Bond Pricing at Par, Premium, and Discount

Par, premium, and discount

Par value, also called face value or principal, is the amount the issuer promises to repay at maturity under the bond's terms. A bond trades at par when its market price equals that amount. It trades at a premium when price exceeds par and at a discount when price is below par.

A bond's market price reflects the present value of expected coupon payments and principal, discounted at a yield investors require for the bond's risk and cash-flow features. If a new bond offers a coupon that matches market yields for comparable risk and maturity, its price is near par. If the coupon is less attractive, the price generally falls below par.

A bond with a coupon above current required yields may trade at a premium because its scheduled payments are more attractive than new issues. The investor pays more than face value but receives the higher coupon. At maturity, the issuer generally repays face value, so the premium is gradually reflected in the bond's return if held to maturity.

A discount bond sells below face value. The investor receives coupon payments and, if the issuer pays as promised, the par amount at maturity. The difference between purchase price and maturity value contributes to the investor's return. A discount can result from higher market yields, issuer credit concerns, liquidity, or security-specific terms.

Par, premium, and discount describe price relative to face value, not credit quality. A premium bond can be risky, and a discount bond can be creditworthy but have a low coupon relative to market rates. Identify why the price differs before interpreting the label.

Why price and yield move inversely

A fixed-rate bond's cash payments are set by its terms. When market yields rise, investors can buy new securities with higher required returns. Existing lower-coupon payments become less attractive, so their market price generally falls. When market yields decline, existing payments look more attractive and the price generally rises.

The relationship comes from discounting. A future dollar is worth less today when discounted at a higher rate. Raising the discount rate lowers the present value of the bond's coupons and principal. Lowering the rate raises present value. This is the same time-value-of-money principle used in other bond valuation problems.

The price change depends on more than the coupon. Remaining maturity, payment frequency, yield level, credit spread, call features, and liquidity all matter. A longer bond's cash flows extend farther into the future and are generally more sensitive to a yield change, all else equal. Embedded options can alter the pattern.

A change in benchmark Treasury yields is only one driver. A corporate bond's credit spread can widen or narrow independently. The price can fall if the issuer's credit risk worsens even when Treasury yields are stable. A municipal bond may also move because tax demand or liquidity changes.

Calculate a bond's present value

For a plain fixed-coupon bond, price is the present value of its coupon stream plus the present value of principal at maturity. If coupons are annual, price = Σ[C ÷ (1 + y)^t] + F ÷ (1 + y)^n, where C is annual coupon dollars, y is the required annual yield, F is face value, and n is years to maturity.

Suppose a $1,000 bond pays a 5% annual coupon, or $50 per year, and has 10 years remaining. If comparable market yield is 5%, discounting the $50 payments and $1,000 principal at 5% gives a price of $1,000. The coupon and required yield match, so the bond is at par under these assumptions.

Now suppose required yield rises to 6%. If coupons are paid semiannually, each coupon is $25, each period's yield is 3%, and there are 20 periods. Price equals $25 times the present-value annuity factor for 20 periods at 3%, plus $1,000 discounted for 20 periods at 3%. The result is about $925.60, below par.

If required yield falls to 4%, each semiannual period uses a 2% yield. The same $25 coupons and $1,000 maturity payment now have greater present value. The price is about $1,081.75, above par. The fixed $50 annual coupon did not change; only the discount rate changed.

The formula assumes the issuer makes promised payments, the bond has no embedded options, and the yield convention matches the payment schedule. Many bonds use semiannual payments and annualized yields, so the periodic coupon and periodic discount rate must be used consistently. Accrued interest can also make the settlement amount differ from a quoted clean price.

Premium pricing and pull to par

When a premium bond approaches maturity, its market price tends to move toward face value if required yields and credit conditions remain stable and the issuer pays. The investor receives coupon cash flows and ultimately par, which is less than the premium purchase price. The high coupon is balanced by the expected decline toward par.

For example, an investor buys a $1,000-face bond for $1,080. If it is held to maturity and repays $1,000, the investor faces an $80 principal difference, offset by coupon income over time. Yield to maturity incorporates both the coupon stream and the movement from price to maturity value. Coupon alone overstates the return if viewed as total yield.

A callable premium bond has additional risk. If the issuer calls it at a specified price before maturity, the investor may receive less than the purchase price sooner than expected. The high coupon may not continue for the full stated term. Compare yield to call and yield to worst, using the actual call dates and prices.

Premium amortization and tax accounting can affect after-tax return for taxable investors. The treatment depends on bond type, account, and applicable rules. A market price premium does not by itself determine taxable income or the investor's basis.

Discount pricing and pull to par

A discount bond purchased below face value may rise toward par as maturity approaches if its yield and credit remain unchanged. The investor can receive coupon income plus a difference between purchase price and the principal paid at maturity. If the issuer defaults or the bond is sold early, that expected path may not occur.

A $1,000-face bond bought at $920 has an $80 difference to par. This is not an $80 guaranteed profit if held; payments depend on issuer performance, the bond terms, and whether the investor actually holds to maturity. The discount may reflect credit deterioration, illiquidity, or a higher market yield.

If market rates rise further after purchase, the discount bond's price may fall more even though time passes. If its credit spread widens, price can also decline. Pull to par is a conditional relationship, not a floor or promise.

Zero-coupon bonds are issued or trade at a discount and pay no periodic coupon. Their return comes from the difference between purchase price and maturity value, assuming payment. Because all value is received at maturity, price can be sensitive to yield changes and there is no coupon income to reinvest.

