Fixed Derivatives Flashcards
Carried Interest
When Fund value > Invested Capital
- Tag Long - Drag Along (Private Equity)
- Ratchet
- Distribution Waterfall
- Tag along: When management has the ability to get equity upon sale of the company
- Ratchet: Specifies equity allocation between LPs and Management
- Distribution Waterfall: How profits flow to the LPs and whether the GP may receive carried interest
Calculate % Ownership of Private Equity First Round Investors (F1)
F1 = Investment / Post1
- Investment: amount invested at the beginning (before post1)*
- Post 1: The present value of the company post the first round of financing*
Calcuate # of Shares F1 Investors need to get % ownership (Spe1)
Spe1 = Se(F1 / 1-f1)
- Spe1: Shares for Private Equity Investors in round 1*
- F1: Fractional ownership of first round investors*
- Se: Number of shares owned by management*
Determine Stock Price after First Round of Private Equity Financing
P1 = Inv1 / Spe1
- P1: share price post financing first round*
- Inv1: Initial financing*
- Spe1: Shares owned by Private Equity investors after round 1*
Loss Given Default / Recovery Rate
- Loss given default: overall position (remaining coupon + Principle) that would be lost
- Recovery rate: The remaining amount after LGD is taken out.
Credit Risk: Expected Loss
= Probability of Default X Loss Given Default
PV of Expected Loss: Credit Risk
- Highest fee holder would pay to remove the risk of the bond
1. Risk neutral probabilities are used instead of actual probs like expected loss - time value of money is considered*
Credit Evaluation Models
- Credit ratings: Consistency over accuracy
- Structural models: Options based with unrealistic expectations
- Reduced Form Models: Variable Rf%, economic factors, etc.
Yield Curve Shapes (4)
Normal: Positive slope
Flat: Horizontal
Inverted: Negative slope
Humped: Positive slope then negative
Yield Curve Shifts (3)
- Parallel: All maturies move by same bps
- Twist: change in YC slope
- Butterfly: + shift YC becomes less humped / - shift YC becomes more humped
Expectation Theories: Term Structure of Interest Rates
- Pure expectations: Geometric mean of short term yields
- Liquidity preference: Long bond holders require higher premium
- Preferred habitat: YC is determined by expectations that can be different for different maturities
Annualizing the Standard Deviation: Fixed income
ASd = Daily Sd X √# of days in year
Nominal / z-Spread / OAS (fixed income)
Benchmark / Compensation for risk
- Nominal: T-yield curve / credit, liquidity, option (a one time view on the spread)
- z-Spread: T-spot rate curve / credit, liquidity, option (the spread that when added to the treasury spot rate will make the bond’s value equal to the price of the bond)
- Option-adjusted: T-spot rate curve / credit, liquidity
Price of Callable Bond
Price of a Putable Bond
Callable Price = Price of Option Free Bond - Price of embedded call
Putable Price = Price of option free bond + price of embedded put
Effective Bond Duration
Duration = (V₋ - V₊) / [2V₀(Δy)]
- used to value bonds with or without an option
- This models a change in price if the yield shifts by 100 bps regardless of the amount of change used to calculate V- and V+
Effective Bond Convexity
Convexity = [V₋ + V₊ - 2V₀] / [2V₀(Δy)²]
Most Appropriate Model for Valuing Models on Mortgage Backed Securities
Monte Carlo for bonds with options
Bond Equivalent Yield
BEY = 2[(1+i)^n -1]
- i: interest rate per period… monthly would lead to an exponent of 6
Appropriate Spread Measure for…
- Callable corporate bonds and MBS
- Credit card or autoloan ABS
- Option free corporate bonds
- Callable = Option adjusted/removed spread
- Credit card or auto loan ABS = Z-spread
- Option free corporate bonds = Z-spread
Market Conversion Premium Per Share (convertible bond)
MCPPS = (Market price of bond / Conversion factor) - Market price of stock
- Bond = 950 / converts to 10 shares / stock trading at 50*
- MCPPS = (950/10) - 50 = $45*
Steps to Value Bond (1 year) with %r Tree
1: Value of bond with up and down change in interest rate one period forward.
2. If value of bond in one branch is greater than callable value… use that
3. Add coupon payment(s) in branches
4. Discount at one-year treasury rate
[(V1+C$)/(1+%r) + (V2+C$)/(1+%r)] / 2
Use OAS spread to determine of callable bond is over, under or fairly valued.
