Finance Formulas

Fixed Income

core

Bond Price & Yield to Maturity

Builds onPresent Value of an Ordinary Annuity · Present Value of a Single Sum — if this page feels steep, start there.

P=C×1−(1+y)−ny+F(1+y)nP = C \times \frac{1-(1+y)^{-n}}{y} + \frac{F}{(1+y)^{n}}

Reading the notation

PP
the bond's price today — what you pay for all the promised cash
CC
the coupon payment per period (coupon rate × face value)
yy
the yield to maturity per period — the market's required return, as a decimal
nn
number of coupon periods until the bond matures
FF
the face (par) value repaid at maturity — usually 1,000
1−(1+y)−ny\frac{1-(1+y)^{-n}}{y}
the annuity factor: today's value of 1 per period for n periods

Play with it

Drag the yield and watch price respond — steeper at low yields, flatter at high ones. That changing slope is duration; its curvature is convexity.

$0$500$1000$15002%5%8%11%14%yieldpriceactual priceduration tangent

$926

Price

8.02y

Macaulay dur.

7.57

Modified dur.

73

Convexity

ScenarioDuration-only est.Actual priceConvexity gap
-200bp$1067$1081+$14
-100bp$997$1000+$3
+100bp$856$860+$3
+200bp$786$799+$13

The gap is positive in BOTH directions — the curve always bows above its tangent. That bow is convexity.

Why it must be true

A bond is nothing new — it is two instruments you already know, stapled together: an annuity (the coupon stream) plus a single future sum (the face value at maturity). Price each with the market's required yield yy and add.

The deep consequence: price and yield move in opposite directions. Raising yy shrinks every discount factor, so the price falls — and read backwards, the market price IS the yield: quote one and you have quoted the other. When c=yc = y the bond prices exactly at par; coupon above yield → premium; below → discount.

The derivation

Discount each promised cash flow at the market yield and add — present values are additive:

P=C(1+y)+C(1+y)2+⋯+C(1+y)n⏟coupon annuity+F(1+y)n⏟redemptionP = \underbrace{\frac{C}{(1+y)} + \frac{C}{(1+y)^2} + \cdots + \frac{C}{(1+y)^n}}_{\text{coupon annuity}} + \underbrace{\frac{F}{(1+y)^n}}_{\text{redemption}}

The coupon leg collapses with the annuity factor you already have:

P=C×1−(1+y)−ny+F(1+y)nP = C \times \frac{1-(1+y)^{-n}}{y} + \frac{F}{(1+y)^{n}}

The yield to maturity is defined implicitly: it is the single rate yy that makes this equation hold for the observed market price — found numerically (the calculator's CPT I/Y).

When to reach for it

Any bond with level coupons and a final redemption: price it from a given yield, or run it backwards to find the yield implied by a given price.

Listen for

“annual-pay bond with a coupon of …”“yield to maturity / priced to yield”“face (par) value of 1,000”“what is the bond worth / fair price”“trading at … what yield does that imply”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

  • ≈

    Par anchor: if coupon rate = yield, price = face. Exactly. Coupon above yield → premium (price > par); below → discount. This instantly eliminates half the choices.

  • ≈

    Rough size of the discount/premium: (coupon rate − yield) × annuity factor ≈ deviation from par, per 1 of face. A 1-point coupon shortfall on a 7-year bond (factor ≈ 5.4) prices ≈ 5.4% below par.

  • ≈

    For YTM: it must land between the current yield and the coupon rate for premium bonds (and above both for discounts) — bracket before computing.

Traps in applying it

  • ✗Semiannual bonds: halve the coupon AND the yield, double n. Mixing annual yield with semiannual periods is the classic bond-desk interview fail.
  • ✗Discounting at the coupon rate instead of the market yield — that always returns par and answers nothing.
  • ✗Forgetting the face value: the annuity factor prices only the coupons; the redemption is a separate discounted lump sum.

Limits & criticisms

The formula assumes a flat yield: one rate discounting every cash flow, when the real term structure slopes — that's why spot-curve (no-arbitrage) pricing exists as the rigorous alternative. It also assumes the cash flows are certain: callable, putable or defaultable bonds violate that, which is where option-adjusted spreads and credit analysis take over. And YTM's realized return requires reinvesting every coupon at that same yield — rarely true.

Where it came from

Bond mathematics is the oldest quantitative finance there is: Venetian prestiti traded at yield-implied discounts in the 1300s, and Dutch and British consols made yield arithmetic a daily trade skill centuries before equities had any theory at all. The yield to maturity concept was formalized alongside actuarial present-value methods in the 19th century, and standardized bond yield tables were the "calculators" every bond desk carried until the 1970s.

Today this identity prices the largest securities market on earth — roughly $140 trillion of global debt — and its inverse (price → YTM) is how every bond you'll ever see is actually quoted.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Price the promise

P=C×1−(1+y)−ny+F(1+y)nP = C \times \frac{1-(1+y)^{-n}}{y} + \frac{F}{(1+y)^n}

The forward face: coupons as an annuity plus the redemption as a lump sum — two formulas you already own, added.

Drill this face →

Yield to maturity from a price

P  ⇒  y  (solved numerically)P \;\Rightarrow\; y \;\text{(solved numerically)}

The market face: given the traded price, back out the single discount rate it implies. There is no closed form — this is what [CPT] [I/Y] exists for.

Drill this face →

On the BA II Plus

Worked example: A 2.5% annual-pay bond (face $1,000.00, 9 periods to maturity) trades at $761.94. What yield to maturity does that price imply?

  1. 1.[2ND] [CLR TVM]always clear the worksheet first
  2. 2.9 [N]
  3. 3.761.9408 [+|-] [PV]cash paid out is negative
  4. 4.25 [PMT]
  5. 5.1,000 [FV]
  6. 6.[CPT] [I/Y]compute the unknown

→ 6%

Where it leads

Master this and the following come almost for free:

Macaulay DurationImplied Forward Rate
Drill this formulaBack to the library