Fixed income Flashcards
Number of paths for binomial tree
2^(n-1)
Upper node interest rate
Lower node interest rate*e^2σ
Lower node interest rate
((Upper node interest rate))/e^2σ
P. of bond
VND – CVA
CVA
PV (Expected loss) = LGD * prob of default*discount factor
Dove–> LGD = VND * (1-Recovery rate)
Prob of default = hazard rate
Sommo tutti i T
VND
puo essere anche semplicemente l’exposure = par value + coupon
PV of bond
LGD
VND* ( 1- recovery rate)
Prob of survival
(1- p of default)^N
Risk neutral default probability P
((VND(1-P))+((VND-LGD)(P)/(1+r)
Dove–> LGD = VND * (1-Recovery rate)
CF anno default
Exposure – LGD
Δ%P
(modified duration of the bond) × (Δ spread)
Se ho anche probability la devo molitplicare ad ogni risultato
Delta spread = credit spread from other letter - credit spread from my letter —> da dove vado a dove sto
E poi sommo tutti i risultati
Credit spread
YTM (risky) – YTM (riskfree)
Upfront payment (by prot. Buyer)
PV (protection leg) – PV (premium leg)
Upfront premium
(CDS Spread – CDS Coupon) * CDS duration
V of capped floater
Straight floater value – Embedded cap value
V of floored floater
Embedded floor value - Straight bond
Minimum value of convertible bond
Greater of conversion value or straight value
Conversion value of convertible bond
Stock market price * Conversion ratio
dove conversion ratio = n. of common shares for which a convertible bond can be exchanged
Market conversion price
Market bond price / Conversion ratio
Conversion ratio = Par value / Conversion price
Market conversion premium per share
Market conversion price – Stock’s market price
Owning stock or Equity
Value of long call = max (S-K)
Owning debt
Value of short put = min (K-S)
Swap spread
Swap rate - treasury yield
info on supply and demand –> swap rate - treasury yield
TED sread
3 monts MRR - 3 months T bill rate
info on credit risk in the economy–> 3 months MRR rate - 3 months T bill rate
Cosa indica MRR OIS spread
MRR rate - overnight rate
well being of banking system –> low spread 0 high liquidity–>MRR rate - overnight spread
OAS
credit and liquidity risk
when volatility ↑, computed OAS for callable bond ↓ and putable bond ↑
o Callable –> if volatility increase –> OAS decrease–>
o Putable –> if volatility increase –> OAS increase–>
cosa indica structural model
o explain why default occurs
o Estimation of default barrier and default occurs if the value of firms assets are below this barrier
o They require info of company –> Default is an endogenous variable (inside the company)
o Option pricing theory
o Use traded market price
Cosa indica Reduced model
o explain when default occurs
o Default = exogenous variable
o Parameter estimation: Default intensity
o Useful for Off-BS
o The probability of default (default intensity) and the recovery rate depend on the state of the economy and are not constant.
o Use historical variables (financial ratios and macroeconomic variables) known by everyone
V of callable bond
v straight bond - v call
V putable bond
V straight bond + v put
Flat yield curve- cosa indica?
all spot = all forwards –> expect decrease in inflation
- If expectation of Upward sloping credit curve - cosa indica?
expectation of recession reduce duration of pf
*
Perchè Swap rate curve preferita?
o Swap rates reflect the credit risk of commercial banks rather than governments.
o The swap market is not regulated by any government.
o The swap curve typically has yield quotes at many maturities indicates the premium for time value of money at different maturities.
o Institutions like wholesale banks use swap curves to value their assets and liabilities.
o Indicate time value of money
- Unbiased expectations th
o Forward rates = unbiased predictor of future spot rates.
o Interest rates expected to increase in future
o Upward sloping yield curve = int rates are expected to increase in the future
o Forward = Break even rate = investor is indifferent between investing for the full term of their investment horizon or investing in part of the horizon and rolling the investment over at the “break-even” forward rate for the remainder of the term.
- Local expectations th
o In the short term= risk-free rate.
o In the long term = risk premium
- Liquidity preference th
o Longer maturity = higher liquidity premium
o Current forward rate > Future spot rates
- Segmented markets th
interactions of supply and demand for funds in different market (i.e., maturity) segments.
o Interest rate are determined by supply/demand for a given maturity sector–> High demand = Lower interest rates
- Preferred habitat th
market participants will deviate from their preferred maturity habitat if compensated adequately.
Effective duration
[(PV-) – (PV+)] / (2 x (curve change) x (PV0))
o economic expansions–> effect on yield curve
rising inflation cb increase ST rates bearish flattening of the yield curve. L’economia va bene in trentino e rispondono con gli orsi
o recessionary times
cb reduce st rates bullish steepening. l’economia va male in spagna e rispondono con I tori
o market turmoil
reduce LT government bond yields bullish flatteningLT bullet is appropriate non si capisce niente in turmoil il toro impazzisce e si sdraia
- Equilibrium Term Structure Models
mean reversion
Cox-Ingersoll-Ross (CIR) = volatility varies with rates (aggiungo la σ√r) Vasicek model. volatility in this model does not increase as level of interest rates increase
o Arbitrage-free models
bonds in the market are correctly priced
Ho-Lee model–>time-dependent drift; noise component
Kalotay-Williams-Fabozzi–>constant volatility
Gauss + model
LT rate / volatility depends on macroec. variables (economy inflation…) and is mean reverting
stripping
o If the principle of value additivity does not hold
buying a bond and then selling off its parts
reconstitution
buying the parts to sell a reconstituted bond
When is convexity positive or negative
o Straight and putable bonds = positive convexity.
o Callable bonds = positive convexity when rates are high. However, at lower rates, callable bonds =negative convexity.
ricorda disegno
- ABS
o RISKS: Operational and counterparty risk of servicer
o Short term granular and homogenous vehicles are evaluated using statistical based approach
o Medium term and homogenous obligations are evaluated using portfolio based approach
o Credit enhancement and distribution waterfall relevant characteristics
o Credit rating agencies do not use same credit ratings for ABS as corporate debt
Effective duration
((PV-) - (PV+)) / (2* curve change * PV0)
Duration theory
o Callable bonds –> lower one sided down duration than one sided up duration
o Putable bonds –> higher one sided down duration than one sided up duration
o Callable bond –>high duration (perchè non viene chiamata) when low coupon and high yield
o Putable bond –>high duration when high coupon and low yield
credit spread
o La credit spread term può essere o positive o flat
o Più è alto il rating più è flat la curva del credit spread
o Economia va bene –> credit spread diminuisce
o Securities with lower credit quality –> greater sensitivity to credit cycle
Portfolio based approach –> a lot of small loans very dynamic
Loan-by loan approach –> few big loans
Statistics based approach –> for static loans
