The price-yield relationship · RiskAtoms Academy

By the end you will be able to

1Explain why a bond's price falls as its yield rises, and why the price-yield curve bends.
2Explain why longer maturities and lower coupons make prices more sensitive to yields.
3Describe pull to par as a bond nears maturity.

Why it matters

Silicon Valley Bank failed in March 2023. Its depositors' money had gone into long-dated, safe bonds. When interest rates rose fast, those bonds fell in value by billions of dollars, and when depositors rushed to withdraw, the losses became real. Nothing defaulted. The price-yield relationship did the damage.

Core explanation

A seesaw with a bend in it

Why price falls when yield rises. A bond's cash flows are fixed. Its price is those cash flows discounted at the yield (L1.4.03). Raise the yield and every cash flow is discounted more heavily, so the total is smaller. Price and yield sit on a seesaw.

There is a market way to see it too. Say you hold a bond paying 4%, and new bonds now pay 5%. Nobody will pay you 100 for your 4% bond when 100 buys a 5% one. Your bond's price must fall until its yield matches the market's 5%.

Some bonds sit further from the pivot. Price sensitivity depends on how far away the cash comes in:

Longer maturity, more sensitive. A payment in 10 years is discounted ten times over; a payment in 2 years only twice. A change in the yield compounds over more years.
Lower coupon, more sensitive. A low-coupon bond gets more of its value from the face value at the end. A zero-coupon bond gets all of it there, so it is the most sensitive of all for its maturity.
Lower yield, more sensitive. At low yields, distant cash flows are discounted less, so they carry more weight in the price.

Think firstWhich moves more?

A 5-year 4% bond or a 20-year 4% bond, when yields rise by 1%?
A 10-year 8% bond or a 10-year zero-coupon bond, both starting at a 4% yield?
For the 10-year 4% bond at 4%: the gain if yields fall 1%, or the loss if they rise 1%?

Why the curve bends. Plot price against yield and you get a curve, not a straight line. It is steep at low yields and flattens out at high yields. Each extra 1% on the yield knocks a little less off the price than the one before, because it is cutting a price that is already smaller. That bend is called convexity, and it works in the holder's favour: for equal moves, gains are bigger than losses. L1.4.09 measures it.

Pull to par. Time matters too. A bond repays 100 at maturity whatever happens, so as that date nears, its price is drawn towards 100. A premium bond drifts down; a discount bond drifts up. This is pull to par, and it happens even if yields never move.

Step throughPulled to par

Three 10-year bonds, all priced at a 4% yield that never changes.

Years since issue0

6% coupon

116.22

4% coupon

100.00

2% coupon

83.78

The formal version

Write the price as a function of the yield, as in L1.4.04, and look at one term at a time:

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

Every term has $(1+y)$ in the denominator, raised to a positive power. Raise $y$ and every denominator grows, so every term shrinks: $P(y)$ is a decreasing function. The term for year $t$ has the power $t$ , so later cash flows react more: that is the maturity effect.

The bend shows up in the numbers. Here is the 10-year 4% bond, priced at yields 1% apart:

Yield move	Price change
2% → 3%	−9.43
3% → 4%	−8.53
4% → 5%	−7.72
5% → 6%	−7.00
6% → 7%	−6.35

The same 1% step costs less each time. A straight line would give equal steps; a curve that flattens gives shrinking ones. In the language of graphs, $P(y)$ is decreasing and convex.

Try it

Bond A starts as a 2-year 4% bond and Bond B as a 10-year 4% bond. Compare their moves for ±1%. Then drop Bond B's coupon to 0 and watch its curve get steeper.

Try itTwo bonds, one chart

Open the full bond pricer →

Bond A

Bond A: years to maturity2

Bond A: coupon rate4.00%

Bond B

Bond B: years to maturity10

Bond B: coupon rate4.00%

Market yield (both bonds)4.0%

Bond	Price	Yield −1%	Yield +1%	Gain − loss
Bond A	100.00	+1.91	-1.86	0.05
Bond B	100.00	+8.53	-7.72	0.81

The shaded band is ±1% around the market yield. The steeper curve is the more sensitive bond. A positive “gain − loss” is the bend in the curve at work.

Worked example

Two 4% bonds, three yields

Two bonds with face value 100 and a 4% annual coupon: one matures in 2 years, the other in 10. Yields are annual. Try predicting each step before revealing it.

Short bond vs long bond

Step 1 of 4

1
Price both at three yields
Yield 2-year price 10-year price
3% 101.91 108.53
4% 100.00 100.00
5% 98.14 92.28

Yield	2-year price	10-year price
3%	101.91	108.53
4%	100.00	100.00
5%	98.14	92.28

Read the curve

Steeper curve, more sensitive bond. Flatter curve, safer price. Same coupon and yield, different maturity: the longer bond always has the steeper curve at that point.

Regulatory lens

Banks must measure how changes in interest rates affect the bonds and loans they hold for the long term. This is IRRBB, interest rate risk in the banking book. Basel standards ask banks to revalue their positions under a set of rate shocks and report the change in economic value. Supervisors look hard at banks whose value would fall a lot. L3.5 and L4.5 cover the rules.

Sparks

SparkSilicon Valley Bank, 2023

SVB had put much of a deposit boom into long-dated US Treasuries and agency mortgage bonds when yields were low. As the Federal Reserve raised rates in 2022, their market value fell well below what SVB had paid. When SVB sold some of them in March 2023 and booked a loss, depositors took fright and withdrew billions in a day. The Federal Reserve's own review, published in April 2023, criticised both the bank's management of interest rate risk and its supervision.

SparkSafe is not the same as stable

A government bond held to maturity pays every cent it promised. But until then, its price moves with yields, and a long one can move a lot. “Safe” describes the credit, not the price.

Key takeaways

Bond prices and yields move in opposite directions, because a higher yield discounts every cash flow more.
Longer maturity, lower coupon and lower yield all make a bond's price more sensitive to yield changes.
The price-yield curve bends (convexity): for equal moves, price gains are larger than price losses.
As maturity approaches, prices are pulled towards par, even if yields never change.
Interest rate risk can cause large losses on bonds that never default, as SVB showed in 2023.

Reaction

Check what stuck

Glossary

Interest rate risk: The risk that a change in interest rates changes the value of what you hold.
Price sensitivity: How much a bond's price changes for a given change in its yield.
Convexity: The bend in the price-yield curve: prices rise more when yields fall than they drop when yields rise by the same amount.
Pull to par: The drift of a bond's price towards its face value as maturity approaches, even if yields do not change.
IRRBB: Interest rate risk in the banking book: the risk to a bank's earnings and value from rate changes on loans, deposits and bonds it holds rather than trades.

Deep Dives & Origins

Deep Dives

Frank J. Fabozzi and Francesco A. Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021, chapter on the price volatility characteristics of bonds. Intermediate.
Board of Governors of the Federal Reserve System, Review of the Federal Reserve's Supervision and Regulation of Silicon Valley Bank, April 2023. Intermediate.

Origins

Fabozzi and Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021.
Board of Governors of the Federal Reserve System, Review of the Federal Reserve's Supervision and Regulation of Silicon Valley Bank, April 2023.
Basel Committee on Banking Supervision, Interest rate risk in the banking book, April 2016.

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