1. Fixed-income theory and interest rates

Two rates are equivalent if applied to the same capital, for the same period of time, they give the same interest.

Spot rate (=zero rate): rate of interest earned on an investment that provides payoff only at T

—> expressed using continuous compounding

Yield curve: representation of relationship between zero rates and remaining time to T of zero-coupon debt securities

Discount each cashflow (earned coupons, face value at T,…) at corresponding zero rate. = fair price of financial instruments 

Discount rate that, when applied to all cashflows, gives bond price equal to market price (search for a single discount rate equal to several zero rates) (=solve for r)

Coupon rate that causes the bond price to equal its face value (use initial zero rates, not bond yield)(=solve for c)

e.g.: with 4 semi-annual coupons:

Method for estimating the yield curve:

Compute each point of the yield curve as yield (zero rate) of a ZCB with corresponding T.

Set by Nymand-Andersen (2018):

-          Flexibility & goodness of fit

-          Robustness

-          Smoothness

—> Best method: Parsimonious functional forms, i.e. Svensson-Nelson-Siegel

(alternatives: Spline based or bootstrapping, both have risk to overfit)

Svensson-Nelson-Siegel

Zero rates:

—> fit by minimizing quadratic difference between yields computed from curve and yields measured in the market

• Treasury rate (considered artificially low)

• Repo rates (based on repurchase agreements) (agreement where a financial institution that owns securities agrees to sell them for X and buy them back in the future (usually the next day) for a slightly higher price, Y)

• EURIBOR (rate at which AA-Bank estimates it can borrow money on unsecured basis from other bank at 11am) (LIBOR discontinued Dec. 2021)

• Overnight rates (unsecured borrowing & lending between banks, computed on previous day activity)

Instrument whose value depends on, or is derived from value of another asset.

Uses: Hedging risk, speculation, lock in arbitrage, change nature of investment/liability

Exchanges, OTC markets

before 2008 largely unregulated, since then regulated to reduce systemic risk, increase transparancy

= majority of global derivative market; mostly interest rate swaps & forward rate agreements

Exchange at time T the difference between a fixed rate defined in T (forward rate) and a variable rate (settlement rate) given at time T, based on notional value (fictional value, never actually exchanged

• Seller makes payment based on variable rate, earns fixed rate

• Buyer other way round

—> Buyers will always realize forward rate

Present value of difference between forward rate agreed on in FRA and current EURIBOR rate calculated today (assuming that EURIBOR rate is certain to be realized)

= “Portfolio of FRAs” with different maturities à fixed & floating reference rates, regular interest exchange

• Initially worth close to zero

• Later, valuated as portfolio of FRAs

• Procedure: calculate floating forward rates; based on this, calculate cash flows; discount swaps