The bond price-yield relationship

What you will be able to do

After this lesson, you should be able to:

• explain why the price of a bond with fixed positive cash flows decreases when the yield increases;
• hold contract cash flows fixed while changing only the discounting input;
• recognize that the price-yield relationship is curved rather than linear.

The price decreases because every denominator increases

Write the bond-payment-frequency as $\explain{bond-payment-frequency}{m_{\mathrm B}}$. In this lesson, $\explain{yield-to-maturity}{y}$ denotes the yield-to-maturity $\explain{yield-to-maturity}{y^{(\explain{bond-payment-frequency}{m}_{\mathrm B})}}$ without its superscript, as the argument of a function. Write the price-yield-curve as $\explain{price-yield-curve}{P(y)}$.

Use the bond-payment-index $\explain{bond-payment-index}{k}$ to select one of the number-of-bond-payments $\explain{number-of-bond-payments}{n}$, and write each promised positive bond-cash-flow as $\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}$. The price at yield $\explain{yield-to-maturity}{y}$ is the sum of the promised cash flows, each discounted at the yield:

$$\explain{price-yield-curve}{P(y)}= \sum_{\explain{bond-payment-index}{k}=1}^{\explain{number-of-bond-payments}{n}} \frac{\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}} {\left(1+\explain{yield-to-maturity}{y}/\explain{bond-payment-frequency}{m_{\mathrm B}}\right)^{\explain{bond-payment-index}{k}}}$$

(2.4.1)

Assume that the yield stays non-negative and that every cash flow stays fixed. If the yield increases, every denominator increases. So every discounted cash flow decreases, and the bond-price decreases. Table 2.4.1 also assumes that the new yield is non-negative.

Define the local yield-shock $\explain{yield-shock}{\Delta y}$ as new yield minus old yield, and define the local price-change $\explain{price-change}{\Delta P}$ as new price minus old price.

Table 2.4.1Signs of a yield shock and the price change. For a rise and a fall in the yield, the direction of every discount denominator and the sign of the price change, with the promised cash flows held fixed.

Yield shock	Denominators	Price change
$\explain{yield-shock}{\Delta y}>0$	Increase	$\explain{price-change}{\Delta P}<0$
$\explain{yield-shock}{\Delta y}<0$	Decrease	$\explain{price-change}{\Delta P}>0$

The table holds only if the promised cash flows stay fixed.

Example 2.4.1 Price-yield comparisons

Open the set, then choose a price-yield comparison.

Three points form a curve

Consider a five-year bond with an annual coupon rate of 5%, semiannual payments, and a face value of 100. Its prices at three yields are:

Nominal annual yield	Price per 100 face
5%	100.0000
6%	95.7349
7%	91.6834

The price decreases as the yield increases. Each yield increase is one percentage point, but the two price changes are different: 4.2651 and 4.0515. So the three points are not on a straight line.

Equal shocks, unequal price changes

Start from a yield of 5%:

• a yield decrease to 4% increases the price from 100.0000 to 104.4913, a change of +4.4913;
• a yield increase to 6% decreases the price from 100.0000 to 95.7349, a change of −4.2651.

The two yield shocks have equal size, but the absolute price changes are different. This difference is a result of the curvature of the price-yield curve. This lesson does not calculate duration, DV01, or convexity.

Hold other inputs fixed

Suppose the issuer changes a coupon, a default occurs, or the settlement date changes. Then more than the yield has changed, and the comparison no longer isolates the effect of the yield.

Explore the bond price–yield curve

Annual coupon rate (%)

Yield to maturity (%)Exact yield to maturity (%)

Maturity (years)

Price: 95.73 per 100 face value

For positive fixed cash flows, moving right to a higher yield lowers every discount factor and therefore lowers the price.

View assumptions and a text alternative

Settlement is on a coupon date. The model uses a flat nominal yield compounded at the coupon frequency, redemption at par, and no accrued interest, default, liquidity effect, tax, or embedded option.

Selected points from the current price–yield curve

Yield	Price
0.0%	125.00
3.0%	109.22
6.0%	95.73
9.0%	84.17
12.0%	74.24
15.0%	65.68

The chart, the live result, and the data table all use the same tested domain function. The notation panel only explains the symbols; it does not calculate prices.

Check your understanding

These direct and transfer items cover direction and nonlinearity. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 2.4.1 Bond price-yield shape

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Model boundary and review note

The plotted curve summarizes a toy model with one yield. It is not a credit curve and not a recommendation. It omits accrued interest, term-structure effects, default, recovery, liquidity, tax, funding, and embedded options.

The inverse direction and the curvature both follow Tuckman & Serrat[1]; Hull notes the same negative price-yield relationship[2]. FINRA states that price and yield are inversely related[3]. The lesson stays draft pending human confirmation of the printed locators.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §4.1, “Price-Rate Curves”, Figure 4.2, prices decrease as rates rise; §4.5, “Convexity”, price-rate curves of coupon bonds are convex. draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). §4.8, Duration. draft ↩
3. FINRA, Understanding Bond Yield and Return. “Price and yield” section. https://www.finra.org/investors/insights/bond-yield-return draft ↩