Last updated: 2026-05-19

Bond yield defined

Bond yield is the annualized return an investor earns by purchasing a bond at its current market price and holding it to maturity. Because bonds trade at prices that differ from their face value — rising when rates fall and falling when rates rise — the stated coupon rate is rarely the actual return. This calculator computes three yield measures from a bond's face value, market price, coupon rate, maturity, and payment frequency: annual coupon, current yield, and yield to maturity (YTM).

The three yield measures

Bonds come with three distinct "yield" numbers that are frequently confused:

The coupon rate is fixed — it is printed on the bond and never changes. The current yield and YTM change every time the market price moves.

Current yield

Current yield is the simplest calculation:

For a $1,000 bond with a 5 % coupon ($50/year) trading at $950, the current yield is $50 / $950 = 5.26 %. It measures the coupon income earned per dollar invested today — nothing more. It ignores the capital gain or loss realized when the bond matures and the issuer repays the face value.

Yield to maturity: the complete return measure

Yield to maturity (YTM) captures both the periodic coupon income and the price-to-face convergence at maturity. It is defined as the discount rate $r$ that satisfies the bond pricing equation:

where $P$ = current price, $F$ = face value, $C$ = annual coupon, $m$ = coupon payments per year, and $N = T \times m$ = total periods. There is no algebraic solution for $r$ — it must be solved numerically.

The textbook approximation

A widely taught closed-form approximation avoids the iterative solver:

where $T$ = years to maturity. The denominator $(F + P)/2$ is the average of face and price — a simple stand-in for the average capital invested. This formula is accurate to within ±0.1–0.5 % for most bonds and is useful for quick estimates or mental math.

The exact solution (Newton-Raphson)

This calculator also provides the numerically exact YTM. Starting from the approximation above as an initial guess, one Newton-Raphson iteration refines the result:

where $f(r) = P(r) - P$ is the pricing error and $f'(r)$ is the derivative of the bond price with respect to the periodic rate. One iteration from a good seed gives roughly eight decimal places of accuracy — well beyond the precision of any published yield table.

Worked example: a discounted semi-annual bond

Scenario: Face value $1,000, market price $950, 5 % annual coupon, 10 years to maturity, semi-annual payments.

Step 1 – Annual coupon: $C = 1{,}000 \times 0.05 = \$50\text{ per year}$

Step 2 – Current yield: $\text{CY} = \frac{50}{950} \approx 5.26\%$

Step 3 – YTM approximation: $\text{YTM}_{\text{approx}} = \frac{50 + (1{,}000 - 950)/10}{(1{,}000 + 950)/2} = \frac{55}{975} \approx 5.64\%$

Step 4 – Exact YTM (Newton-Raphson): approximately 5.66 % — the discount applied to 20 semi-annual $25 payments plus the $1,000 face value at maturity that produces a present value of $950.

Note that YTM (5.66 %) exceeds the coupon rate (5 %) because the bond was bought at a discount: the investor earns the coupon plus gains $50 ($1,000 − $950) at maturity.

The price-yield inverse relationship

Bond prices and yields always move in opposite directions. The mechanics are simple: a bond's coupon payments are fixed. When prevailing market rates rise, newly issued bonds offer higher coupons, making existing lower-coupon bonds less attractive. The only way to make an existing bond competitive is to lower its price. A lower price means a higher current yield and a higher YTM.

The reverse applies when rates fall: existing bonds with higher coupons become attractive, investors bid up prices, and yields compress.

This inverse relationship has a practical implication for bond investors: buying at a discount (price < face value) raises the return above the coupon rate; buying at a premium (price > face value) lowers it below the coupon rate.

Par, discount, and premium bonds

These relationships are exact when coupon and YTM share the same compounding frequency, and approximately true in general.

Zero-coupon bonds

A zero-coupon bond pays no periodic coupons — the entire return comes from buying below face value and receiving face at maturity. The pricing equation simplifies to:

Zero-coupon bonds have the highest interest-rate sensitivity (duration) of any bond class because all cash flow arrives at maturity. A 1 % rise in rates causes a larger price drop for a 20-year zero than for a 20-year coupon bond.

Assumptions and limitations

YTM assumes:

1. Reinvestment at YTM. All coupon payments are reinvested at the same rate as the YTM. In reality, reinvestment rates vary over time, so the actual realized return may differ.
2. Held to maturity. A sale before maturity makes the actual return depend on the sale price, not the face value.
3. No default. YTM ignores credit risk. Compare YTM to the risk-free rate (U.S. Treasury yield) to gauge the credit spread.
4. No call provisions. Callable bonds may be redeemed early, making yield to call (YTC) more relevant than YTM for premium-priced callable bonds.