who buys it? Why is the 10% interest not multiplied separate per year to the 1000$ why is it not even..

  34. So why buy a zero coupon bond for $1000 if rates go up or down you dont get to sell that bond for more than you paid ?
    And if you keep the bond you get no interest.
    Unless yield is something different again ?

  35. The interest rate in the video refers to Which Interest rate ?The interest rate at which central govt lends to bank or the interest charged by banks to its lenders? Please clarify…

  36. Great info but watching you draw/write is such a drag to watch (even at 1.5 speed) … (do the drawings before the video)

  37. wait i thought the interest rate going up actually favors the holder, since this gives them a higher semi annual return or Im i wrong here

  38. Hello, If Company payout the interest semi-annually, then isn't it supposed not to be compounded. Cause I think Principle could only be compunded if the interests adds to the original princple next time, but you said they payout interests on six months????

  39. Thanks dude this helps me with my digital nomad adventures. Knowledge is power :D. Cant donate but ill toss up a sub. Cheers!

  40. If the bond is a zero coupon rate one and if I want an interest rate, who's going to pay the interest? The price will decrease I understood but who will pay me every year, the interest, the company won't pay right.

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