Foundations

This part builds the valuation tools that the rest of the course reuses. The first three chapters set up the inputs: a timeline of signed, dated cash flows, events and expectations over a finite partition, and rate quotes with their compounding convention and units. The next two chapters turn a rate into a discount factor and add discounted cash flows into a present value. The last two chapters value one payoff from another. A replicating portfolio fixes a price by no-arbitrage, and risk-neutral pricing states the same price as a discounted expectation under pricing weights, which are not forecasts.

Part 1 of 77 chapters7 numbered equations7 knowledge checks

1. Cash-flow timelines and perspective

   Place signed payments on one time axis before valuing them.

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2. Events, conditional probability, and expectation

   Calculate expected values from disjoint events, including values that vary inside an event.

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3. Rate quotes, compounding, and basis points

   Separate an annualized quote from its periodic rate and its units.

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4. Discount factors

   Convert one deterministic future unit into its value at time zero.

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5. Present value of a cash-flow schedule

   Discount each signed dated amount and then add the comparable values.

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6. No-arbitrage and replication

   Match every dated state-contingent cash flow before using one portfolio to value another.

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7. Risk-neutral pricing is not a risk-free probability

   Separate real-world forecasts from no-arbitrage pricing weights, then discount a finite-state expected payoff.

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The argument

1. A valuation starts from signed cash flows placed on one time axis
2. An expectation over disjoint events weights each value by its probability
3. A rate quote needs its compounding convention to give a periodic rate
4. A discount factor converts one future unit into its value at time zero
5. Present value discounts each dated amount before adding
6. A portfolio that matches every future cash flow has the same price
7. Pricing weights state that price as a discounted expectation

After this part you can

• Read the amount and timing of each cash flow from a stated valuation-time timeline.
• Assign positive and negative signs to cash flows from an explicitly stated perspective.
• Identify events and a disjoint exhaustive partition in a stated probability model.
• Calculate the probability of one event conditional on another positive-probability event.
• Calculate a finite expected value from event probabilities and either event values or conditional event means.
• Interpret a nominal annual rate together with its stated compounding frequency.
• Calculate the per-period rate j^(m)/m from a nominal annual rate j^(m) compounded m times per year.
• Convert rate changes among decimal, percentage-point, and basis-point units.
• Explain a discount factor as today's value of one unit paid at a future date.
• Calculate D(0,T) = (1+j^(m)/m)^(-mT) from a nominal annual rate j^(m) compounded m times per year.
• Explain present value as the valuation-time sum of discounted cash flows under stated assumptions.
• Discount and add dated deterministic cash flows to calculate present value.
• Explain why two portfolios with identical dated state-contingent cash flows must have the same value under the lesson's no-arbitrage assumptions.
• Calculate a claim's no-arbitrage value from the current values and signed quantities of a portfolio that exactly replicates its future cash flows.
• Interpret risk-neutral probabilities as no-arbitrage pricing weights rather than real-world forecasts.
• Calculate time-zero value from finite-state risk-neutral probabilities, signed terminal payoffs, and a deterministic discount factor.

Leads to

• Bonds
• Rates and curves
• Derivative foundations
• Credit risk
• CDS