# MidtermTwoAnswers_Fall2009 - ACTUARIAL SCIENCE 445/845...

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ACTUARIAL SCIENCE 445/845: Asset Liability Management Midterm Two - Fall 2009 Aids: Calculator, Answer all questions in the space provided. Show your work. Name: ____________________ Marks: _____________________/35 ID#: _____________________ Course (445 or 845):___________ Concept Questions 1. [5 marks] Briefly explain what the option adjusted spread of an interest rate derivative security is, and describe how it is used to calculate effective duration. Answer: (From the lecture notes: Unit 8 (Part I) Discrete Time Interest Rate Models; see pages 17‐19 of these notes for more details). The option‐adjusted spread (OAS) for a security is the fixed spread (usually measured in basis points) over the benchmark rates that equates the theoretical price and the market price. The effective duration is: D m e = V V + 2 V 0 Δ y where V 0 is the current market price, V + /V are the price after a (small) upward/downward shift in the spot‐rate curve. In practice, the prices used to compute these quantities should take the OAS into account. This can be done as follows: • Step 1: Given the security’s market price, find the OAS. • Step 2: Shift the spot‐rate curve by a small quantity _y. • Step 3: Compute a binomial interest‐rate lattice based on the shifted curve obtained in Step 2. • Step 4: Shift the binomial interest‐rate lattice obtained in Step 2 by the OAS. • Step 5: Compute V + based on the lattice obtained in Step 4. Similar steps are done to compute V .

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2. a) [1 mark] Give the stochastic differential equation for the Vasicek interest rate model.
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