Adjusted Cash Flow Approach

1
Adjusted Cash Flow Approach

• Fixed income valuation has two parts
• Estimating cash flows
• Placing a present value on those cash flows
• We will concentrate on the first
• Adjusted Cash Flow (ACF) Approach
• Single cash flow estimate for each date
• Riskless discount rate (i)

2
Estimating Mortgage Portfolio Cash Flows

• Example fixed rate mortgage portfolio with
  current principal balance of 20,000,000.
• Coupon rate? Use weighted-average coupon (WAC)
• Example with 3 mortgages

3
Estimating Mortgage Portfolio Cash Flows

• Maturity? Use the weight-average maturity (WAM).
• Example with 3 mortgages

4
Estimating Mortgage Portfolio Cash Flows

• ECF1 has 3 parts expected PI payment,
  expected prepayment, and expected recovery from
  default.
• Expected PI cash flow
• PV 20,000,000
• N 332
• I/YR 7.875
• Implies PMT 148,136 ECF1,PI
• Interest 0.0785/12 20,000,000 131,250
• Principal 148,136 131,250 16,886

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Estimating Mortgage Portfolio Cash Flows

• Prepayment rate
• Typically provided as an annual Cumulative
  Prepayment Rate or CPR
• Need to convert to Single Monthly Mortality or
  SMM
• SMM 1 (1 CPR)1/12
• Example CPR 22.0 implies SMM 1 (1
  0.22)1/12 0.0205
• Estimated Prepayment Cash Flow
• Remaining principal after PI payment
  20,000,000 16,886 19,983,114
• ECF1,prepay 19,983,114 0.0205 409,654

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Estimating Mortgage Portfolio Cash Flows

• Default and recovery rates
• Annual default rate estimate needs to be
  converted to monthly
• monthly 1 (1 annual)1/12
• Example annual 3.0 implies monthly 1 (1
  0.03)1/12 0.0025
• Recovery rate is the percentage of defaulted
  balance that is expected to be received as a cash
  flow
• Example 85.0
• Estimated Default Amount and Recovery Cash Flow
• Edefault1 19,983,114 0.0025 49.958
• ECF1,recovery 49,958 0.85 42,464

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Estimating Mortgage Portfolio Cash Flows

• Expected Cash Flow
• ECF ECFPI ECFprepay ECFrecovery
• ECF1 148,136 409,654 42,464 600,254
• Remaining Principal Balance
• New Balance Old Balance scheduled principal
  prepayment default
• Balance1 20,000,000 16,886 409,654 49,958
  19,523,502
• Iterative Process
• Now that we know the (expected) remaining
  balance, we can start the process over again and
  calculate the second expected cash flow (and the
  third, and the fourth, and so on.)

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Estimating Mortgage Portfolio Cash Flows

• 2nd expected cash flow
• PV 19,523,502 N 331 332 1 I/YR
  7.875
• Implies PMT 144,729 ECF2,PI
• Interest 0.0785/12 19,523,502 128,122
• Principal 144,729 128,122 16,606
• Remaining principal after PI payment
  19/523/502 16,606 19,506,896
• ECF1,prepay 19,506,896 0.0205 399,891
• Edefault2 19,506,896 0.0025 48,767
• ECF1,recovery 48,767 0.85 41,452
• ECF2 144,729 399,891 41,452 586,072
• Balance2 19,523,502 16,606 399,891 48,767
  19,058,238

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MBS example annual cash flows
10
CMO Cash Flows

• Passthrough would get total expected cash flow.
• IO and PO Strip Structure
• IO gets interest less servicing
• PO gets the rest (scheduled principal plus
  prepayment plus recovery from default)

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CMO Cash Flows

• Tranche A
• 500,000 principal balance
• 4.50 coupon rate

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CMO Cash Flows

• Tranche B
• 250,000 principal balance
• 6.00 coupon rate

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CMO Cash Flows

• Tranche Z
• 150,000 initial principal balance
• 7.50 earnings rate
• Residual (R-tranche) gets remaining cash flows

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Problems with ACF Approach

• Doesnt consider interest rate volatility
• As we will learn, the value of any option grows
  as volatility increases.
• Mortgage Loan Prepayment Option
• Implies that mortgage value should decline as
  interest rate volatility increases.
• This cant happen if the model doesnt have
  volatility as an input.

