16.2 Interest Rates

Interest Rates

Discount factors are the truth; interest rates are convenient quotes of those factors under a compounding convention. VRM–10 trains you to move fluently among spot, forward, and par rates, to see how maturity maps into price, and to read curve shape trades—including the fact that interest-rate swaps are living definitions of par rates.

Compounding Impact

The same DF(t) can be written with many rate quotes:

• Annual compounding: DF(t) = 1 / (1 + z_a)^t
• Semi-annual bond basis: DF(t) = 1 / (1 + z_sa/2)^(2t)
• Continuous: DF(t) = e^(−z_c t)
• Money-market simple Actual/360: DF(t) = 1 / (1 + z_mm × τ_360)

Never compare raw quotes across conventions. Convert each quote to DF(t), then to a common compounding basis.

Worked conversion

DF(2) = 0.9426. Annual spot: (1 + z_a)^2 = 1/0.9426 ⇒ 1 + z_a = √(1/0.9426) ≈ 1.0300 ⇒ z_a ≈ 3.00%. Continuous: z_c = −ln(0.9426)/2 ≈ 0.02955 (2.955%). Semi-annual bond equivalent: 1 / (1 + z_sa/2)^4 = 0.9426 ⇒ (1 + z_sa/2)^4 ≈ 1.0609 ⇒ 1 + z_sa/2 ≈ 1.01489 ⇒ z_sa ≈ 2.978%. Same discount factor; three different headlines.

Spot Rates and Discount Factors

The spot rate (zero rate) to maturity t is the yield on a zero-coupon claim maturing at t. The map DF ↔ spot is one-to-one once compounding is fixed. Bootstrapping builds DF(t) from liquid instruments (bills, bonds, swaps, futures) ordered by maturity; each new maturity solves for one new DF given prior DFs.

Spots are not averages of forwards in level terms without care—but the discount factor is the product of one-period forward discount factors (see below).

Forward Rates from Spots

The simply compounded forward rate from T to S (agreed today) satisfies, under annual periods for illustration,

DF(S) = DF(T) / (1 + F(T,S) × (S−T))

so

F(T,S) = (1/(S−T)) × [DF(T)/DF(S) − 1]

With annual spots z(T), z(S) and annual compounding,

[1 + z(S)]^S = [1 + z(T)]^T × [1 + F(T,S)]^(S−T)

for the one-year forward when S = T+1:

1 + f_T = [1 + z(T+1)]^(T+1) / [1 + z(T)]^T

Worked forward

Annual spots: z(1) = 3.00%, z(2) = 3.50%. Then 1 + f_1 = (1.035)^2 / (1.03) = 1.071225/1.03 = 1.040024 ⇒ f_1 ≈ 4.00%. Intuition: the second year must earn about 4% to raise the 1y 3% zero up to a 2y 3.5% zero.

Forward curves can lie above the spot curve when the spot curve is upward sloping (as here), and below when the spot curve is inverted—because forwards are marginal rates embedded in the spot curve’s slope.

Par Rates

A par rate c(T) is the coupon rate that makes a T-maturity fixed coupon bond price at par (dirty price = face) given today’s discount curve. For annual coupons and face 1,

1 = Σ_{k=1..T} c(T) × DF(k) + 1 × DF(T)

hence

c(T) = [1 − DF(T)] / Σ_{k=1..T} DF(k)

Par rates sit near spots but are blended: each par rate averages discount factors across the coupon schedule (a duration-weighted blend of zeros).

Worked par rate

DF(1)=0.9709, DF(2)=0.9426, DF(3)=0.9151. Sum of DFs = 2.8286. c(3) = (1 − 0.9151) / 2.8286 = 0.0849/2.8286 ≈ 3.00%. A 3y annual 3% coupon bond is approximately par on this curve (small rounding).

Spot / Forward / Par Relationships

All three curves are transforms of one DF curve:

• Spots ↔ DF by compounding inversion.
• Forwards ↔ ratios of DFs.
• Pars ↔ linear combinations of DFs (annuity formula).

Ordering on an upward-sloping curve typically reads forward > spot > par for a given maturity region (par is “softer” because it mixes shorter DFs). On an inverted curve the inequality flips. Exam items love asking which curve is highest given a shape.

Maturity and Price

For a fixed coupon bond, extending maturity while holding the coupon and yield curve fixed changes price through added distant cash flows and through whatever spots apply at those dates. At a flat yield equal to the coupon, price stays near par as maturity grows. If yields exceed the coupon, longer maturity usually lowers price (more years of below-market coupons). If yields are below the coupon, longer maturity usually raises price. With an upward-sloping curve, “yield” is ambiguous—use DF pricing instead of a single YTM story when precision matters.

Zero-coupon prices fall monotonically in maturity when all spots are positive: DF(t) decreases in t. That is the cleanest maturity–price link.

Flatten and Steepen Trades

Curve shape trades bet on the spread between sectors (e.g., 2s10s = 10y par − 2y par):

• Flattener: profit if the curve flattens (long-end yields fall relative to short-end, or shorts rise relative to longs). Classic: short the long bond / pay fixed on long swap, long the short bond / receive fixed on short swap, duration-weighted.
• Steepener: opposite—profit if long yields rise relative to short yields.

These are not pure duration bets if hedged to near-zero DV01; they isolate slope. Risk remains in butterfly reshapes, convexity differences, and roll of the points chosen.

Mini numeric intuition

2y par = 3.0%, 10y par = 4.0%, 2s10s = 100 bp. If 2y rises to 3.3% and 10y rises only to 4.1%, spread = 80 bp → curve flattened 20 bp → flattener wins (after hedge accounting).

Swaps Defining Par Rates

A plain-vanilla fixed-for-floating interest-rate swap with notional 1 has zero value at inception when the fixed rate equals the swap par rate for that tenor and payment frequency. That swap par rate is exactly the par coupon on a fixed bond versus floating, bootstrapped from the same discount (and projection) curves used for swaps.

In a single-curve textbook world,

S_par(T) = [1 − DF(T)] / Σ δ_k DF(t_k)

matching the bond par formula with year fractions δ_k. Multi-curve reality separates discounting (OIS) from forwarding (SOFR term structure), but the idea remains: the swap’s mid fixed rate is the market’s par rate for that schedule.

Why risk managers care

Trading desks quote swaps more liquidly than cash bonds in many tenors. The swap curve often defines the par curve used for marking and for extracting DF(t) after bootstrap. Bond YTMs then trade at a spread to the swap or government par curve—bridge topics for VRM–11.

Bottom line for VRM–10: pick a compounding convention, convert everything through DF(t), extract spots, forwards, and pars as different views of one curve, and treat flatteners/steepeners and swap par rates as applications of that geometry—not as separate theories of interest.