Summary of "Ses 2: Present Value Relations I"

This lecture, part of an MIT OpenCourseWare finance series, introduces foundational concepts in financial valuation, focusing on the present value of assets defined as sequences of cash flows. The instructor uses real-world examples, current events, and theoretical frameworks to motivate and build a systematic approach to valuing financial assets.

Main Ideas and Concepts

1. Motivation: Real-World Financial Context

The lecture begins by referencing a classroom auction exercise illustrating how information asymmetry affects asset valuation.
It then connects to a major contemporary event: the U.S. government takeover of Fannie Mae and Freddie Mac during a financial crisis.
- These government-sponsored enterprises (GSEs) support mortgage markets by buying loans from banks, enabling credit availability.
- Their near-collapse created systemic risk, requiring government backing to prevent market panic.
- This event motivates the need for tools to understand valuation and risk in financial markets.

2. Definition of an Asset

An asset is defined abstractly as a sequence of current and future cash flows at a given point in time \( T \):
- This excludes past cash flows.
- The asset's value depends on the timing and amount of these cash flows.
- Cash flows can be positive or negative (liabilities are just assets with negative cash flows).
Examples include physical assets (property, equipment), financial instruments (stocks, bonds), intangible assets (patents, trade secrets, reputation).
The instructor emphasizes the importance of thinking about assets in terms of their cash flows rather than their physical or legal form.

3. Present Value and the Value Operator \( V_t \)

The value operator \( V_t \) is introduced as a function that takes a sequence of cash flows and returns a single number: the value of the asset at time \( t \).
One practical example of \( V_t \) is the market price of the asset at time \( t \).
The lecture stresses the importance of timing and the order of cash flows — cash flows at different points in time are like different currencies and cannot be simply added.

4. Time Value of Money and Exchange Rates Across Time

Cash flows at different times are analogous to cash flows in different currencies.
To add or compare cash flows occurring at different times, they must be converted to a common "currency" — usually today's dollars (time zero).
This conversion uses exchange rates across time, also called discount factors, which reflect the value today of one unit of currency received at a future date.
These discount factors come from the market, e.g., prices of securities paying known amounts in the future.

5. Net Present Value (NPV)

The Net Present Value operator (NPV) is defined as the sum of all cash flows converted into today's dollars using discount factors.
The NPV accounts for any initial investment (usually a negative cash flow at time zero).
Decision rule: if NPV > 0, the investment/project is financially worthwhile.

6. Source of Discount Rates / Exchange Rates

discount factors or exchange rates are derived from market prices of securities.
The market provides the "wisdom of the crowds" to determine these rates.
discount factors are typically less than one because of impatience (preference for consumption today) and inflation.
The reciprocal of discount factors gives the interest rate or opportunity cost of capital \( r \).

7. Mathematical Formulation of Present Value

Discount factor for a cash flow at time \( t \) is \( \frac{1}{(1+r)^t} \).
Present value of a sequence of cash flows \( \{CF_0, CF_1, ..., CF_T\} \) is: \( V_0 = CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \cdots + \frac{CF_T}{(1+r)^T} \)
\( r \) is the discount rate, also called interest rate, cost of capital, user cost, etc.
This formula applies under certainty (no uncertainty about cash flows).

8. Assumptions and Limitations

The current lecture assumes:
- Cash flows are known with certainty.
- Exchange rates/discount factors come from frictionless markets.
- No transaction costs or currency conversion fees.
These assumptions will be relaxed in later lectures to incorporate uncertainty, market imperfections, and more complex financial instruments.

9. Practical Examples

Calculation of NPV for a project with an initial investment and future cash inflows discounted back to today.
How to price the project if payment is deferred to a future date.
Emphasis on always drawing timelines to avoid