Yield to maturity · RiskAtoms Academy

By the end you will be able to

1Calculate a bond's yield to maturity from its price, and interpret what it means.
2Find the yield by trial and error, and with a spreadsheet's RATE or YIELD function.
3State the assumptions behind yield to maturity and say when they break.
4Compare the coupon rate, the current yield and the yield to maturity.

Why it matters

“The 10-year Treasury yield rises to 4.5%.” Headlines like this appear every week, and a whole market of prices sits behind the number. What does it actually measure?

Core explanation

The pricing problem, run backwards

In L1.4.03 you knew the yield and worked out the price. In the market it is the other way round. Traders see a price on the screen, and they want to know what return it implies.

The yield to maturity (YTM) is the answer. It is the one rate that, used to discount every coupon and the face value, gives exactly the market price. If you buy at that price and everything goes to plan, the YTM is your annual return.

Think of it like this

Pricing: cash flows + yield → price. YTM: cash flows + price → yield. Same formula, solved for a different unknown.

No neat formula. For a bond with several coupons, you cannot rearrange the price formula to get $y$ on its own. Instead you search: guess a yield, price the bond, compare with the market price, and adjust. Because price falls as yield rises, every guess tells you which way to go. A price that is too high means the guess was too low.

In practice a spreadsheet does the searching. RATE(5, 4, -95.67, 100) returns 5.00% for the bond below. YIELD does the same from dates, with the coupon frequency and day count, which matters between coupon dates (L1.4.05).

Three yields, one bond. The coupon rate is fixed on the term sheet. The current yield is the coupon divided by today's price: a quick measure of income. The YTM adds the gain or loss as the price is pulled to 100 at maturity. For a bond below par, the order is always coupon rate < current yield < YTM. Above par it flips.

Coupon rate

4.00%

Coupon ÷ face value. Set at issue, never changes.

Current yield

4.18%

Coupon ÷ price = 4 ÷ 95.67. Income only.

Yield to maturity

5.00%

Income plus the pull to 100 at maturity.

The 5-year 4% bond from L1.4.03, priced at 95.67.

The small print. The YTM is your return only if three things hold: you keep the bond to maturity, you reinvest every coupon at the same yield, and the issuer pays in full. Each can fail. Think about each one before you reveal the answer:

Think firstWhat if the plan changes?

You buy the bond at 95.67 and hold it for five years, but rates fall and you can only reinvest the coupons at 2%. Do you still earn 5.00%?
After 2 years you need the cash and sell. Yields on similar bonds have risen to 6%. What happens?
The issuer stops paying in year 4. Does the YTM tell you anything?

Callable bonds. If the issuer can repay early (L1.4.02), investors also work out the yield to call, as if the bond is repaid at the first call date. The lowest of these yields is the yield to worst: the least you can earn if the issuer acts in its own interest.

The formal version

Write the price from L1.4.03 as a function of the yield, $P(y)$ . The YTM is the value of $y$ that solves one equation:

P_{\text{market}} = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T}

On a graph, $P(y)$ is a curve that falls as $y$ rises, and $P_{\text{market}}$ is a flat line. The YTM is where they cross. Because the curve only ever falls, they cross exactly once.

Narrowing it down. If two trial yields $y_1$ and $y_2$ give prices $P_1$ and $P_2$ on either side of the market price, draw a straight line between them:

y \approx y_1 + \frac{P_1 - P_{\text{market}}}{P_1 - P_2} \times (y_2 - y_1)

The curve bends, so this is a first guess, not the answer. Price the bond at the guess, and repeat until the price matches.

The current yield needs no search:

\text{current yield} = \frac{C}{P_{\text{market}}}

With two coupons a year, the search finds a six-month rate, and the market quotes the YTM as twice that rate. The 5-year 4% bond with semi-annual coupons priced at 95.62 has a six-month rate of 2.50%, quoted as a YTM of 5.00%.

Try it

The market price is 95.67 for a 5-year, 4% bond. Move your guess until the dot lands on the dashed line. Then change the price and do it again.

Try itFind the yield

Open the yield toolkit →

Market price95.67

Coupon rate (annual)4.00%

Years to maturity5

Your guess for the yield4.00%

Price at your guess

100.00

vs market price 95.67

At your guess the bond is worth more than its market price. Raise the yield.

Coupon rate4.00%

Current yield4.18%

Yield to maturity

Price as a function of yieldThe curve is P(y). The dashed line is the market price. The YTM is where they cross.

Worked example

The 5-year 4% bond at 95.67

Face value 100, annual coupon 4, five years to maturity, market price 95.67. Yields are annual and compounded once a year. Try each step before revealing it.

Find the YTM by trial and error

Step 1 of 6

1
First guess: the coupon rate
At 4% the bond prices at 100.00 (par, as in L1.4.03). That is above 95.67, so the yield must be higher.

Regulatory lens · looking ahead

No rules apply yet. Yields are the raw material for much of what follows: credit spreads (L1.4.07) are differences between yields, and the yield curve (L1.3.08) is built from them. Later levels use yields to measure interest rate risk in banks' balance sheets (L3.5).

Sparks

SparkPaying to lend

For several years after 2014, investors paid governments in Germany, Switzerland and Japan to hold their money: yields on some of their bonds were below zero. A negative YTM means the price is higher than everything the bond will ever pay back. At the peak, in late 2020, around $18 trillion of bonds worldwide traded at negative yields.

SparkYield is the headline, price is the trade

Bond traders often talk in yields and ignore prices. A move from 4.50% to 4.55% is “5 basis points”. Quoting yields lets you compare bonds with different coupons and maturities on one scale.

Key takeaways

Yield to maturity is the single rate that makes the present value of a bond's cash flows equal its market price.
There is no closed-form formula: find it by trial and error, interpolation, or a spreadsheet's RATE or YIELD.
Price too high at your guess means the guess is too low, and the other way round.
YTM is your return only if you hold to maturity, reinvest coupons at the YTM, and the issuer pays in full.
Below par: coupon rate < current yield < YTM. Above par the order flips. At par all three are equal.

Reaction

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Glossary

Yield to maturity: The single annual discount rate that makes the present value of a bond's remaining cash flows equal to its market price.
Current yield: The annual coupon divided by the bond's price. It ignores any gain or loss on the way to maturity.
Reinvestment risk: The risk that coupons can only be reinvested at a lower rate than expected.
Yield to call: The yield calculated as if a callable bond is repaid on its first call date at the call price.
Yield to worst: The lowest of a callable bond's yield to maturity and its yields to each call date.
Basis point: One hundredth of a percentage point: 0.01%. 100 basis points make 1%.

Deep Dives & Origins

Deep Dives

CFA Institute, Fixed Income Analysis (CFA Institute Investment Series), “Introduction to Fixed-Income Valuation”. Wiley. Intermediate.

Origins

Frank J. Fabozzi and Francesco A. Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021.
CFA Institute, Fixed Income Analysis, “Introduction to Fixed-Income Valuation”.
Bloomberg, Global Aggregate negative-yielding debt data (December 2020).

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