Lesson 2 of 8
Price and yield
One equation read two ways. Discount the cash flows to get a price, or solve for the rate that produces the price you were quoted.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- A bond's price is the present value of its own coupons and principal at some rate.
- The yield is that rate, found by solving the same equation backwards from a quoted price.
- Price above face means the coupon is above the market rate. Price below face means the opposite. There is no third case.
Pricing a bond needs nothing that was not in the last course. It is a factor table, five multiplications and one addition. The only new idea is that in practice you are usually given the answer and asked for the rate.
The same five-year 5 percent bond at three market rates
- At 4.00 percent$1,044.52
- At 4.78 percent$1,009.58
- At 6.00 percent$957.88
Pricing it
Take a five-year bond with a 5 percent annual coupon and a face value of 1,000. It pays 50 at the end of each of five years, and 1,000 more at the end of the fifth. Discount all six amounts at the market rate and add them.
Priced at 4.78 percent, the five-year US Treasury par yield on 11 September 2026
- Year 1: 50 multiplied by 0.9544 equals 47.72.
- Year 2: 50 multiplied by 0.9108 equals 45.54.
- Year 3: 50 multiplied by 0.8693 equals 43.46.
- Year 4: 50 multiplied by 0.8296 equals 41.48.
- Year 5: 50 plus 1,000 is 1,050, multiplied by 0.7918 equals 831.38.
- Add all five: 1,009.58.
The bond is worth 1,009.58, which is more than its face. That is not an opinion. Its coupon of 5 percent is above the 4.78 percent the market wants, so a buyer pays a premium for the difference.
The same equation, solved the other way
In a real market nobody hands you the rate. You see a price and you want to know what return it implies. The equation is identical and the unknown has moved.
There is no closed-form solution for a coupon bond, so you search. Try 6 percent and the price is 957.88, which is too low. Try 4 percent and it is 1,044.52, which is too high. Narrow until you reach the price you were quoted. That number is the yield, and the next lesson is about what it assumes.
Four fifths of it is one payment
In the ladder above, 831.38 of the 1,009.58 price is the final year. Four fifths of the value of this bond is one payment, five years out, and the four coupons before it contribute 178.20 between them.
That concentration is why a long bond's price moves so much when rates move, and it is the fact duration measures three lessons from now. A bond is not evenly spread across its life, and the closer its cash is to the end, the more a rate change matters.
Check yourself
A bond trades above its face value. What do you know about its coupon?
- It is below the current market rate
- It is above the current market rate
- Nothing, without knowing the maturity
Show the answer
Answer: B.
Paying more than face for a fixed set of payments means those payments are generous relative to what the market currently requires. Maturity changes how much the premium is, not its direction.Why is a coupon bond's yield found by searching rather than by a formula?
- Because coupons are not known in advance
- Because the rate appears in several powers at once and cannot be isolated
- Because market prices are inaccurate
Show the answer
Answer: B.
The price equation has the unknown rate raised to every power from one to the number of periods. There is no general algebraic rearrangement, so a solver iterates.In the five-year bond above, which single cash flow carries most of the price?
- The first coupon
- The final payment of coupon plus principal
- They are all about equal
Show the answer
Answer: B.
The final payment of 1,050 discounts to 831.38 of a 1,009.58 price. The concentration of value at the end is what makes a bond's price sensitive to rates.Sources
- US Department of the Treasury, Daily Treasury Par Yield Curve Rates, session of 11 September 2026: the 4.78 percent five-year par yield used as the market rate.
- Williams, The Theory of Investment Value, 1938: the present value of a dated payment stream, applied here to a bond rather than a business.
What this leaves out
- The bond is invented, with round figures so every line can be recomputed. Only the discount rate is a real published number.
- Coupons are treated as annual. US corporate and Treasury coupons are usually semi-annual, which changes the arithmetic slightly and none of the reasoning.
- Accrued interest between coupon dates is ignored here. A real quoted price is usually a clean price with accrued interest added separately.
- Nothing here is advice about any security.