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

Question

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?

Answer 1

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.

Answer 2

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.