What is a bond yield?
– A bond yield is the annual return an investor receives from holding a bond, expressed as a percentage of invested capital. It is a measure of the bond’s income-generating power and can be calculated several different ways depending on what you want to measure (current income, total expected return to maturity, or an annualized rate that accounts for compounding).
Key yield definitions (short)
– Coupon rate: The fixed annual interest the bond pays, stated as a percentage of the bond’s face (par) value. It is set when the bond is issued.
– Current yield: Annual coupon payment divided by the bond’s current market price. Measures the income return at today’s price; ignores capital gains or losses and time value of money.
– Yield to maturity (YTM): The single discount rate that makes the present value of all future coupon payments and the principal repayment equal to the bond’s current market price. YTM is the most complete single-number estimate of the bond’s expected annualized return if held to maturity and if coupons are reinvested at the YTM rate.
– Bond equivalent yield (BEY): A convention that annualizes a semi‑annual yield by multiplying the semi‑annual yield by two. It is a simple annualization, not an effective (compounded) rate.
– Effective annual yield (EAY): The true annual return after taking compounding into account. For semiannual compounding, EAY = (1 + semiannual rate)^2 − 1.
Why yields matter
– Bond yields move inversely to bond prices: when market interest rates rise, existing bond prices fall (and yields rise); when rates fall, bond prices rise (and yields fall).
– Different yield measures answer different investor questions: coupon rate tells you the stated interest; current yield tells you the income relative to today’s price; YTM estimates the total annualized return to maturity, accounting for price paid and time value of money.
Core formulas (concise)
– Coupon rate = Annual coupon payment / Face value
– Current yield = Annual coupon payment / Bond market price
– Price = Σ_{t=1 to N} [C / (1 + y)^t] + FV / (1 + y
^N]. (Here C = coupon per period, FV = face (par) value, y = period yield, N = number of periods.)
– Yield to maturity (YTM) — the periodic rate y that makes the present value of a bond’s cash flows equal to its market price; in other words, YTM is the internal rate of return (IRR) on the bond if you hold to maturity and reinvest coupons at the same rate. There is no simple closed-form algebraic solution for y; you solve the price equation numerically (financial calculator, spreadsheet RATE/IRR, or iterative methods).
– Useful approximation for annual YTM (when exact solution is not needed):
YTM ≈ [Annual coupon + (Face value − Price) / Years to maturity] / [(Face value + Price) / 2]
(This gives a quick, rule-of-thumb annual rate; it assumes annual coupons and linear average pricing.)
– Semiannual coupon notes — common in U.S. corporate and Treasury bonds. If coupons are paid twice a year:
– Compute the per-period yield y_semi by solving the price equation with C = annual coupon/2 and N = years×2.
– Convert to a quoted (nominal) annual YTM (bond-market convention) by: YTM_nominal = 2 × y_semi.
– Convert to the effective annual yield (accounting for compounding) by: YTM_effective = (1 + y_semi)^2 − 1.
– Yield to call (YTC
: YTC) — Yield to call (YTC) is the yield calculated assuming the issuer redeems (calls) the bond at the earliest allowable call date and at the stated call price. Callable bonds let the issuer retire the bond early, typically when interest rates fall. To compute YTC, use the same present-value equation as for YTM but replace:
– N (periods) with the number of coupon periods until the call date, and
– FV (future value) with the call price (the price issuer pays at call).
Formula (annual-pay bond, for clarity):
Price = sum_{t=1}^{N_call} (C / (1 + y)^{t}) + (CallPrice / (1 + y)^{N_call}),
where C = annual coupon, N_call = years to call, y = YTC.
Practical steps (worked example):
– Bond face value = $1,000, coupon = 6% (annual), call price = $1,020, callable in 5 years, market price = $1,050.
– Solve for y in: 1,050 = sum_{t=1}^{5} (60 / (1+y)^t) + (1,020 / (1+y)^5).
– Use a financial calculator or spreadsheet function (Excel: =RATE(nper, pmt, -pv, fv)). In Excel: =RATE(5, -60, 1050, -1020) returns y ≈ 3.11% (annual).
Interpretation: If y (YTC) < YTM, the bond is more likely to be called in a falling-rate environment; investors use YTC to assess reinvestment and call risk.
Yield to put (YTP) — similar to YTC but for putable bonds (investor has the right to sell back to issuer at put date/price). Compute by replacing N and FV with put terms.
Yield to worst (YTW) — the most conservative (lowest) yield an investor can receive, assuming the issuer exercises options in the way that delivers the lowest yield to the bondholder. Calculation checklist:
1. Compute YTM.
2. Compute YTC for every permitted call date (if callable).
3. Compute YTP for every permitted put date (if putable).
4. YTW = minimum of all those yields.
Use YTW when evaluating bonds with embedded options; it is a common disclosure for bond funds.
Current yield — a simple, short-term measure of income:
Current yield = Annual coupon payment / Current market price.
Example: A bond paying $60/year and trading at $1,050 → Current yield = 60 / 1,050 = 5.71%.
Limitation: ignores capital gain/loss at maturity and time value of money.
Coupon (nominal) rate — the periodic coupon divided by face value. Fixed at issue and does not change with price. Example: 6% coupon on $1,000 face = $60/year.
Real yield (inflation-adjusted) — converts nominal yield into purchasing-power terms using the Fisher relation:
1 + real = (1 + nominal) / (1 + inflation).
Approximation for small rates: real ≈ nominal − inflation.
Example: nominal yield 5.0%, expected inflation 2.0%:
Real yield = (1.05 / 1.02) − 1 ≈ 2.94%.
Tax-equivalent yield — useful for comparing tax-exempt municipal bond yields with taxable bonds:
Tax-equivalent yield = Tax-exempt yield / (1 − marginal tax rate).
Example: muni yield = 3.00%, marginal federal tax rate = 24% → Tax-equivalent = 3.00% / 0.76 = 3.95%.
Zero-coupon (spot) yields and the spot curve — a zero-coupon or spot rate is the yield on a risk-free zero-coupon instrument for a given maturity. Bootstrapping is the method to derive spot rates from coupon-bearing bond