# Subject: Bonds - Relationship of Price and Interest Rate

Last-Revised: 28 Oct 1997

Contributed-By:
Rich Carreiro (rlcarr at animato.arlington.ma.us),
Chris Lott (contact me)

The basic relationship between the price of a bond and prevailing market interest rates is an inverse relationship. This is actually pretty straightforward. For example, if you have a 6% bond (this means that it pays $60 annually per $1000 of face value) and interest rates jump to 8%, wouldn't you agree that your bond should be worth less now if you were to sell it?

If this isn't clear, think about it this way. If the rate of interest
being paid on newly issued bonds stands at 8%, a bond buyer would get
paid $80 annually for each $1,000 investment in one of those bonds.
If that bond buyer instead bought your old 6% bond for the price you
originally paid, that bond would yield $20 less per year when compared
to bonds on the market. Clearly that's not a very attractive offer
for the buyer (although it would be a *great* deal for you).

To quantify the inverse relationship between the price and the interest rate, you really need the concept of the present value of money (also see the article elsewhere in the FAQ on this topic). Computing present value figures helps you answer questions like "what's better, $95 today or $100 one year from now?" The beginnings of it go something like this.

Pretend that you have $100. Also pretend that you can invest it in something that will pay a 5% annual return. So, one year from now you have:

$100 * (1 + 0.05) = $105

This can be turned around. Let's say that you want to know how much money you need to have today in order to have $200 a year from now, if you can earn 5%:

X * (1 + 0.05) = $200 or X = $200/(1 + 0.05) = $190.48

Therefore, we can say that the *present value* of $200 one year
from now, assuming a "discount rate" (this is what the assumed
interest rate in a present-value calculation is called) of 5% is
$190.48.

But what if you wanted to know how much you needed today to have $200 two years from now, again assuming you could earn 5%? Here's the computation.

[ X * (1 + 0.05) ] * (1 + 0.05) = $200

X represents the original amount, and the quantity "X * (1 + 0.05)" represents the amount after 1 year. Solving for X we get:

X = 200/(1 + 0.05)^2 = $181.41

So, the present value of $200 two years hence, at a discount rate of 5% is $181.41. It should be clear that the present value of $200 N years from now at a discount rate of 5% is:

PV = 200/(1 + 0.05)^N

And this can be generalized to the present value of an amount C, N years from now, at a discount rate of r:

PV = C/(1 + r)^N

Now you can combine these. Let's say I promise to pay you $300 a year from today and $500 two years from today. What could I have paid you today that would have made you just as happy as what I promised? Assume you can earn 7% on your money. To solve this, just sum the present value of each payment. This sum is called the "net present value" (NPV) of a series of cash flows.

NPV = $300/(1 + 0.07) + $500/(1 + 0.07)^2 = $717.09

So, given the 7% discount rate, the payments I scheduled are equivalent to a payment of $717.09 made today.

Let's get a little fancier. What if I'm willing to promise to pay you $50 per year for 4 years, starting a year from now, and further promise to pay you $1000 five years from now. What's the most you'd be willing to pay me now to make you that promise. Assume a discount rate of 6%.

NPV = $50/(1 + 0.06) + $50/(1 + 0.06)^2 + $50/(1 + 0.06)^3 + $50/(1 + 0.06)^4 + $1000/(1 + 0.06)^5 = $920.51

Let's say you want to wait until tomorrow. You have a dream that night that makes you believe that you'll now be able to earn 10% on your money. When I come back to you, you now tell me you'll only pay me

NPV = $50/(1 + 0.10) + $50/1.1^2 + $50/1.1^3 + $50/1.1^4 + $1000/1.1^5 = $779.41

My promise is now worth quite a bit less. You should be able to see that if your dream had led you to believe you could earn less on your money, then my promise would have been worth more to you than it did yesterday.

At this point, it's probably clear that my "promise" is effectively what a bond is -- I'm agreeing to pay you a fixed amount each year (actually, the bond would pay half that fixed amount twice a year) and then the principal amount at maturity. Given what you think you can earn on your money, the price you should pay for the bond is well-defined. The question is what affects what you think you'll be able to earn on your money? Fed policy might. What you think the chances of inflation are might. Lots of other things might. This is where the fun starts. :-)

Also note that you can turn the equation around. Let's say that you have a $1000 bond paying $75 per year. The bond matures in 10 years. Someone is willing to sell it to you for $850. What will I have earned on my investment? The net present value equation always holds, so $850 equals the net present value of the yearly payments and principal payment.

Obviously, since we know everything except the discount rate, this equation must define the discount rate that makes it true. The problem is that the rate cannot be simply calculated. You must make a guess, compute the net present value, see how different it is from $850, use that to adjust your guess, and try again until the sides of the equation balance. The discount rate you come up with is called the "internal rate of return" (IRR) and in the bond world is called the "yield to maturity" (YTM). In fact, if you know the initial value of some portfolio, all cash flows into and out of the portfolio, and the final value of the portfolio, you can compute your IRR, thus answering the common misc.invest.* question of "I put $N into a fund on date X, but then added $D on date Y and $F on date W. My account is today worth $B. What's my return?"

As a final note, here's a bit of a stumper to spring on someone:
Assuming you could earn 5% on your money, would you rather be paid
$1000 annually (first payment is today, next is a year from now, etc.)
*forever* (assume you are immortal :-) or $25000 today? Believe
it or not, you should take the $25000 today. Here's the analysis why.

NPV = $1000 + $1000/1.05 + $1000/1.05^2 ... or NPV = $1000*(1 + 1/1.05 + 1/1.05^2 + ...)

A math reference book can tell you (or you might remember or derive it) that the infinite sum:

1 + x + x^2 + x^3 + ... = 1 / (1 - x) if |x| < 1

In this case, x = 1/1.05, so

NPV = $1000 * [ 1 / (1 - 1 / 1.05) ] = $21000

So believe it or not, you'd be better off taking $25000 today then taking $1000 per year forever, given the 5% discount rate assumption.

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