- OAS is negative compared to corp spot rate curve = overvalued
- OAS is zero compared to corp spot rate curve = Fairly valued
- OAS is positive compared to corp spot rate curve = undervalued
Delta Hedging
Number of shares in contract = Investment / Delta
Gamma is Greatest for…
At the money options
Put-Call Parity
Po = Co - So + x/e^(r)(t)
- t should be in decimal form (90 days = .25)*
- this formula assumes continuous compounding*
Put-Call Parity for Futures and Forwards
P0 = Co + [(X - F(0,T) / 1+r^t]
- F(0,T) = value of future now that expires in time T*
- The numerator could be negative*
Value of Call Option for one period (Binomial Model) including probability calculation
Step 1: Calculate max gain for up and down move
Step 2: prob = (1 + Rf% - d) / u - d
Step 3: Co = [prob(call+) + 1-prob(call-)] / (1 + Rf%)
- d = 1 - the likely decline in value in percentage terms*
- u = 1+ the likely rise in value in percentage terms*
- call+ = The maximum value of the call assuming the positive forecast comes true at expiration*
Value of Call (binomial model) - Two Periods
Step 1: Calculate max gain for up, down, and up/down-down/up moves - up and down move must be squared
Step 2: prob = (1 + Rf% - d⁻) / u⁺ - d⁻
Step 3a: C⁺⁺ = [prob(call⁺⁺) + 1-prob(call⁺⁻)] / (1 + Rf%)
Step 3b: C⁻⁻ = [prob(call⁺⁻) + 1-prob(call⁻⁻)] / (1 + Rf%)
Step 4: C = [prob(C⁺⁺) + 1-prob(C⁻⁻)] / (1 + Rf%)
- d = 1 - the down move (or 1/1+upmove if no down move given)
- u = 1+ the up move
- call+ = The maximum value of the call assuming the positive forecast comes true at expiration
Assumptions about Black Scholes Merton (BSM) Formula
- Volatility of the return on the underlying stock is known and constant
- Stock prices are lognormally distributed
- Continuous Rf% is known and constant
Fixed Rate for Swap Contract
Fixed Rate = DFn / (DF1 + DF2 + DF3… + DFn)
- DF: Discount Factor*
- If annualized then Fixed rate must multiplied by the appropriate factor - semi-annual x2. *
No arbitrage price on forward contract on treasury bond
Step one: NPV of next coupon payment PMT/1+r^t
Step two: (Vo - NPVcoupon) X 1+r^t’
- t = time to next coupon/365*
- t’ = forward length/365*
- Vo = price of bond now*
- r = effective rf%*
Loss (or value of short position) on no arbitrage forward contract treasury bond
Step one: Current NPV of next coupon payment PMT/1+r^t*
Step two: V1 - NPVcoupon - (Original forward value) / 1+r^t*’
t* = Current time to coupon payment / 365
t*’ = Current time to expiration / 365
- Original calculation to determin forward’s first price…*
- Step one: NPV of next coupon payment PMT/1+r^t*
- Step two: (Vo - NPVcoupon) X 1+r^t’*
- t = time to next coupon/365*
- t’ = forward length/365*
- Vo = price of bond now*
- r = effective rf%*
Determine Forward Rate Agreement rate from Spot Rate Curve (2X5)
= ({[(1+150d%)^150/360] / [(1+60d%)^60/360]} - 1) / 360/90
- curve moves in 30 day increments
Loss (or gain) on short position on 2x5 FRA 30 days later
Step one expected Payoff = (FRAo - FRA1)(notional value)(90/360)
Step two PV of expected Payoff = payoff/1+120day%^(120/360)
- note: the PV is probably not going to be much less than step 1.
Calculate the payoff on an interest rate floor
Floor Payoff = Principle X (Floor rate - Market rate)
Definition of a Swaption
The option to enter into an interest rate swap
Payer Swaption definition
Allows buyer to enter into a swap as the fixed rate payer/ variable rate reciever
Receiver Swaption definition
Allows buyer to enter into swap as fixed rate receiver and floating rate payer
Interest Only Strips (MBS)
- Collect only interest payments from mortgage pool
- If rates rise, prepayments drop and interest payments stay high for a longer period
- Worried about contraction risk
Principle Only Strips (MBS)
- Collect only principle payments from mortgage pool
- Discount security so falling interest rates = quicker return of capital
- Worried about extention risk
Single Month Mortality Rate/ Conditional Prepayment Rate / Estimated Monthly Prepayment
- SMM = 1 - (1 - CPR)^1/12
- CPR = 1 - (1 - SMM)^12
- EMP = SMM X (MBSo - Principle)
PSA (Has to do with )
PSA benchmark for conditional prepayment rate (CPR)
- Assumed to be .02% in month one and increases .02% per month until month 30.
- Remains at 6% after month 30 until 100% PSA
PAC Tranche vs. Support Tranche
Planned Amortization Class
- They have priority on principle payments over other classes
- The non PAC classes are “support” tranches with uncertain paybacks
Effective Collar on PAC Tranche
The PSA speeds within which the PAC tranche has protection from the support tranches.
