Bonds, Yields, and Duration

Rates are the water everything else swims in — discounting in option pricing, the risk-free leg of every Sharpe ratio, the carry in every forward. This lesson is the fixed-income minimum: enough to price a bond, reason about its risk, and survive the rates questions embedded in general quant interviews.

Price is discounted cash flows

A bond paying coupon $c$ annually and principal at $T$, priced with yield $y$:

$$P = \sum_{t=1}^{T} \frac{c}{(1+y)^t} + \frac{100}{(1+y)^T}$$

Yield to maturity is the single rate making discounted cash flows equal the market price — the bond's internal rate of return, and (like implied vol) a quoting convention extracted from price rather than an input. The instant relations: price and yield move inversely (the seesaw); coupon above yield ⇒ premium bond, below ⇒ discount; and a zero-coupon bond is the atom — one cash flow, price $100/(1+y)^T$ — from which the yield curve (zero rates by maturity) prices everything by decomposition. Forward rates fall out of the curve by no-arbitrage: locking in borrowing from year 1 to 2 must cost what the two zero rates imply, $\frac{(1+y_2)^2}{(1+y_1)} - 1$ — the same replication logic as the forwards lesson, in time rather than assets.

Duration: the bond's delta

Modified duration $D$ measures price sensitivity: $\frac{\Delta P}{P} \approx -D \,\Delta y$. A duration-7 bond loses ~7% when yields rise 1%. Anchors worth carrying: duration of a zero equals its maturity; coupons shorten duration (cash arrives earlier); higher yields shorten it too. Duration is the fixed-income delta — portfolio durations add value-weighted, hedge ratios are duration ratios, and "DV01" (dollar value of a basis point, $\approx P \times D \times 0.0001$) is how desks actually quote the risk.

Convexity is the gamma: the Taylor lesson's second-order term, $\frac{\Delta P}{P} \approx -D\Delta y + \tfrac12 C (\Delta y)^2$. Positive for ordinary bonds — prices rise more when yields fall than they drop when yields rise — so convexity is good for the holder and is priced (barbell vs bullet portfolios of equal duration differ in convexity, and the barbell costs more). Negative convexity exists and matters: callable bonds and mortgage securities shorten exactly when you'd want them long — the issuer holds the option, and the Greeks lesson's short-gamma discomfort translates directly.

The curve and what it whispers

Upward-sloping normally (term premium: lock-up risk demands yield); inversion — short rates above long — is the classic recession signal, readable as the market pricing future cuts. Curve shapes are traded in their own right (steepeners/flatteners: duration-neutral pairs of maturities), and the PCA lesson's level/slope/curvature decomposition is the standard risk language for them — three factors, ~99% of curve variance, one of the cleanest empirical facts in finance.

The interview version

*"Yields rise 50bp; your 10-year-duration portfolio of \$100M does what?"* —$-10 \times 0.005 = -5%: down ~\5M, softened slightly by convexity. "Two bonds, same maturity, different coupons — which has more duration and why?" — the lower coupon: more of its value sits at the far cash flow. "Why can a bond fund lose money if 'bonds always pay back'?" — mark-to-market duration losses, plus the reinvestment/horizon distinction — held to maturity the yield is locked, but the fund's NAV walks the price path. Mechanical, quick, and assumed knowledge at every rates-adjacent seat.