S1,S2,...Sn

P = 1/(1 Sn)^n

P = C1/(1 S1)^1 C2/(1 S2)^2 ... (Cn Par)/(1 Sn)^n

(1 S3)^3 = (1 S1)*(1 f(1,1))*(1 f(2,1))

(1 S3)^3 = (1 S1)*(1 f(1,2))^2

F(m,n) = 1/(1 f(m,n))^n = P(m n)/P(m)

P = C1/(1 S1) C2/(1 S1)(1 f(1,1)) ... (Cn Par)/(1 S1)(1 f(1,1))...(1 f (n-1,1))

premise

in conclusion

forward contract arbitrage

futures arbitrage

The zero-coupon bond yield can be derived step by step from front to back based on the parity rate.

P = C1/(1 YTM)^1 C2/(1 YTM)^2 ... (Cn Par)/(1 YTM)^n

1. Hold until maturity

2. Repay principal and interest on schedule

3. The reinvestment rate of return is YTM

1-year bond par rate=YTM=coupon rate=S1

N-year bond par rate=YTM=(S1 S2 ... Sn)/n≈Sn

YTM>coupon rate →discount

YTM<coupon rate →premium

YTM=coupon rate → par price

swap rate = par rate

Compared with treasury bonds, swap rates have a certain risk premium.

Compared with treasury bonds, there are many types of swap interest rates and maturities.

Not subject to government regulation

Investors are risk neutral

Expected future short-term interest rates determine the shape of the yield curve

Can explain all interest rate curve shapes

Derived theory: local expectation theory

The forward interest rate f(m,n) is the biased expectation of the expected future short-term interest rate E(Sm n)

The forward interest rate f(m,n) is the expected future short-term interest rate E(Sm n) plus rp(bias)

rp is positively related to deadline

Explain that in most cases the interest rate curve slopes upward

The interest rate curve is determined by the supply and demand for funds of different maturities.

Can explain all interest rate curve shapes

The forward interest rate f(m,n) is the biased expectation of the expected future short-term interest rate E(Sm n)

Bias is the cost of causing investors to deviate from a finite horizon

Can explain all interest rate curve shapes

CIR model

Vasicek model

Ho-Lee model

Δ%P/Δy

Individual bond risk measurement

Portfolio Risk Measurement

Duration

The probability of the interest rate going up or down at each point in time is 50%

Vu=Vd*e^2σ

V=(coupon 0.5VU 0.5VL)/(1 r)

Simulate 1,000 interest rate change paths

Calculate the sum of the present value of cash flows for each path

Average 1000 paths

Issuer Rights Vcallable=Vpure-Vcall

When interest rates fall, issuers will redeem early (reinvestment risk)

OAS<Z-spread

Negative convexity may occur when interest rates are low

Investor Rights Vputable=Vpure Vput

When interest rates rise, investors will terminate early

OAS>Z-spread

extendable bonds

After the investor dies, the heirs have the right to recover the investment

The issuer repays principal annually starting in a specific year

sinking fund provision has no options

Accelerated sinking fund provision has options

Each node needs to determine whether it will exercise its rights

The influence of σ

The interest rate curve becomes flat, Vcall↑, Vput↓

callable duration:one side up>one side down

putable duration:one side up<one side down

KDR matching expiration date is greater than 0

All other KDRs are 0

Maximum KDR matching expiration date

Zero coupon bonds and KDRs shorter than maturity may be less than 0

When interest rates are low, the KDR matching the exercise date is the largest.

When interest rates are higher, the KDR matched to the maturity date is the largest

When interest rates are lower, the KDR matched to the maturity date is the largest

When the interest rate is higher, the KDR matching the exercise date is the largest.

issuer

investor

The risk of the borrower being unable to pay principal and interest in full on time

PD (Probility of Default) probability of default

Loss Given Default Loss due to breach of contract

EL loss expectation

CVA

IRR

credit scoring

credit rating

structural models

reduced form models

credit spread = YTM of risky bond - YTM of benckmark

Positive correlation

negative correlation

1. credit quality

2.financial conditions

3. market demand and supply

4. equity market volatility

short-term granular and homogeneous →statistical-based approach

medium-term granular and homogeneous →portfolio-based approach

discrete and non-granular portfolio →evaluated at the individual loan level

operational and counterparty risk

credit enhancement

Not only the underlying assets but also the company’s credit

Short credit risk, pay CDS spread each period

Benefit if the bond defaults

long credit risk, receive CDS spread each period

Loss if bond defaults

fixed coupon on CDS

= PV(protection leg) - PV(premium leg)

≈(CDS spread - CDS coupon) * CDS duration

The fixed coupon rate and the actual interest rate are settled at the beginning of the period, and any excess will be refunded and any excess will be compensated.

1% - Investment Grade

5% - Speculative grade

Compensation time

Compensation amount

As long as the bond with a rating ≥ your own defaults, you can claim compensation.

cheapest-to-deliver, the bond with the largest loss in the same class pays the compensation

the protection for each issuer is equal

The higher the correlation between the entities of the portfolio internal standard, the higher the CDS spread.

Delivery of underlying bonds, compensation par value

Cash delivery, the amount is determined based on the bond with the largest loss of the same grade

profit for CDS buyer% ≈ change in spread% * duration

Buy CDS on a short bond

Buy one company's CDS and sell another company's CDS. You can make a profit when the credit spread of the two companies changes in the future.

curve-steeping trade

curve-flattening trade

Take advantage of the different credit spreads of bond market and CDS market to make profits