Chapter 4: Interest Rates

Types of Rates

Treasury Rates

Rates earned on government bonds. Considered approximately risk-free in their home currency.

LIBOR (Being Phased Out)

The London Interbank Offered Rate — the rate at which large banks are prepared to lend to each other. Being replaced by overnight reference rates due to manipulation scandals and lack of underlying transactions.

Repo Rates

Short-term secured borrowing rates where government bonds are used as collateral. The repo rate is slightly below the risk-free rate.

Reference Rates (New)

• SOFR (Secured Overnight Financing Rate) — US dollar
• SONIA (Sterling Overnight Index Average) — British pound
• €STR (Euro Short-Term Rate) — Euro
• SARON (Swiss Average Rate Overnight) — Swiss franc
• TONAR (Tokyo Overnight Average Rate) — Japanese yen

These are overnight rates based on actual transactions, not expert judgment like LIBOR.

Measuring Interest Rates

Compounding Frequency

$A{\left(1+\frac{R}{m}\right)}^{mn}$

where $A$ is the principal, $R$ is the annual rate, $m$ is compounding frequency per year, $n$ is number of years.

Continuous compounding ($m\to \infty$):

$A{e}^{Rn}$

Converting from $m$-times-per-year compounding to continuous:

$R_c=m\cdot \ln \left(1+\frac{R_m}{m}\right)$

Zero Rates

A zero rate (spot rate) for maturity $T$ is the rate of interest earned on an investment that provides a payoff only at time $T$ (no intermediate payments).

Bond Pricing from Zero Rates

$B={\sum }_{i=1}^{n}C\cdot e^{-R_i{t}_i}+P\cdot e^{-R_n{t}_n}$

where $C$ is the coupon, $P$ is the principal, $R_i$ are the zero rates for times $t_i$.

Determining Zero Rates (Bootstrap Method)

Starting from short maturities and working forward:

1. Use short-term instruments to get near-term zero rates
2. For each subsequent bond, all previous zero rates are known
3. Solve for the single unknown zero rate that makes the bond correctly priced

Forward Rates

A forward rate is the future zero rate implied by today's term structure:

$R_F=\frac{R_2{T}_2-R_1{T}_1}{T_2-T_1}$

where $R_1$ and $R_2$ are zero rates for maturities $T_1$ and $T_2$ with $T_2>T_1$.

Example: If the 1-year zero rate is 3% and the 2-year zero rate is 4%, the 1-year forward rate starting in 1 year is:

$R_F=\frac{0.04\times 2-0.03\times 1}{2-1}=0.05=5\%$

Forward Rate Agreements (FRAs)

An FRA is an OTC agreement that a certain interest rate will apply to a certain principal during a future time period.

If the actual rate $R_M$ exceeds the agreed rate $R_K$, the lender compensates the borrower. The payoff at the start of the period:

$\text{Payoff}=L\cdot (R_K-R_M)\cdot \tau$

where $L$ is the principal and $\tau$ is the period length. This is discounted to present value at settlement.

Duration

Duration measures the sensitivity of a bond's price to yield changes. Macaulay duration:

$D={\sum }_{i=1}^n{t}_i\cdot \left(\frac{P{V}_i}{B}\right)$

where $P{V}_i$ is the present value of the $i$-th cash flow, $B$ is the bond price.

Modified duration:

$D^{\ast }=\frac{D}{1+y}$

The percentage price change ≈ $-D^{\ast }\cdot \Delta y$.

For continuous compounding: $D^{\ast }=D$.

Duration-based hedging: To match the durations of assets and liabilities, the face value of futures needed is:

$N=\frac{D_A\cdot V_A}{D_F\cdot V_F}$

Convexity

Duration is a first-order approximation. Convexity captures the second-order effect:

$C=\frac{1}{B}{\sum }_{i=1}^n{t}_i^2\cdot P{V}_i$

The full price change approximation:

$\frac{\Delta B}{B}\approx -D\cdot \Delta y+\frac{1}{2}C\cdot (\Delta y{)}^2$

Bond portfolios should ideally have positive convexity — bond A with higher convexity than bond B will outperform regardless of whether rates rise or fall (all else equal).

Theories of the Term Structure