12.4 Properties of Interest Rates

Properties of Interest Rates

FMP–16 is the rates toolkit that underpins derivatives discounting, FRAs, and bond risk. FRM candidates must move fluently among benchmarks, compounding conventions, spot and forward curves, duration/convexity, and term-structure stories.

Treasuries, SOFR, Repo, and the Risk-Free Rate

Post-crisis practice often discounts collateralized derivatives on OIS/SOFR-type curves, not on unsecured LIBOR. Treasuries remain the sovereign curve for cash bonds. When a question says “risk-free rate,” identify whether it means Treasury, OIS/SOFR, or a generic continuous r in Black–Scholes pedagogy.

Repo matters for bond relative value: if financing (repo) is cheap, leveraged long-bond trades earn more carry. General collateral (GC) vs special repo: specials have lower repo rates (harder to borrow the bond).

Compounding Frequencies

A quoted rate R with compounding m times per year means $1 grows to (1 + R/m)^(mT) in T years. Continuously compounded rate r satisfies e^(rT) = (1 + R/m)^(mT), so r = m ln(1 + R/m).

Worked conversion

5% with continuous compounding for 1 year → e^0.05 ≈ 1.05127 (effective 5.127%). The equivalent annual rate is 5.127%. A 5% semiannual bond yield means (1 + 0.05/2)^2 − 1 = 5.0625% effective annual. Never subtract 5% − 5% across conventions and call the difference “spread” without converting.

Bond Price from Coupons and Yield

A fixed coupon bond paying c per year (or c/m per period) with face 100 and y the yield per the bond’s compounding convention has price equal to the PV of coupons plus principal. For annual coupons:

P = Σ_{t=1..N} C / (1 + y)^t + 100 / (1 + y)^N

where C is the annual coupon dollar amount.

Worked yield price

3-year annual 6% coupon bond, face 100, yield 5%.

P = 6/1.05 + 6/1.05^2 + 106/1.05^3 = 5.714 + 5.442 + 91.567 = 102.72.

If yield rises to 6%, P = 100 (par). Duration will tell you the sensitivity of that 102.72 to yield shocks.

Spot-Rate Pricing

Spot rate Z(T) is the yield on a zero maturing at T. Coupon bonds should satisfy:

P = Σ C_i e^(-Z(t_i) t_i) (continuous) or Σ C_i / (1 + Z(t_i)/m)^(m t_i)

using each cash flow’s own maturity spot—not a single y—when the curve is not flat. YTM is the single flat rate that recovers market price; spots are the true building blocks.

Worked spot pricing

Cash flows: $5 in 1y, $105 in 2y. Continuously compounded spots: Z(1) = 4%, Z(2) = 5%.

P = 5 e^(-0.04×1) + 105 e^(-0.05×2) = 5 × 0.9608 + 105 × 0.9048 = 4.804 + 95.007 = 99.81.

Macaulay, Modified, and Dollar Duration

Macaulay duration D_mac = weighted average time of cash flows (weights = PV of CF / price), in years.

Modified duration D_mod = D_mac / (1 + y/k) for discrete yields with k payments per year; with continuous compounding, D_mod = D_mac.

Dollar duration = D_mod × P (sometimes per 1% or per 01—read the convention). DV01 ≈ dollar duration × 0.0001 for a 1 bp move when D_mod is per 1.00 (100%) yield change.

Interpretation: modified duration is the semi-elasticity: dP/P ≈ −D_mod dy.

Worked duration (annual)

2-year 10% annual coupon bond at y = 10%, price 100. PVs: 10/1.1 = 9.0909; 110/1.1^2 = 90.9091.

D_mac = [1 × 9.0909 + 2 × 90.9091] / 100 = 1.9091 years.

D_mod = 1.9091 / 1.10 = 1.7355.

If y rises 10 bp (0.001), ΔP ≈ −1.7355 × 100 × 0.001 = −0.1736.

Convexity and Price-Change Approximation

Convexity captures curvature: for annual yield,

Convexity ≈ (1/P) Σ t(t+1) CF_t / (1+y)^(t+2) (discrete textbook form; continuous forms use t^2 weights).

Second-order approximation:

ΔP ≈ −D_mod × P × Δy + (1/2) × Convexity × P × (Δy)^2

Duration alone understates price increases when yields fall and overstates losses when yields rise for standard fixed-coupon bonds (positive convexity). For large Δy, include convexity.

Worked convexity adjustment

P = 100, D_mod = 1.7355, Convexity = 4.0 (in the units matching Δy in decimal). Δy = +0.01 (100 bp).

Duration-only: ΔP ≈ −1.7355 × 100 × 0.01 = −1.7355.

With convexity: ΔP ≈ −1.7355 + 0.5 × 4.0 × 100 × (0.01)^2 = −1.7355 + 0.02 = −1.7155.

Forward Rates

The forward rate from T1 to T2 locked today satisfies no-arbitrage between zeros:

Continuous: e^(Z2 T2) = e^(Z1 T1) e^(f (T2−T1)) ⇒ f = (Z2 T2 − Z1 T1) / (T2 − T1).

Worked forward

Z(1) = 4%, Z(2) = 5% continuous. One-year forward starting in one year: f(1,2) = (0.05×2 − 0.04×1) / 1 = 0.06 = 6%.

Investing two years at 5% matches investing one year at 4% and the second year at 6%.

FRA Valuation

A forward rate agreement settles the difference between a contracted forward rate K and the fixing L on a notional for an accrual fraction τ, typically paid at the start of the loan period and therefore discounted:

Settlement ≈ Notional × (L − K) × τ / (1 + L τ) (for the party borrowing at floating / paying floating—sign depends on long/short).

Value mark-to-market before fixing replaces L with the current forward for that period and discounts on the appropriate curve.

Worked FRA settlement

Notional $10m, τ = 0.25, K = 3%, fixing L = 3.6%. Long FRA (receive floating conceptually / benefit when L > K):

Payoff at period start ≈ 10m × (0.036 − 0.030) × 0.25 / (1 + 0.036×0.25) = 10m × 0.0015 / 1.009 ≈ $14,866.

Bootstrapping Zero Rates

Bootstrapping extracts successive zeros from liquid instruments (deposits, FRAs/futures, swaps or coupon Treasuries):

1. Shortest maturity zero from the shortest bill/deposit.
2. Use the next coupon instrument: strip known PV of earlier coupons using already-found zeros; solve for the final zero that matches the instrument’s market price.
3. March out the curve.

Worked bootstrap step

Annual 6% coupon 2y bond priced at 101.5; Z(1) already 5% continuous. Coupon 6, final 106. Let Z2 be continuous 2y zero:

101.5 = 6 e^(-0.05×1) + 106 e^(-Z2×2) 101.5 = 5.707 + 106 e^(-2 Z2) 95.793 = 106 e^(-2 Z2) e^(-2 Z2) = 0.9037 ⇒ 2 Z2 = −ln(0.9037) ≈ 0.1012 ⇒ Z2 ≈ 5.06%.

Term-Structure Theories

Upward-sloping curves can mean expected rate rises, liquidity premia, or pension demand for long bonds—exam answers should name the theory that matches the vignette’s assumption.

Synthesis

Quote conventions before arithmetic; price on spots when the curve is not flat; risk-manage with modified duration and convexity; read forwards and FRAs as locked future borrowing rates; bootstrap zeros from liquid strips of the curve; explain shapes with expectations, liquidity preference, or segmentation.