Bond Price vs Time Graphs
Bond Price vs Time Graphs
A bond's price follows a predictable path as maturity approaches: if purchased at a discount, price rises toward par; if purchased at a premium, price falls toward par. Time and yield combine to shape this path.
Key takeaways
- When a bond is purchased at a discount, its price rises along a curve toward par as maturity approaches (the pull to par).
- When purchased at a premium, its price falls along a curve toward par.
- A bond held to maturity will always converge to par, regardless of interim price movements.
- Yield changes shift the entire price curve; rising yields move the curve down, falling yields move it up.
- The shape of the curve (steepness) depends on the bond's coupon, maturity, and the current yield.
The standard price-time path: discount bond
Imagine a 10-year Treasury bond with a 3% coupon, purchased when yields are 5%. The bond is worth less than par:
Using the present-value formula, the price is approximately $926. This is a discount bond because the coupon (3%) is below the market yield (5%).
As time passes and the bondholder does nothing (holding it, collecting coupons), what happens to the price? Each year, time advances and one year of cash flows has been received. The remaining bond is a 9-year bond, then 8-year, then 7-year, and so on. At the same 5% yield, the price of a 9-year bond with 3% coupon is about $935. A 5-year bond is about $954. A 2-year bond is about $981. Finally, a bond one day before maturity is worth $999.99, and at maturity, it is worth exactly $1,000.
The price rises smoothly from $926 to $1,000 over the 10-year period. This is the pull to par, visualized. The curve is not straight but curved (convex): the price rises slowly at first, then more quickly as maturity approaches.
The standard price-time path: premium bond
Now consider the same 10-year Treasury, but purchased when yields are 2%. The coupon (3%) is above the yield (2%), so the bond trades at a premium, worth about $1,078.
As time passes at the 2% yield, the bond prices are:
- 10-year bond: $1,078
- 9-year bond (one year later): $1,071
- 5-year bond (halfway to maturity): $1,043
- 2-year bond: $1,015
- At maturity: $1,000
The price falls smoothly from $1,078 to $1,000, again in a curved path. The premium erodes over time, pulled down to par by the certainty of $1,000 repayment. This is the pull to par working in the premium direction.
Shifting the curve: the impact of yield changes
The path above assumes yields remain constant. But yields change, shifting the entire price curve. Suppose the bond bought at a discount ($926 at 5% yield) experiences a yield increase to 6% after one year. The price path changes.
Before the yield change, the 9-year bond (one year remaining) is priced at $935 (assuming 5% yield unchanged). After the yield increase to 6%, the same 9-year bond is worth less, maybe $910. The price curve has shifted down. The pull to par is still operating (the bond converges to $1,000), but the path is lower.
Conversely, if yields fall to 4%, the 9-year bond might be worth $945. The price path has shifted up.
This is why bond funds can post negative returns in high-rate-hike years (like 2022) even though the bonds are being held. The yield-driven shift in the price curve overcomes the pull-to-par effect in the short term. The pull to par reasserts itself over longer horizons.
Comparing different coupons: same maturity, different curves
A high-coupon bond and a low-coupon bond of the same maturity follow different price paths when purchased at the same yield. The high-coupon bond is purchased closer to par (maybe $1,020 vs. $1,000 for the low-coupon bond). As time passes and maturity approaches, both converge to par. But the high-coupon bond descends more steeply (because it started higher), while the low-coupon bond rises more gently (because it started lower).
The key point: they converge to the same terminal value (par), but from different starting points and along different curves. This is why coupon matters for interim price behavior, even though the terminal value is the same.
Zero-coupon bonds: a linear path to par
A zero-coupon bond offers a special case. A 10-year zero purchased at $600 (roughly a 6% yield) has a simple path to par. With no coupons, the only return is the difference between purchase price and face value.
The price-time path is linear (a straight line) if you assume the yield remains constant. Each year, the bond "accretes" in value by a constant amount, reaching $1,000 in exactly 10 years. The path is:
Year 0: $600 Year 1: $636 Year 2: $675 Year 3: $716 Year 4: $759 ... Year 10: $1,000
(The exact numbers depend on the compounding convention, but the idea is the same: a straight line from purchase price to par.)
In reality, yield changes will perturb this linear path, just as they do for coupon bonds. But the long-term trend is a certain convergence to par.
Using graphs to estimate bond prices
In practice, bond traders use price-yield curves (not price-time curves) to estimate prices. A price-yield curve plots the price of a bond (y-axis) against different yield levels (x-axis), holding maturity constant. These curves are downward-sloping (inverse relationship) and convex (curving inward). The shape of the curve indicates the bond's price sensitivity: steeper curves indicate more sensitive bonds.
Price-time curves, on the other hand, show the evolution of price for a specific bond under the assumption that yields do not change. They are useful for illustrating the pull to par and for understanding how much of a bond's return comes from coupons versus price appreciation.
Real-world example: a Treasury bond from 2013 to 2023
Consider a 10-year Treasury bond issued on January 1, 2013, with a 2% coupon. It matures on January 1, 2023.
On issuance, the yield is 2%, so the price is par ($1,000). As time passes and yields fluctuate:
- In 2013–2014, yields fall, and the bond price rises to $1,030+.
- In 2015, yields stay low, and the bond gradually pulls back toward par over time.
- In 2016, yields rise, and the bond price falls to $990.
- In 2017–2019, yields are volatile, but the bond gradually approaches par.
- In 2020, yields fall sharply (pandemic response), and the bond price rises to $1,010.
- In 2021, yields remain low, and the bond price is near par.
- On January 1, 2023 (maturity), the price is exactly $1,000 and the bondholder receives principal.
Over 10 years, the price fluctuated due to yield changes, but the pull to par ensured convergence. The bondholder who held to maturity was indifferent to interim price movements (except for opportunity cost—could they have done better by selling and buying a different bond?).
Flowchart
Next
So far, we have priced bonds using yield to maturity and understood the impact of coupon and maturity on price behavior. But in real markets, bonds are quoted using a different metric: the clean price. The clean price excludes accrued interest, which creates a disconnect between the price you see and the price you actually pay. Understanding this distinction is essential for trading or understanding bond fund quotes.