CFA Level I Exam · Fixed-Income Bond Valuation: Prices and Yields

Pull to Par and Constant-Yield Price Trajectory Explained

Updated 7 October 2026 · Fact-checked

Pull to par is the tendency of a bond's price to move toward its par value as maturity nears, if its yield to maturity stays constant. A discount bond rises and a premium bond falls. To solve it, reprice the bond with the remaining periods at the same yield, or use new price = old price × (1 + r) − coupon.

Understand Pull to Par and Constant-Yield Price Trajectory

A bond's price is the present value of its remaining cash flows, discounted at its yield to maturity (YTM). Time passes, so fewer cash flows remain. On the maturity date the only cash flow left is the final coupon and par, so the price must equal par (plus that coupon).

This is the pull to par effect. Suppose the yield does not change. A discount bond (coupon rate below YTM) was priced below par. Its price rises toward par. A premium bond (coupon rate above YTM) was priced above par. Its price falls toward par. A par bond (coupon rate equal to YTM) stays at par. A zero-coupon bond is always a discount bond, so its price rises toward par every period.

The path of prices at a constant yield is the constant-yield price trajectory. It is not a straight line. Each period the price grows at the yield and drops by the coupon paid: price now × (1 + r) − coupon. For a discount bond the price gain is small at first and larger later. For a premium bond the price drop is small at first and larger later.

The same idea sits behind accounting for bonds under the effective interest rate method. The carrying value of a bond follows the constant-yield path set at issuance. Interest expense is the yield times the opening carrying value. The difference between the coupon and that interest amortizes a premium or accretes a discount.

This only holds when the yield is constant and the issuer does not default. In practice yields change, and a change in yield moves the price in addition to the pull to par. The exam usually separates the two effects: first the time effect at the same yield, then the yield effect.

Key formulas to remember

Bond price at any time: PV = Σ C ÷ (1 + r)^t + FV ÷ (1 + r)^N; Use r as the yield per period and N as the periods remaining. For the trajectory, keep r fixed and reduce N.
Price roll-forward (just after a coupon): P(t+1) = P(t) × (1 + r) − C; Use per-period r. The price grows at the yield, then the coupon is paid out. It is a quick check on any trajectory.
Premium amortization per period: Amortization = C − r × P(t); Positive for a premium bond. The price falls by this amount.
Discount accretion per period: Accretion = r × P(t) − C; Positive for a discount bond. The price rises by this amount.
Direction rule at constant yield: Coupon rate < YTM: price rises. Coupon rate > YTM: price falls. Equal: price stays at par.; Price equals par at maturity in every case, ignoring the final coupon.

How to solve Pull to Par and Constant-Yield Price Trajectory questions

Use this method for any question on how a bond's price or carrying value changes with time when the yield stays the same.

1Compare the coupon rate with the YTM per period. This tells you whether the bond is a discount, premium or par bond and the direction of the price move.
2Check the claim. A discount bond must rise toward par and a premium bond must fall toward par. Eliminate any option that moves the wrong way.
3Write down the remaining number of periods N and the periodic yield r. Convert annual figures to the coupon frequency.
4Reprice the bond with the reduced N at the same r. Or use P(t+1) = P(t) × (1 + r) − C when you know the earlier price.
5If the question asks for amortization or accretion, compute |C − r × opening price|. Interest is always yield × opening price.
6Check the answer: it should lie between the old price and par, and it should not cross par.
7Round at the end and match the option to your answer.

Quickest way: Roll-forward shortcut

When to use it: Use it when you are given or can quickly compute today's price and the question asks for the price after one or two periods, or for amortization.

Get the opening price P0, the periodic yield r and the coupon C per period.
Compute interest = r × P0.
Next price = P0 + interest − C. If C is more than interest the price falls. If less, it rises.
Sanity check: the next price lies between P0 and par.
Amortization or accretion is simply |C − interest|. No repricing is needed.