Direct Capitalization Method (real estate valuation)
- Calculate average cap rates from recent sales (Price/NOI)
- Apply average cap rate to expected NOI
Direct Sales Valuation Method (real estate)
Step 1: Calculate Price per SqFoot for each comparable
Step 2: Multiply step 1 by 1+(age of building (depreciation), location, condition, age of transaction (appreciation))
Step 3: Apply average of step 2 to investment
Equity Dividend Rate (real estate investment)
- Equity (use LTV to calculate)
- Annual debt service costs (12 X Monthly service costs)
- Net operating income
- EDR = (NOI - ADSC) / Equity
Levered Rate of Return (real estate)
PMT = NOI - Debt Service
PV = - Equity (use LTV to calculate)
FV = Expected Sale price - Loan payoff
N = Expected holding period
CPT I/Y = Levered IRR
Value of Property Cost Approach (real estate)
Replacement cost (include builder profit)
- Curable deterioration
- Incurable deterioration (Vo X (age/total age))
- Total Obsolescence
Building value
+ Land value
Total cost estimate
Incurable / Curable Deterioration (real estate)
- Curable: Deferred maintenance that is curable
- Incurable: Depreciation of the building.
Appraisal Lag
Appraisal values lag changing transaction values. If prices are rising, they rise first and then appraisals come up. Combat appraisal lag by unsmoothing the index or relying on transaction indexes.
Assets included in CDO’s
- HY Corp bonds
- Structured products (MBS, ABS)
- Emerging Market bonds
- Bank loans
- Special situation and distressed debt
Most appropriate spread measure for home equity loan ABS
OAS spread with monte carlo simulation
Gains/Losses on bonds held that are available for sale are reported where on financial statements?
On the BS in other comprehensive income. Versus held for trading where gains/losses are shown on income statement. Or held to maturity where amortized cost and interest income is only thing reported on BS and IS respectively.
Economic Factors that Affect REIT investments
- Industrial Properties
- Apartments
- Office properties
- Industrial - Retail Sales
- Apartments - Population and job growth
- Office - job growth
Value of REIT (Asset Value Approach)
NOI
- Non-cash Rents
+ Acquisition Adjustment
Proforma NOI
PFNOI (1+g) = Projectied NOI
PrNOI / cap rate = estimated value of assets
+ other assets
- Liabilities
NAV
Reason to choose investing in real estate directly versus REIT, or REOC
Avoid structural conflicts of interest that may exist in REOC and REIT
Value of REOC using Discounted Cash Flows Approach
- Determine present value of dividends over forecasting period.
- Determine terminal value (look at instructions)
- Discount rate = Cap rate + Expected growth rate
Effect of a negative butterfly shift in yield curves on a short term callable bond
- It would experience price compression
- Short term rates are likely falling more than intermediate or long term rates
- However, prices of callable bonds are capped. - limiting the upside.
Calculate the effect of a change in yield on a portfolio duration
- Durations of different maturities are negative - multiply them by the interest rate increase (+) or decrease (-).
- Add the changes determined in step 1 to find change in portfolio duration.
Determine the fixed rate for a semi-annual swap
- Calculate discount factors for each semi-annual period
ie 1 / 1 + (.0245 x 180/360) = .9879 for the first 6 months or 180 days. - Determine fixed rate using annualized interest rates (assume this is a 2-year swap)
[(1 - 720discount) / (180discount + 360discount + 540discount + 720discount)] X 2 = fixed rate
Payoff for Quarterly Interest Rate Cap
- Doesn’t matter how long cap has been in place.
- (Current R% - Cap R%) X 90/360 X Notional = Payoff
Calculate Futures Price
FP = So X [e^(r-d)(t)]
- e = ln or 2.71818
- the exponent is the Rf% - Dividend rate times time
- t = days to expiration / 360
Convenience Yield Definition
Is the non-monetary benefits of holding an asset that is in short supply. if there is a shortage of a commodity, your storage costs as the holder may be less than the costs of the future’s price.
Price of Futures vs. Forward in flat (constant) interest rate environment
If the interest rate is constant then there will be no difference between futures and forward price.
OAS provides insight into what?
- the liquidity risk of the bond with options when compared to similar bonds with options.
- Different volatility assumptions will affect OAS
Post money VC valuation using single period NPV
= Expected future value / 1+r^n
- Don’t worry about initial investment since this is post money.
Calculate carried interest at the end of a series of years
- called down capital - management fees + performance for each year = end of year NAV
- Difference between total NAV
CMBS’s are riskier than MBS’s because…
They tend to be non-recourse and are only supported by the value of the property and its cash flows. The lender cannot generally go after the company’s other assets like you can with a residential MBS.
