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Outline the Monte Carlo simulation steps used to value an MBS.

FRM® Part I · Financial Markets and Products · Mortgages and Mortgage-Backed Securities · FRM1.FM.18.i

The idea

Monte Carlo valuation handles the path dependence of MBS cash flows. Prepayments depend not only on today's rate but on the route rates took, through burnout. Simulate many interest rate paths from an arbitrage-free model calibrated to the current curve and volatilities. On each path derive the mortgage rate and refinancing incentive month by month. Run the prepayment model to project interest, scheduled principal and prepayments. Discount each path's cash flows at that path's own simulated short rates, plus any spread. Average the present values across paths. The average is the model value. Solving for the spread that makes it equal the market price gives the OAS. Revaluing under shifted curves gives effective duration and convexity. Standard error falls with the square root of the number of paths. The exam point is to discount along each path, never on one average curve.

Formulas

Effective duration

EffDur = (PVdown − PVup) ÷ (2 × Δcurve × PV₀)

PVdown
the price after the benchmark curve is shifted down
PVup
the price after it is shifted up by the same amount
Δcurve
the size of that shift, as a decimal
PV₀
the price before either shift

For an MBS, hold the OAS constant and rerun the prepayment model on each shifted curve, so the cash flows change with rates.

Standard error of a Monte Carlo value

standard error = σ ÷ √N

σ
the standard deviation of the path present values
N
the number of independent paths

Halving the error needs four times the paths. It measures sampling noise only, not error in the rate or prepayment model.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which sequence correctly describes the Monte Carlo valuation of an MBS?

  1. Simulate rate paths, project cash flows on each path, discount each at its own rates, then average. - correct
  2. Average the simulated rate paths into one path, project cash flows on it, and discount them.Why not: Averaging paths first destroys the nonlinear option effects; averaging is done on values.
  3. Project one prepayment curve, discount it at today's spot curve, and then simulate shocks for duration.Why not: Does not simulate paths for valuation, so it ignores option and path-dependent effects.
  4. Simulate paths, discount at path rates, and only then run the prepayment model on the average value.Why not: Wrong order: the prepayment model needs path rates to produce cash flows before discounting.

Cash flows depend on the path of rates through the prepayment model, so cash flows are projected path by path, discounted with the same path's rates, and only then averaged.

A harder one

In the simulation, each path's cash flows are discounted using:

  1. The short rates simulated along that same path, plus any spread. - correct
  2. Today's observed spot curve, applied to every simulated path alike.Why not: Ignores the simulated rates, which differ between paths.
  3. The weighted average coupon of the pool, used as a fixed discount rate.Why not: The WAC is a cash flow rate, not a discount rate.
  4. The average simulated rate across all paths at each date.Why not: Averaging the rates before discounting ignores the convexity of discount factors.

Each path is a scenario with its own discount factors. Using the same path for projection and discounting is consistent, and the spread is added to the short rates.

The hardest tier

A Monte Carlo valuation with 10,000 paths gives a standard deviation of path present values of 6.0 points, so the standard error of the estimate is 0.06. To halve the standard error to 0.03 with independent paths, the number of paths needed is closest to:

  1. 20,000Why not: Doubles the paths, assuming standard error falls linearly with N.
  2. 40,000 - correct
  3. 14,142Why not: Multiplies by the square root of 2 instead of squaring the factor.
  4. 80,000Why not: Multiplies by 2³ instead of 2².

Standard error = sigma / sqrt(N). Halving it requires four times the paths: 4 x 10,000 = 40,000.

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