11 - Asset Pricing I - The Uniform Annuity Framework

Asset Pricing I: The Uniform Annuity Framework

• Presenter: S. Levkoff, PhD, CAP®

The Classical Theory of Asset Prices

• Any financial asset provides rights to future cash flows (CFS).

  • Cash Flow Stream (CFS): CFS = (C0, C1, C2, …, CT)

• Value future payments using Net Present Value (NPV) formulation.

  • NPV is the present worth of future cash flows discounted at a specific rate.

• Equilibrium market price equals NPV: P(t) = NPV.

  • Important connection linking asset pricing and present value.

Market Valuation Situations

• If P(t) < NPV:

  • Market undervalues the asset; an individual can buy at P(t) and earn arbitrage profit, pushing price closer to NPV.

• If P(t) > NPV:

  • Market overvalues the asset; an individual can sell at P(t) to earn profit, driving price down towards NPV.

Equilibrium in Asset Pricing

• Equilibrium occurs when no agent wants to change their behavior (P(t) = NPV).

  • No arbitrage condition ensures riskless profit cannot be made.

• Assumptions of the classical theory:

  1. Markets are competitive.

  2. All agents have perfect information about asset prices.

Agenda of the Presentation

• Price the uniform annuity flow.

  1. Apply classical theory of asset pricing.

  2. Use geometric series for simplification.

• Applications include:

  1. Amortizing mortgage payments.

  2. Pricing an infinite uniform annuity flow (perpetuity).

Valuation of Cash Flow Streams

• General cash flow stream is given as CFS = (C0, C1, C2, …, CT).

• Valuation via discounting payments to present values:

  • NPV Formula: NPV(i, T) = C0 + C1/(1+r) + C2/(1+r)^2 + … + CT/(1+r)^T

  • Aggregate to find present value: NPV(i, T) = ∑Ct/(1+r)^t.

Discount Factors and NPV

• Single period discount factor represented as 1/(1+r).

• Using this factor:

  • NPV(i, T) = ∑(C0 * (1+i)^t).

• Connect market price P to NPV: P = NPV.

Changes in Cash Flow Application

• For pricing, no payment is collected in period zero; only the asset price is paid as cash outflow.

• Benefits stream: CFS = (C1, C2, …, CT).

• Focus on uniform annuity where each cash flow is the same; CFS = (P, P, …, P).

  • Simplified to uniform annuity flow accounting for T payments starting from period 1.

Pricing the Uniform Annuity

• Pricing using classical theory results in:

  • P(i, T) = C1/(1+r) + C2/(1+r)^2 + … + CT/(1+r)^T.

• Impose uniformity condition:

  • P(i, T) = P + P + … + P.

• Factoring out P:

  • P(i, T) = P(1 + 1 + 1 + … + 1).

Computational Considerations

• Summing can be intensive with many terms; use geometric series for simplification.

  • Geometric series result aids in reducing computational burden:

    • P(i, T) = P * [(1 - (1+r)^-T) / r].

Pricing Formula in Terms of Interest Rate

• The condensed uniform annuity pricing formula:

  • P(i, T) = P(1 - (1+r)^-T)/r.

• Where r is the interest rate; can also express in terms of discount factors.

Example: Pricing a Uniform CFS

• Example of an annuity with payments of $20,000 per year over 30 years at an interest rate of 5%:

  • Calculation yields total present value of approximately $307,449.

Amortization and Time Value

• Amortization involves converting a lump-sum value (LSV) into future CFS.

  • Balances profits over time to allow payments.

• Amortization finds uniform annuity size for given LSV and interest rate.

Example: Mortgage Payment Calculation

• Mortgage amount of $400,000 over 30 years at 4% APR requires amortization:

  • Monthly payment can be calculated using the inverted formula.

  • Monthly interest rate derived from APR and total number of payments leads to:

    • Monthly payment calculated to be approximately $1,909.66.

The Perpetuity: Infinite Annuity

• Consider an infinite payment stream of equal cash flows forever.

• Price is finite, defined as:

  • Price of perpetuity P = Cash Flow / Discount Rate.

• We can derive this from the finite uniform annuity pricing formula by taking limits.

Example: Comparing Perpetuity and Uniform Annuity

• A previous example regarding a finite uniform annuity priced $307,449.

• Valuing perpetuity at a 5% interest rate gives $400,000 using perpetuity formula.

Concluding Remarks

• The uniform annuity pricing formula developed from classical asset pricing theory.

• Pricing functions analyzed for finite and infinite uniform cash flow streams.

• Revealed numerical examples demonstrating application of the concepts.

Next Steps:

• Topics to be explored next include:

  • Pricing Bonds and Structure.

  • Pricing Common Stock and its determinants.