Fixed Income

Swap Fixed Rate give SPOT Rate

SFR₂ / (1 + S₁) + SFR₂ / (1 + S₂)² + 1 / (1 + S₂)² = 1

1-year forward rate in one year f(1,1)

(1+S₂)²/(1+S₁) – 1

1-year forward rate in two years [ƒ(2,1)]

(1+S₃)³/(1+S₂)² – 1

Swap Spread

Swap Rate - Treasury Rate

I Spread

Bond’s Yield - Swap Fixed Rate

Z Spread

constant spread added to benchmark spot rates

TED Spread

MRR - TBill Yield

Market Segmentation Theory

contends that lenders and borrowers have preferred maturity ranges, and that supply and demand forces in each maturity range determines yields

Pure Expectation / Unbiased Expectation Theory

The yield curve slopes upward because short-term rates are lower than long-term rates.

Local Expectation Theory

Market evidence shows that short-term holding period returns from investing in long-maturity bonds exceed the short-term holding period returns from investing in short-maturity bonds.

Liquidity Theory

The liquidity theory of the term structure proposes that forward rates reflect investors' expectations of future rates plus a liquidity premium to compensate them for exposure to interest rate risk, and this liquidity premium is positively related to maturity.

Preferred Habitat theory

Inclination to chase higher yields in the longer maturity spectrum is consistent with the preferred habitat theory whereby investors will leave their preferred habitat if they are compensated with higher returns.

Gauss+ is a multi-factor model

incorporates short-, medium-, and long-term rates where the short-term rate is devoid of a random component—consistent with the role of the central bank controlling short-term rate.

Ho-Lee model

calibrated to the current term structure using the time-dependent drift term θ_t and has a random noise component σdz_t.

Kalotay-Williams-Fabozzi (KWF) model.

Like the Ho-Lee model, the KWF model uses a random noise component (but assumes that the short rate is lognormally distributed).

Swap Fixed Rate when Prices are provided

SFR₃ (P₁ + P₂ + P₃) + P₃ = 1

Credit Valuation Adjustment(CVA)

Sum of present values of expected losses

Bond Value when CVA is provided

VND - CVA

Value of Non Default - Credit Valuation Adjustment

Loss Given Default

Exposure × (1 - Recovery Rate)

Probability of Default

Hazard Rate × (1 - Hazard Rate )^t-1

Under structural model the put option value

value of risk-free bond – value of the risky bond = CVA

Recovery Cash Flow

Exposure × Recovery Rate

Under structural models value of risky debt

value of risk-free debt – value of put option on company assets

Expected Loss

LGD × Probability of Default

Probability of Survival (PS)

(1 - Hazard Rate )^t

Option-free convertible bond value

straight value + value of the call option on the stock.

upfront premium (%)

(credit spread – CDS coupon) × duration

profit for protection buyer

change in spread × duration × notional principle

Exposure (zero-coupon Bond)

PAR / (1+r_f)^{yrs to mature}

Discount Factor

1 / (1+r_f)^t

CVA (Credit Valuation Adjustment)

PV(Expected Loss)