Fixed Income

Government Bond

Also known as: Treasury, Bund, Gilt, Sovereign

A loan to a state, and the reference price of money itself — the yardstick every other asset is measured against.

Asset class
Fixed income
Instrument type
Sovereign coupon bond
Traded
OTC dealer market, huge and liquid
Typical users
Central banks, pensions, banks, everyone
The fundamental curve: bond price falls as yield rises, with convex (curved) shape.
y₀Price–yieldYield to maturityBond price
BeginnerWhat is it, really?

A government bond is an IOU from a country: lend the state money today, and it promises fixed interest payments (coupons) every period and your money back (the principal or face value) at maturity.

Because rich-country governments can tax — and, in their own currency, print — their bonds are treated as the closest thing markets have to a risk-free asset. That makes their yields the baseline price of money: mortgages, corporate loans and stock valuations all key off them.

One rule to internalise: prices and yields move in opposite directions. If new bonds pay 4% and yours pays 2%, nobody buys yours at full price — it must get cheaper until its effective return matches. Rates up, bond prices down; rates down, prices up.

Key intuition: a bond is a promise of fixed cash flows. Its price is simply what those fixed flows are worth when the going interest rate changes.
IntermediateHow it works in practice

Reading a bond quote

  • Coupon: annual interest as % of face (paid semi-annually for Treasuries, annually for Bunds).
  • Yield to maturity (YTM): the single discount rate equating price to promised cash flows — the "internal rate of return" if held to maturity.
  • Clean vs. dirty price: quotes exclude accrued interest; you pay clean + accrued ("dirty").

The yield curve

Plot yields against maturities and you get the yield curve — normally upward-sloping (long lending pays more), inverting when markets expect rate cuts, which is why inversion is a famous recession signal. The curve embodies expected policy rates plus a term premium.

Duration: the risk number

Duration measures price sensitivity: a bond with duration 7 loses ≈ 7% of its price per 1 percentage point rise in yield. Longer maturity and lower coupons → higher duration. This is the single number bond portfolios are managed around.

Worked example: a 10-year bond with 2% coupon trades at 100 when yields are 2%. Yields jump to 3%: with duration ≈ 8.9, price falls to roughly 91.4. "Safe" assets can still move a lot — safety here means default risk, not price stability.
AdvancedPricing & valuation

Pricing off the curve

The price is the sum of discounted cash flows; the honest version discounts each flow at its own maturity's rate (the zero/spot curve \(z_t\)):

$$ P = \sum_{t=1}^{T} \frac{c}{(1+z_t)^t} + \frac{100}{(1+z_T)^T} \qquad\text{vs.}\qquad P = \sum_t \frac{CF_t}{(1+y)^t} $$

YTM \(y\) is the flat-rate shortcut. Zero curves are bootstrapped from bond prices; forward rates \(f(t_1,t_2)\) follow from no-arbitrage between zeros.

Duration and convexity, precisely

$$ D_{mod} = -\frac{1}{P}\frac{\partial P}{\partial y}, \qquad C = \frac{1}{P}\frac{\partial^2 P}{\partial y^2}, \qquad \frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2} C (\Delta y)^2 $$

Convexity is why the price–yield curve bows: losses decelerate and gains accelerate — valuable, hence low-coupon long bonds (high convexity) trade at slightly lower yields, other things equal.

Beyond one factor

Curve risk is multi-dimensional; desks hedge key-rate durations (sensitivity to each maturity bucket) and trade level/slope/curvature factors, which explain ~99% of curve variance in PCA. Term-structure models (Vasicek, Hull–White, HJM) impose no-arbitrage dynamics for pricing derivatives on these curves.

The plumbing premia

Even sovereigns have basis effects: on-the-run vs. off-the-run liquidity, repo specialness, and the futures delivery cycle all create yield wedges unrelated to credit — the raw material of relative-value trading.

Practitioner note: "the" yield is a summary statistic. The tradeable objects are the curve, its forwards, and the financing of the position — three markets wearing one bond's clothes.