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Monte Carlo Simulation(aka Random Paths Model)

• Idea / Concept
• Randomly generate a large number of equally
  possible future interest rate paths
• Calculate the value of the security or portfolio
  on each path
• The Value of the security or portfolio is the
  average value across the random paths
• Step 1 Determine a single market or economic
  driving factor
• In fixed income valuation, this is an market
  interest rate.
• Usually the short-term riskless rate.
• Step 2 Develop or borrow a mathematical model
  that describes how interest rates change over time

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Monte Carlo Simulation(aka Random Paths Model)

• Step 2 Develop or borrow a statistical model
  that describes how interest rates change over
  time
• e.g., Federal Reserve chairperson flips a coin
  each morning. If its head, rates go up 5 bps.
  If its tails, rates go down 5 bps.
• Formerly, this is a binomial distribution with
  50 probability for each outcome.
• Alternatively, one might believe that changes in
  the short-term rate is normally distributed with
  some mean and standard deviation.
• More complex models abound.

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Monte Carlo Simulation(aka Random Paths Model)

• Step 3 Use interest rate model to randomly
  generate a series of interest rate changes,
  called an interest rate path
• Starts from current interest rate
• Goes forward in even increments that match cash
  flow frequency and maximum maturity (e.g., 360
  monthly steps for a 30-year mortgage pool)
• Not a prediction of the future or expected path
  of interest rates simply a possible future
• REPEAT Step 3 many, many times (500 or more)
• This create the Monte Carlo rate cloud
• Steps 1-3 are done one time for the entire fixed
  income portfolio

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Monte Carlo Simulation(aka Random Paths Model)

• Step 4 Develop a model of expected cash flows
  for a particular type of security
• e.g., for a mortgage pool, the model would relate
  the level (and history) of interest rates to
  prepayment expectations
• This cash flow model will be path dependent
  must look at the history of interest rates on a
  path and build the cash flows iteratively
• Step 5 On each interest rate path
• Generate ECFt for each date
• Iterative process (start at t1 and move to
  tmaturity of security/portfolio)
• Discount back the ECFs using the interest
  rates on the interest rate path.
• Repeat on each interest rate path

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Monte Carlo Simulation(aka Random Paths Model)

• Step 6 PV of the security/portfolio
• Equal to the average PV across the randomly
  generated interest rate paths
• Clearly impacted by changes in volatility
  assumptions (as well as changes in assumptions
  about prepayment, default, recovery, etc.)
• Calibration issues test using a set of
  securities with fixed cash flows and
  well-established market prices
• Usually calibrated against Treasury securities
• Calibration does not ensure that the cash flow
  model is correct
• Complexity of Monte Carlo valuation
• Computation rigor

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Binomial Tree (Lattice) Models

• Simple Set-up
• Focused on short-term interest rate
• Assumed to move up or down each step
• Must provide size of up movement, size of down
  movement, and probability of moving up
• Number of steps related to maturity of securities
  to be evaluated and number of steps between each
  cash flow date
• Step 1 select parameters
• Choose rup, rdown, Pr(up), and number of steps.
• Pr(down) 1 Pr(up)
• Problem
• 30-year mortgage portfolio with monthly steps

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Binomial Tree (Lattice) Models

• Step 2 For each possible path
• Estimate cash flows at each cash flow date
• Discount back at risk interest rate
• Step 3 Value
• Equals the probability-weighted value across the
  paths.
• Probability may be different for each path (not
  equally weighted)
• Problem
• 30-year mortgage portfolio with monthly steps
  requires 2360 2.35 10108 paths
• Binomial (lattice) valuation techniques are used
  for very short-term derivative securities