Clean price, dirty price, and accrued interest

A bond quote may show a clean price that excludes accrued interest since the last coupon date. The buyer usually pays the clean price plus accrued interest at settlement, called the dirty or full price. The seller receives accrued interest for the time they held the bond since the prior payment.

This convention avoids confusing the market price movement with the coupon accrual between payment dates. If a bond is quoted at 101, that usually means 101% of face value before accrued interest, subject to market convention. A $1,000 bond would have a clean quoted value of $1,010, with accrued interest added to the settlement amount.

Accrued-interest day counts vary by instrument and market. The trade confirmation identifies price, accrued interest, settlement date, and total consideration. Do not assume the quoted price is the full cash outlay.

Worked case: compare coupons and prices

Bond A has $1,000 face value, a 4% annual coupon, and trades at $920. Bond B has $1,000 face value, a 6% coupon, and trades at $1,080. Both have similar remaining maturity and credit for this simplified comparison. A pays $40 annual coupon and is at a discount; B pays $60 and is at a premium.

The current market environment requires a yield around 5% for similar bonds. Bond A's lower coupon is compensated by its discount toward par; Bond B's higher coupon is offset by its premium. Their yields to maturity may be near each other even though coupons and prices differ. Calculate cash flows rather than ranking by coupon.

If Bond B is callable at $1,000 soon, the $80 premium may be at risk of being paid back only to par on a call. If Bond A's issuer credit weakens, the $80 discount may reflect expected loss, not simply attractive pull to par. Structure and credit must be compared.

If the investor sells after one year, the realized return depends on sale price, coupons received, accrued interest, fees, and market changes. The initial premium or discount does not reveal the one-year outcome. A client who may sell early should focus on market-value risk.

A practical bond price analysis

Confirm face value, coupon rate, payment frequency, maturity, call schedule, current price, accrued interest, and the quoted yield. Determine whether the bond is fixed-rate, floating-rate, zero-coupon, or callable. Identify issuer credit, seniority, and collateral.

Compare coupon with required yield. If the coupon matches the market yield for comparable risk and cash flows, price is near par. If coupon is lower, price generally needs to be below par; if higher, price generally rises above par. Optionality, credit, and liquidity can shift the price.

Use present-value math with periodic cash flows. Convert annual yields and coupons to the correct per-period amounts. Include accrued interest when calculating purchase cost. Use yield to maturity or yield to call for total return comparisons rather than relying on coupon or current yield alone.

For a client, consider maturity, call risk, duration, credit, liquidity, tax effects, and need for cash. A premium bond's high coupon can be attractive but may be called; a discount bond may offer pull to par but can also signal risk. Explain assumptions and possible outcomes.

Read a quoted price and calculate the position value

Bond prices are often quoted as a percentage of face value. A quote of 97.25 usually means $972.50 for each $1,000 face amount before accrued interest. A client holding $25,000 face value would multiply $25,000 by 0.9725 to estimate a clean market value of $24,312.50. The quote is not a yield and does not tell you the cash needed at settlement.

If the same bond is quoted at 103.40, the clean value of $25,000 face is $25,850. That premium may reflect a coupon above current market rates, but confirm the bond terms and comparable yields before explaining the cause. A credit event, scarcity, or embedded option can also affect price. Use the quote convention stated in the question if it differs from the usual percentage-of-par format.

A client says a bond price fell from 104 to 98 and concludes that the coupon must have been cut. Separate the price from the contract: a fixed coupon normally remains unchanged while market price moves. Ask whether benchmark rates, credit spreads, or liquidity changed, then calculate the dollar effect. On $10,000 face, a move from 104 to 98 reduces clean value from $10,400 to $9,800, a $600 market-value decline before accrued interest.

The same percentage-point move has a different dollar effect for different positions. A six-point decline on $50,000 face is approximately $3,000; on $5,000 face it is approximately $300. This calculation measures the quoted-price change. It does not estimate future loss or account for coupon income, accrued interest, fees, or a later sale price.

Check the assumptions behind a price estimate

When using a present-value estimate, make the cash-flow dates explicit. If a bond pays twice a year, count each coupon date and the maturity payment separately. Divide the stated annual coupon by two, convert the annual yield to the matching periodic rate under the convention given, and discount each cash flow for its number of periods. Mixing annual cash flows with a semiannual rate produces a price that does not match the contract.

A spreadsheet can verify the arithmetic, but its inputs still need review. Confirm whether the yield is nominal or effective, whether settlement occurs between coupon dates, and whether the quoted price is clean or full. For a callable issue, also price the relevant redemption scenarios. A useful answer explains why the estimate is above or below par and which assumptions could change it.

How to answer Series 65 questions

At par means price equals face value; premium means price exceeds face value; discount means price is below face value. For fixed-rate bonds, price and required yield generally move in opposite directions.

If coupon equals market yield for comparable risk and maturity, price is generally at par. A coupon below market yield generally means a discount; a coupon above market yield generally means a premium. Use the cash-flow present-value relationship if the problem supplies numbers.

Do not confuse coupon rate, current yield, and yield to maturity. Consider maturity value, payment timing, accrued interest, call features, and issuer credit. A premium or discount is a price relationship, not a guarantee of return.

Sources

TreasuryDirect, Understanding Pricing and Interest Rates; OpenStax Principles of Finance, Bond Valuation; NASAA Series 65 Test Specifications, Section I. All example prices and yields are hypothetical.