Definition — “At par”
– “At par” means a security is trading at its face value (also called par or nominal value). For a bond, preferred share, or similar debt instrument, trading at par means the market price equals the amount the issuer promised to repay at maturity.
Key terms (concise)
– Par (face) value: the fixed dollar amount printed on a bond or share when issued (commonly $1,000 or $100 for bonds).
– Coupon rate: the stated annual interest the bond pays, expressed as a percentage of par.
– Market price: the current trading price, which can be above par (premium) or below par (discount).
– Current yield: coupon payment divided by the bond’s market price (coupon / price).
– Yield to maturity (YTM): the total expected return if the bond is held to maturity, accounting for price paid, coupon payments, and repayment of par.
Why bonds rarely stay at par
– Par is fixed at issuance; market price moves. Interest-rate changes, shifts in the issuer’s creditworthiness, and remaining time until maturity push prices away from par.
– When market interest rates differ from a bond’s coupon rate, buyers demand an adjusted price so the bond’s effective return matches market rates.
Issuing at par, discount, or premium
– Issued at par: the issuer receives the face value at sale.
– Issued at a discount: the issuer sells below face value to make the bond’s effective yield competitive with market rates.
– Issued at a premium: the issuer sells above face value when the coupon is attractive relative to current rates.
How coupon, price, and yields relate (simple rules)
– If market yields = coupon rate → bond price ≈ par (trades at par).
– If market yields > coupon rate → bond price < par (discount).
– If market yields par (premium).
– A bond trading at par has current yield equal to its coupon; when price changes, current yield differs from the coupon.
Quick numeric example (one-year bond, simple case)
Assumptions:
– Par value = $1,000
– Coupon rate = 5% → annual coupon = $50
Case A — market yield = 10%:
– Price = (coupon + par) / (1 + market yield) = (50 + 1,000) / 1.10 = $954.55
– Price par → bond trades at a premium.
Short checklist: How to check whether a bond is at par and why
1. Find the bond’s par (face) value — typically $1,000 or $100.
2. Confirm the coupon amount (coupon rate × par).
3. Look up current market price or quote (e.g., quoted as 100 = 100% of par).
4. Compare price to par:
– Price = par → trading at par.
– Price > par → premium; Price < par → discount.
5. Compare coupon rate to prevailing market yields for similar credit quality and maturity.
6. Check credit rating and time to maturity — both affect price sensitivity to rate moves.
7. For valuation, compute current yield and (for multi‑period bonds) yield to maturity.
Practical note about common stock par value
– Par value on common shares is largely a legal formality in modern practice (often set very low). It does not determine market price.
Formulas (compact)
– Current yield = annual coupon / current market price.
– One-period price example: Price = (coupon + par) / (1 + market yield) — for a one‑year bond.
– For multi-year bonds, price is the present
…value of the bond’s future coupon payments and final principal, discounted at the prevailing market yield.
Formal price formula (fixed‑rate bond)
Price = sum_{t=1}^{n} (C_t / (1 + y)^t) + (F / (1 + y)^n)
Where:
– C_t = coupon payment at time t (for level coupons, C_t = C each period)
– F = face (par) value returned at maturity
– y = market yield per period (expressed in same period basis as coupons)
– n = total number of periods remaining
Why a bond trades at par
A bond trades at par (price = F) when its coupon rate equals the market yield for comparable risk and maturity. Algebraic proof for level coupons (annual payments):
If C = F · y, then
Price = C · sum_{t=1}^n 1/(1+y)^t + F/(1+y)^n
Use the geometric sum: sum_{t=1}^n 1/(1+y)^t = [1 − 1/(1+y)^n]/y
So Price = F·y·[1 − 1/(1+y)^n]/y + F/(1+y)^n = F
Hence coupon = market yield ⇒ price = par.
Worked numeric examples
1) Three-year annual bond at par
– Face F = $1,000
– Annual coupon rate = 5% ⇒ C = $50
– Market yield y = 5% (annual)
Price = 50/(1.05) + 50/(1.05^2) + 1,050/(1.05^3)
Compute:
50/1.05 = 47.6190
50/1.1025 = 45.3515
1,050/1.157625 = 907.0295
Sum ≈ 1,000.00 → bond trades at par.
2) Same bond, yield rises to 6%
Price = 50/1.06 + 50/1.06^2 + 1,050/1.06^3
Compute:
50/1.06 = 47.1698
50/1.1236 = 44.4896
1,050/1.191016 = 881.7076
Sum ≈ 973.37 → bond trades at a discount (Price < Par).
3) Semiannual coupon adjustment (common in U.S. corporate and Treasury bonds)
– If coupon is 5% annually but paid semiannually, C_per = 25, y_per = y_annual/2, n = years·2.
Always convert coupon and yield to the same periodic basis before using the formula.
Practical checklist: is a bond at par?
– Compare coupon rate to current market yield for similar credit/maturity. If equal (on same periodic basis), bond should be near par.
– Check time to maturity—short maturities reduce sensitivity to yield changes, so price may be close to par even if yields differ slightly.
– Confirm coupon frequency (annual vs semiannual) and use matching yield periodicity.
– Adjust for accrued interest: the traded "clean" price can be near par while the "dirty" price (clean + accrued interest) differs.
– Consider optional features (callable, putable): these contract terms can push market price away from par even when coupon ≈ yield.
– Validate credit risk and liquidity: downgrades or illiquidity can cause wide deviations from par.
Quick note on clean vs dirty price
– Clean price = quoted market price excluding accrued interest.
– Dirty price (or full price) = clean price + accrued interest since last coupon. Brokers typically quote clean prices; settlement uses dirty price. Accrued interest can make the settlement amount slightly above or below par even when the clean price equals par.
Common pitfalls and assumptions
– The “coupon = yield ⇒ price = par” result assumes identical credit risk and yield term structure (flat yield) and that coupons are level and paid on the same periodic basis as the yield. Nonstandard features (floors, step-ups), embedded options, or changing credit spreads violate these assumptions.
– Yield to maturity (YTM) is the single discount rate that equates price with the present value of promised payments; solving for YTM often requires numerical methods for multi‑period bonds.
Sources for further reading
– Investopedia — “At Par” https://www.investopedia.com/terms/a/at-par.asp
– U.S. Securities and Exchange Commission (SEC) — “Bonds: What You Need To Know” https://www.investor.gov/introduction-investing/investing-basics/investment-products/bonds
– FINRA — “Bond Prices and Yields” https://www.finra.org/investors/learn-to-invest/types-investments/bonds/bond-prices-and-yields
Educational disclaimer
This explanation is for educational purposes and does not constitute individualized investment advice. Always consider consulting a licensed professional before making investment decisions.