# Bond Math Formula Sheet

Fixed Income Interview Prep, fixedincomeinterview.com. Free to keep and share.
Check any number below on the bond calculator: https://fixedincomeinterview.com/tools/bond-calculator

Conventions on this sheet: a fixed-coupon bullet bond priced on a coupon date (so clean price equals
dirty price), yields quoted annually and compounded m times a year, prices per 100 of face value.
Desks quote their own conventions (day counts, compounding); ask which one the interviewer means.

## Notation

    F   face value (100 on this sheet)
    c   annual coupon rate            m   coupons per year (1, 2 or 4)
    C   coupon per period = c x F / m n   periods remaining = years x m
    y   annual yield to maturity      t_k time of the k-th cash flow in years = k / m
    P   price                         CF_k the k-th cash flow (C, plus F at the end)

## 1. Price from yield

    P = sum over k = 1..n of  CF_k / (1 + y/m)^k

    Closed form:  P = C x [1 - (1 + y/m)^(-n)] / (y/m)  +  F x (1 + y/m)^(-n)

- Coupon equal to yield: the bond prices at par (100).
- Coupon above yield: premium. Coupon below yield: discount.
- Price and yield move in opposite directions, and the curve between them bends (convexity).

## 2. Yields

    Current yield = annual coupon / price

Current yield ignores the pull to par. A bond bought at 80 that matures at 100 earns more than its
current yield, which is the trap in "a 5% bond at 80: what do you earn?"

    Yield to maturity: the y that solves the price formula above. No closed form: solve by iteration.

    Mental-math approximation:  YTM  ~  [annual coupon + (F - P) / years]  /  [(F + P) / 2]

Check: 5% coupon, 5 years, price 80. Approximation: (5 + 20/5) / 90 = 10.0%. Solving exactly with
annual coupons gives about 10.32%, because the approximation understates a deep discount.

## 3. Duration and DV01

    Macaulay duration (years):  D_mac = [ sum of t_k x CF_k / (1 + y/m)^k ] / P

    Modified duration:          D_mod = D_mac / (1 + y/m)

    Price change for a small yield move:  dP / P  ~  -D_mod x dy

    DV01 per 100 face    = D_mod x P x 0.0001
    DV01 per $1mm face   = D_mod x P            (P per 100; the result is in dollars per basis point)

- A zero-coupon bond's Macaulay duration equals its maturity.
- Modified duration is a percentage price change per 1.00 (100 basis points) of yield, not a time.
- DV01 (also PV01 or PVBP) is in money per basis point. Traders hedge in DV01, not in notional.

## 4. Convexity

    Convexity = [ 1 / (P x (1 + y/m)^2) ] x sum of  CF_k x t_k x (t_k + 1/m) / (1 + y/m)^k

    Price change with convexity:  dP / P  ~  -D_mod x dy  +  1/2 x Convexity x dy^2

- For an ordinary bond the convexity term is positive whichever way yields move: the bond gains more
  when yields fall than it loses when they rise.
- Mortgage-backed securities and callable bonds show negative convexity: falling yields shorten them
  (borrowers refinance, issuers call) just when a longer bond would gain most.

## 5. Effective duration and convexity (bump and reprice)

For bonds whose cash flows change with rates (callables, mortgage-backed securities), reprice at the
yield plus and minus a small bump dy and use:

    D_eff = (P_down - P_up) / (2 x P_0 x dy)
    C_eff = (P_down + P_up - 2 x P_0) / (P_0 x dy^2)

P_down is the price after yields fall by dy; P_up after they rise by dy.

## 6. Forward rates

    (1 + s_2)^2 = (1 + s_1) x (1 + f_1,1)      so      f_1,1 = (1 + s_2)^2 / (1 + s_1) - 1

s_1 and s_2 are the one- and two-year spot rates; f_1,1 is the one-year rate one year forward.
Check: s_1 = 8%, s_2 = 10% gives f_1,1 = 1.21 / 1.08 - 1 = 12.04%. If you can lock a different
forward rate, one leg is cheap and the other rich: that is the arbitrage question.

## 7. Accrued interest and day counts

    Accrued interest = C x (days since the last coupon / days in the coupon period)
    Dirty (invoice) price = clean price + accrued interest

    Actual/actual   US Treasuries
    30/360          US corporate bonds
    Actual/360      money markets: repo, SOFR, most floating-rate loans

## 8. Spreads

    G-spread   bond yield minus the interpolated government bond yield
    I-spread   bond yield minus the swap rate of the same maturity
    Z-spread   the constant spread over the zero-coupon curve that reprices the bond
    OAS        the Z-spread with the value of any embedded option taken out
    CS01       price change for a 1 basis point move in the credit spread (the credit cousin of DV01)

Write "corporate bond credit spread" in full when searching: "credit spread" alone is mostly an
options strategy.

## 9. Repo and carry

    Repo interest = cash lent x repo rate x days / 360

    Carry over a holding period  ~  coupon income  -  financing (repo) cost
    Roll-down: the price gain as a bond ages down an upward-sloping curve at an unchanged curve

## 10. Curve trades

    DV01-neutral hedge ratio:  face of the short-maturity leg / face of the long-maturity leg
                               = DV01 per unit face (long leg) / DV01 per unit face (short leg)

A 2s10s steepener buys the 2-year and sells the 10-year in that ratio, so a parallel move leaves it
flat and only a change in the curve's slope moves the P&L.

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Standard textbook conventions, set out for interview preparation, not trading or investment advice.
More worked questions: https://fixedincomeinterview.com/bond-math-interview-questions