# Can you calculate 10-year Treasury Note yield from price or vice versa?

I grabbed this screenshot from CNBC this morning: Are the price (\$98.0469) and yield (2.981%) mathematically related? Investopedia says "A 10-year Treasury note pays interest at a fixed rate once every six months, and pays the face value to the holder at maturity." If it didn't pay interest I suppose one could calculate the yield from the price pretty easily. But with the interest payments how can this be calculated?

• Read these for a better understanding of yield calculations: (1) sapling.com/8558435/yield-calculation-10year-treasury and (2) investinganswers.com/calculators/yield/… – Bob Baerker Apr 23 '18 at 15:44
• Ok, so the part I was missing is the coupon payment interest rate. According to your second link, if that rate is currently 2.75%, the yield to maturity would come out to 2.98%. Since the yield to maturity is above the coupon interest rate, the note's value is below face value (\$100), correct? This makes me wonder, why bother showing anything but yield to maturity, because by itself, that tells you everything you need to know doesn't it? – Craig W Apr 23 '18 at 17:28
• @CraigW Price, Coupon, and Yield are mathematically related and are all important from different points of view. So only ever showing one measure would not be appropriate. Many data providers give all three even though you can calculate one knowing the other two just so it's applicable no matter what your point of view is. – D Stanley Apr 23 '18 at 17:58
• @DStanley I see how they are all mathematically related, but it seems like yield is all that really matters, and price and coupon are just the particulars of how it happens. For example, you could instead have a no-coupon bond where you pay \$74.39 and get \$100 10 years later (for a yield of 2.981%, compounded semi-annually). So I'm curious why they don't only show the yield. Adding the price seems kinda meaningless unless you know the coupon rate, or know how to go calculate it from the other two. – Craig W Apr 23 '18 at 18:13
• @CraigW What I'm saying is that they all matter depending on your point of view. Yield matters for comparability. Coupon matters to understand the actual cash flows. Price matters to know what you're actually paying for the bond. Providing only one is not enough information, and providing only two and requiring non-trivial calculations for the third is not always appropriate. The screenshot you show is geared towards traders and economists who do not care what the coupon is, but mortgage consumers care very much what the coupon on 10-year notes is if it affects their mortgage rate. – D Stanley Apr 23 '18 at 19:10

A 10 year bond offers two coupons per year. With a \$1000 face value, at the moment the bond is issued, with coupons of \$15 each, the price, 100, means the yield and YTM are both 3%.

Now, the way the math works. One can calculate the present value of each coupon, sum them up, and see that the sum is the current \$1000, or price of 100.00 (it’s quoted as \$100 even though the full bond is \$1000).

Next, if general rates drop, say to 2.95%, and you discount each of the 21 future payments, you’ll get a number higher than \$1000, and the bond price will be quoted as 101.00 or in that range.

Member Chris D can offer the full set of equations, me, I can write a spreadsheet pretty quickly that would calculate NPV and offer similar results. To be clear, when the bond is issued, the coupons and final payment are known. The present value and rate are what changes, inversely to each other. Rate goes up, present value goes down.

That’s it. Whatever rates do, the value of the bond goes up or down to reflect a YTM of the new rate. “Opportunity cost” is misused in the other posted answer. There is no unknown when it comes to present bond value.

• Nice answer. Do you mean 20 future payments (21 if you include the return of principal)? – Craig W Apr 24 '18 at 1:04
• Fixed. Yes. 20 coupons plus final \$1000. I was thinking 30 year bond. Thx for the correction . – JoeTaxpayer Apr 24 '18 at 1:25

No.

You cannot calculate the price given the information you have given, because you don't know the opportunity cost which is a big thing when dealing with bonds. You can, however, calculate the price of another bond with the same properties but different interest rate (in theory at least).

Consider if you have 2 bonds (A and B), both with a face value of 100 and maturity of 2 years. Bond A has an interest rate of 10% and B an interest rate of 5%. Let's say that the price of A is 100 - then what is the price of B?

``````|  Bond  |  Face value  |  Maturity  |  Interest rate  |  Price  |
|--------|--------------|------------|-----------------|---------|
|   A    |     100      |    2 yr    |      10.0%      |   100   |
|   B    |     100      |    2 yr    |       5.0%      |    ?    |
``````

Let's say you discount rate is 0% for simplicity. We can now calculate the present value of Bond A by setting up the cashflows:

``````| Cash Flow         |  Year 0  |  Year 1  |  Year 2  |
|-------------------|----------|----------|----------|
| A                 |     0.00 |    10.00 |   110.00 |
| Discount factor   |          |     1.00 |     1.00 |
| Present value     |  -100.00 |    10.00 |   110.00 |
|-------------------|----------|----------|----------|
| Net present value |      20  |          |          |
``````

Now let's consider Bond B - if the price was the same a Bond A, you would have below net present value:

``````| Cash Flow         |  Year 0  |  Year 1  |  Year 2  |
|-------------------|----------|----------|----------|
| B                 |     0.00 |     5.00 |   105.00 |
| Discount factor   |          |     1.00 |     1.00 |
| Present value     |  -100.00 |     5.00 |   105.00 |
|-------------------|----------|----------|----------|
| Net present value |      10  |          |          |
``````

Obviously you would not want to engage in Bond B if the price was the same as A, because your net present value is much lower (you lose 10 on bond B compared to A).

Now what do you want to pay for bond B?

It is easy - you want to have the same Net Present Value for the two bonds, so that neither is to prefer over the other. In this case you want the difference in discount to choose Bond B over A. This means that Bond B will be priced at:

``````100 - (20 - 10) = 90
``````

This works in theory only - because reality is much more complex (e.g. what is the discount rate?). Principles are the same though.

• Your example is confusing - you use "zero-coupon" bonds but both have interest rates? – D Stanley Apr 23 '18 at 18:01
• Sorry - changed my mind but not the text :-) – ssn Apr 23 '18 at 18:48
• This answer makes no sense at all. Tbonds are not priced as you described. What does your NPV represent? – JoeTaxpayer Apr 24 '18 at 17:17
• @JoeTaxpayer How much your investment is worth, and how much you are willing to pay for a similar bond but with a different interest rate. – ssn Apr 24 '18 at 19:29