Common mistakes in Pull to Par and Constant-Yield Price Trajectory

Saying every bond's price rises as maturity approaches.
Candidates remember that price converges to par and forget it can converge from above.
Fix: Compare the coupon rate with the YTM first. Below YTM, price rises. Above YTM, price falls. At par, it stays.
Treating the price path as a straight line.
Straight-line amortization is simple, and it is used in some accounting contexts.
Fix: At a constant yield the change each period is C − r × P(t), which differs every period. The path is curved. A straight line is not the effective interest method.
Confusing the time effect with the effect of a yield change.
Questions mention both the passage of time and a change in market yield.
Fix: Separate them. Step one: reprice at the same yield with fewer periods. Step two: reprice at the new yield. The question may ask for only one.
Using the annual yield with semiannual coupons.
The roll-forward formula needs the periodic rate, and candidates forget to halve it.
Fix: Divide the annual yield by the coupon frequency, and count the periods that remain, not years.
Computing interest from the coupon rate times par instead of the yield times the carrying value.
Coupon = coupon rate × par is familiar, so candidates reuse it for interest.
Fix: The coupon is cash paid: coupon rate × par. Interest at the effective rate is YTM × opening price. Amortization is the difference.
Thinking a par bond's price changes with time.
Candidates assume every bond moves with time.
Fix: If the coupon rate equals the yield, the price stays at par at constant yield. There is nothing to pull.

Worked examples

Example 1

A 3-year, 6% annual-pay bond with par of $1,000 is priced to yield 8%. The yield stays at 8%. What is the price one year later, just after the first coupon is paid? A) $948.46 B) $964.33 C) $1,024.33

Show the solution

The coupon rate 6% is below the YTM 8%, so this is a discount bond. The price must rise toward par. The option $948.46 is today's price, so it would not be the answer.
Today's price: 1.08^3 = 1.259712, so 1,000 ÷ 1.259712 = 793.83. The annuity factor is (1 − 0.793832) ÷ 0.08 = 2.57710, so the coupons are worth 60 × 2.57710 = 154.63. P0 = 948.46.
Roll forward: P1 = 948.46 × 1.08 − 60 = 1,024.34 − 60 = 964.33 (using unrounded values).
Check by repricing with 2 years left: 60 ÷ 1.08 + 1,060 ÷ 1.1664 = 55.56 + 908.78 = 964.33. This matches.
$1,024.33 is the value before the coupon is paid, so it is a trap. The price is above $948.46 and below $1,000, as expected.

Answer: B) $964.33

Example 2

A 2-year, 7% annual-pay bond with par of $1,000 is priced to yield 5%, so the price is $1,037.19. If the yield stays at 5%, how much of the premium is amortized during the first year? A) $18.14 B) $51.86 C) $70.00

Show the solution

The coupon rate 7% is above the YTM 5%, so this is a premium bond. The price falls toward par.
Interest at the effective rate = 5% × 1,037.19 = 51.86.
The cash coupon = 7% × 1,000 = 70.00.
Amortization = coupon − interest = 70.00 − 51.86 = 18.14.
Check: the price after one year = 1,037.19 − 18.14 = 1,019.05. Repricing: 1,070 ÷ 1.05 = 1,019.05. This matches, and the price is still above par.

Answer: A) $18.14

Exam tips

Do the direction check first. Comparing the coupon rate with the YTM can eliminate one or two options in seconds.
A price that lands past par is wrong at constant yield. Use par as a boundary to cross out options.
Read carefully whether the price is quoted just after a coupon. Options can include a pre-coupon value as a trap.
For amortization questions, use C − r × P. It is faster than computing two prices on a calculator.
If the question says the yield changes, price the time effect and the yield effect separately.

Practice questions from Fixed-Income Bond Valuation: Prices and Yields

Pull to Par and Constant-Yield Price Trajectory: frequently asked questions

What is the pull to par effect?

It is the movement of a bond's price toward par as maturity gets closer, when the yield does not change. Discount bonds rise and premium bonds fall. At maturity the price equals par.

Does the price path to par follow a straight line?

No. At a constant yield the price change each period is the coupon minus (or plus) the yield times the opening price. That amount changes each period, so the path is curved.

How is this linked to bond carrying value in financial reporting?

Under the effective interest rate method, the carrying value follows the same constant-yield path set at issue. Interest expense is the issue yield times the opening carrying value. The gap to the coupon is the amortization of a premium or discount.

Does pull to par apply to zero-coupon bonds?

Yes. A zero-coupon bond is a discount bond. Its price grows at the yield each period until it reaches par at maturity.