Fixed Income

CFA_LV2_FI_F.hwp

 

 

Fixed Income

 

[ The Term Structure and Interest Rate Dynamics ]

 

○ Bond pricing

- forward rate에서 spot rate(zero-coupon rate) 뽑아내고, spot rate으로 채권 CF 할인

- YTM ≦ spot rate ≦ forward rate

 

○ Spread

- swap spread = swap rate - Treasury yield

- I-spread = risky bond yield - swap rate

- Z-spread = spot rate - default-free spot rate

- Ted spread = 3M Libor - 3M T-bill

- Libor-OIS spread = Libor - OIS(1일물)

- OAS(option adjusted spread) : constant spread added to each forward rate in a benchmark binomial interest rate tree, such that the sum of the present values of a credit risky bond's cash flows equals its market price

 

○ Swap rate vs. government bond

	swap rate	government bond
credit risk	commercial bank	government
comparability in different country	comparable
maturities	many maturities	small maturities
user	wholesale bank	retail bank

 

○ Term structure of interest rates

- pure expectations theory : risk neutrality, forward rates are an unbiased predictor of future spot rate.

- local expectations theory : bond maturity does not influence returns for short holding　period. But risk premiums exist in long time.

- liquidity preference theory : investors demand a liquidity premium that is positively related to a bond's maturity.

- segmented markets theory : the shape of the yield curve is the result of the interactions of supply and demand for funds in different market segments.

- preferred habitat theory : similar to the segmented markets theory, but recognizes that market participants will deviate from their preferred maturity habitat if compensated adequately.

 

○ Modern term structure

- equilibrium term structure models : using fundamental economic variables

⇒ the C.I.R model : economy has a natural long-run interest rate that the short-term rate converges to. high interest rates, the amount of period-over-period fluctuation in rates is also high.

= 추세(a;속도, b;장기평균, r;현재금리) + 변동성

⇒ the Vasicek model : similar to CIR, but interest rate volatility level is independent of the level of short-term interest rates. interests can be negative.

- Arbitrage-free model : begins with observed market prices and the assumption that securities are correctly priced

⇒ the Ho-Lee model : calibrated by using market prices to find the time-dependant drift term that generates the current term structure

 

[ The Arbitrage-Free Valuation Framework ]

 

○ Binomial interest rate free

							UU
		50%		U
									r ×
							UL & LU
	기초								midpoint ⇒ implied forward rate for this period
				L					r ×
		50%
							LL

 

○ Pathwise valuation

- average of the values of the bond at each path. For a n-period binomial tree, there are possible paths.

 

○ Monte Carlo simulation

- incorporates a volatility assumption and assumes probability distribution

- MBS ⇒ path-dependent cash flow ⇒ Monte Carlo simulation

 

[ Valuation and Analysis: Bonds with Embedded Options ]

 

○ Volatility

- volatility↑ ⇒ call & put ↑ ⇒ callable ↓ & putable ↑ ⇒ OAScall ↓ & OASput ↑

 

○ Effective Duration

-

-

- ED(callable or putable) ≤ ED(straight)

 

○ Key rate duration

- interest rate sensitivity of a bond to changes in yields of specific benchmark maturities.

 

○ One-sided Durations

- 금리가 오른/내린 경우 한 쪽만의 duration (양쪽을 합치면 ED)

- callable : one-sided down-duration < one-sided up-duration

⇒ 금리 하락, 채권가치 한계까지 상승(call price), but 금리 상승, 채권가치 fully 하락

- putable : one-sided down-duration > one-sided up-duration

⇒ 금리 하락, 채권가치 fully 상승, but 금리 상승, 채권가치 한계까지 하락(put price)

○ Convertible bond

- market conversion premium per share = market conversion price - stock's market price

- market conversion premium ratio =

- premium over straight value =

 

○ Put-call parity

-

- protective put = fiduciary call

 

[ Credit Analysis Models ]

 

○ Measures of credit risk

- expected Loss = probability of default × loss given default

○ Present value of the expected loss due to credit risk = PV - Risk-free PV

 

○ Option analogy (Structural Model)

- value of debtT = Min(AT, K) ⇒ RF Bond - put option on asset

- value of stockT = Max(0, AT - K) ⇒ + call option on asset

A	Value of Debt		A	Value of Equity
		K			K

○ Credit ratings

- Strengths : simple, summary measures of risk that are easy to communicate.

- Weaknesses : do not adjust with business cycle, and the stability in ratings comes at an expense of reduction in correlation with default probabilities.

 

○ Structural Model (자본구조 및 옵션 이용)

- Assumptions : company's assets are traded in a frictionless market with return μ and variance σ2. The risk-free interest rate (ϒ) is constant. The company has a simple balance sheet structure. (lognormal distribution, only zero coupon bond)

- Strengths : provides option analogy to understand probability of default and loss given default and can be estimated using current market prices.

- Weaknesses : model assumptions of simple balance sheet and traded assets are not realistic. Estimation procedures do not consider business cycle.

 

○ Reduced Form Model (credit spread 이용)

- Assumptions : company has a zero-coupon bond with a maturity at time T, and it trades in frictionless and arbitrage-free market. The risk-free interest rate (ϒ) and the state of the economy are stochastic. The probability of default and recovery rate is not constant and depends on the state of economy.

- Strengths : since model inputs are observable, historical estimation procedures can be used. Credit risk is allowed to fluctuate with the business cycle. Reduced form models do not require specification of the company's balance sheet structure.

- Weaknesses : unless the model has been formulated and back-tested properly, the hazard rate estimation procedures (using past observations to predict the future) may not be valid.

 

○ Reduced form model > structural model > credit ratings model