Calculate Fixed Payment in Fixed for Fixed Currency Swap
- Derive PV Factors (example of a 2-year swap)
* a. Z360 = 1/[1+360Rf%(360/360)] = 360PV Factor
b. Z720 = 1/[1+720Rf%(720/360)] = 720PV Factor*
- 2*. Fixed payment in “pesos”
- a. (1-720PV)/(720PV + 360PV) = Fixed payment per peso*
- Annual fixed payment
* Fixed payment per peso X $Notional / Current exchange rate*
Currency Swap Rate Spreads Reflect the risk of What?
- Global credit risk not the credit risk of the counterparties
Credit Risk peaks at what point for currency and interest rate swaps?
- Currency swaps: peaks at the end because notional amounts must be exchanged.
- Interest rate swaps: peaks at the middle and then declines as swap payments are made and fewer are due.
Present value of currency swap payments and notional principle
- Calculate the Annual Fixed Payment from original rate structure
- Use the new rate structure to find PV factors (adjust for time passing)
- Present Value
* Annual fixed payment X (new360PVF + new 720PVF) + Notional X new 720PVF = PV *
Collateral Changes in an ABS
- Amortizing ABS collateral is not changed after creation (home loans, auto loans, etc)
- Non amortizing (revolving) collateral may change during lockup period. Principle payments are used to buy more collateral
Clean up call provision in ABS
Triggers a provision where cash flows are used to pay down principle if the value of the ABS has been impaired. This applies to credit card ABS in the lockup period.
Excess Servicing Spread (ABS)
Issuer pays a lower coupon than the coupon on underlying collateral. The excess cash is used to pay for losses on the colateral. Internal credit enhancement.
Letter of credit
An agreement from a bank to cover losses on an ABS up to a certain amount. This can be affected by the credit of the bank and is an external credit enhancement.
This is an example of the weak link philosophy.
Overcollateralization (ABS or MBS)
- Internal credit enhancement
- Principle of collateral is greater than principle of tranches
- Absorbes losses before junior traunches
Determine which tranche has best OAS value
- OAS value per Option cost
a. 36 bp OAS for 15bp option cost
b. 36 bp OAC for 19bp option cost
A is the better choice in terms of OAS value.
Trick to determine whether to convert currency transaction at bid or ask.
For triangular arbitrage or covered interest rate parity or whatever, just convert at the bid or ask for the worst deal.
Calculate Cap Rate (real estate)
= Required Return - Growth Rate
Venture Capital Method to Determine Percentage Ownership
Investment(1+r^n) / Expected Vo in future
Value of Equity Forward at Point in Time
Vt = St - F(0,T)/(1+r)^T-t
- Vt = value now*
- St = Spot price now*
- F(0,T) = Forward value at expiration*
- T-t = Time to expiration at the beginning and t is time to expire now. ie 90/360*
Forward Contract Price on Equity Index
F(0,T) = [Soe^-(δc)T] X [e^(rc)T]
δc = Continuously compounded dividend rate
rc = Continuously compounded interest rate
Calculate Forward Price on Currency Rate
= So X (1+rD)^T / (1+rF)^T
- MAKE SURE YOU CHECK FOR 360 vs 365 YEAR
Calculate forward price on dividend paying stock
= So(1+r)T - (div/1+r^t + div/1+r^t…
- *The t in the dividends needs to be time to dividend divided by the time to expiration. *
Synthetic stock
So = C - P + Long bond/(1+r^t)
Accrual Tranche
receives principle and interest after PAC and sequential pay tranches have been paid. Interest owed to accrual tranche will be added to principle.
Sequential Pay Tranche
Each SPT recieves principle payments after the tranches in front of it have been paid off.
Common source of value creation from a leveraged buyout…
Reduction of debt by the time the company is sold.
Total Value to Paid in Capital (private equity)
- Total value = NAV after distributions (last year) + total distributions
- TVPiC = TV/Paid in capital in last year (the sum)
Find market value of equity swap
Step one=1+gain on equity - 360df - (.XX)(Df+Df+df+df)
Step two = Notional x Step one = value
Market Value of a Receiver Swaption
= Max (0), (fR% - Market rate)X(Dfn+Dfn+Dfn…)XNotional
- Remember to use 360 not 720 in your Df calculations
- the market rate needs to be calculated from the rate structure
Spot price versus Futures price
EXPECTED Spot price = Futures price + Risk premium
Based on sharpe ratio which asset should be added to portfolio to achieve greatest mean-variance improvement?
Max of: Sharp of potential asset > Sharpe port X correlation between port and new
Standard Deviation of a Portfolio of two Assets
= √(w1² X σ1²) + (w1² X σ1²) + 2(w1,w2r1) X σ1σ2
Value of Swaption
Max (0, (r-M%) X (DF1 + DF2…)xNP
