Price and yield — the inverse relationship

A bond's price is the present value of its promised cash flows discounted at the prevailing market yield. Price and yield move in opposite directions because the cash flows are fixed at issuance: when the market demands a higher yield, the same cash flows are discounted more heavily and the bond's price falls; when the market accepts a lower yield, the same cash flows are discounted less heavily and the bond's price rises. The inverse price-yield relationship is the single most-important fact in fixed income and the foundation that the duration, convexity, and yield-curve mechanics in subsequent modules build directly on.

The pricing formula in full

                    N
                  ─────       CF_t
        P     =     ∑      ─────────
                    t = 1     (1 + y)^t

For a typical fixed-coupon bond, CF_t expands to:

      CF_t  =  C        for t = 1, 2, … N−1
      CF_t  =  C + F    for t = N    (final coupon plus principal at maturity)

Variable glossary — every symbol explained

• P — the bond's full price today, also called the 'dirty price' because it includes any accrued interest. Reported in currency units per unit of face value (e.g. USD 1,012.50 against a USD 1,000 face).
• N — the total number of cash-flow dates remaining until maturity. A 10-year annual-coupon bond has N=10; a 10-year semi-annual bond has N=20. Doubling N roughly doubles a bond's exposure to long-term yield changes but only when the new cash flows are pushed out in time.
• t — the index running over each cash-flow date, expressed in years from today. For semi-annual bonds, t takes values 0.5, 1.0, 1.5, … N/2. Time enters the formula in the exponent of the discount factor, which is why longer cash flows are penalised more heavily by higher yields.
• CF_t — the cash flow received at time t. For a plain bond this is C (the coupon) for every period except the last, when it becomes C + F. For floaters, sinking funds, or amortising bonds, CF_t becomes more complex but the formula's shape stays the same.
• C — the periodic coupon payment in currency units. For a 6% annual-coupon bond on USD 1,000 face, C = USD 60; for the same bond paid semi-annually, C is USD 30 per period and N doubles.
• F — the face value (par value, principal). The amount returned at maturity. For US corporate bonds typically USD 1,000; for Kenyan T-bonds typically KES 50,000 or KES 1,000,000. F sets the denomination but does not by itself drive value once everything is scaled to it.
• y — the yield to maturity, expressed in decimal form (0.06 for 6%). The single discount rate that equates the present value of all promised cash flows to the bond's current market price. y is set by the market every minute the bond trades.
• (1 + y)^t — the discount factor that brings a future cash flow at time t back to today. A higher y means the discount factor at every date is larger, pulling all PVs down. The exponent t is what makes distant cash flows lose value disproportionately as y rises — and what makes long-duration bonds so sensitive to yield moves.

The semi-annual convention — same formula, halved

US Treasuries, most US corporate bonds, and many emerging-market sovereigns pay coupons twice a year. The pricing formula is unchanged in shape; you simply substitute C/2 for the coupon, 2N for the period count, and y/2 for the per-period discount rate. The mathematical reason: the bond's cash flows arrive every six months, so the periodic compounding rate is half the annualised yield. Forgetting this halving when pricing a semi-annual bond is one of the most common analyst errors and produces a price difference of around 1-2 percent on a mid-tenor bond.

Three yield measures, three different questions

• Yield to maturity (YTM, y): the single discount rate that solves the pricing formula above. Answers 'what total return will I earn if I buy at today's price and hold to maturity, reinvesting coupons at the same y?' The market's go-to measure.
• Current yield: the annual coupon C divided by the current price P. Answers 'what cash income am I receiving as a percentage of my outlay?' Ignores capital gain or loss to maturity. Useful for income-focused investors but misleading for total-return comparison.
• Running yield: synonymous with current yield in UK convention; in US usage sometimes refers to coupon divided by face value (i.e., the coupon rate). Always check the definition before quoting because the same words mean different things across desks.

Why prices and yields move inversely — fully unpacked

Consider a 10-year bond issued at par with a 5% coupon when market yields were 5%. Plugging into the formula at issuance, P = USD 1,000, every cash flow's present value sums to exactly face value. Suppose the market yield rises to 6% the next day. The bond's contractual cash flows have not changed — it still pays 50 per year for 9 years and 1,050 in year 10. But now those cash flows must be discounted at 6% instead of 5%. Each discount factor is larger, each present value is smaller, and the sum (the new price) falls below 1,000. The bond is now a 'discount bond' — its price is below par — because its locked-in 5% coupon is unattractive relative to fresh 6% bonds in the market.

Conversely, if yields fall to 4%, the bond's 5% coupon is suddenly more attractive than what the market is willing to offer on new issues. Buyers bid the price above par; the bond becomes a 'premium bond'. The magnitude of these moves depends on duration, which the next module formalises.

Clean price, dirty price, and accrued interest

Bond markets quote a 'clean price' that strips out the interest accumulated since the last coupon date. The 'dirty price' (also called the full or invoice price) is what the buyer actually pays at settlement: clean price plus accrued interest. The relationship:

Dirty price  =  Clean price  +  Accrued interest

where:
  Clean price       =  the quoted market price, excluding any partial-period coupon
  Accrued interest  =  the share of the next coupon that the seller has 'earned'
                       by holding the bond through part of the coupon period
                     =  C  ×  (days held in current coupon period)
                                 / (total days in coupon period)

Example: a 6% annual coupon bond on a $1,000 face,
  half-way through a coupon period,
  accrued = $60 × (180/360) = $30
  Dirty price = Clean price + $30

The buyer pays the seller the $30 accrued; on the next coupon date the
buyer receives the full $60 coupon, netting to $30 of actual coupon income
over the half-period they have owned the bond. The mechanics keep coupon
income economically aligned with holding period.