Expected Return for Asset Pricing

Business Context

At NorthBridge Capital, an interviewer asks you to explain basic asset pricing using simple probability and statistical reasoning rather than finance jargon. You are given a one-period risky asset and asked to quantify its value and test whether its average return appears meaningfully different from a benchmark.

Problem Statement

A stock will be worth different amounts in one year depending on the state of the economy. Use the state probabilities to compute the asset's expected payoff, expected return, and risk. Then use a small sample of historical annual returns to test whether the stock's mean return differs from a 6% benchmark return.

Given Data

State-Based Pricing Inputs

Historical Annual Returns Sample

Requirements

1. Compute the expected one-year payoff of the asset.
2. Compute the expected return based on the current price.
3. Compute the variance and standard deviation of the state-based return distribution.
4. Test whether the historical mean annual return differs from 6% using a two-sided one-sample t-test at $\alpha = 0.05$.
5. State whether the asset appears attractive relative to the 3% risk-free rate, while distinguishing expected return from realized sample evidence.

Assumptions

• State probabilities are correct and sum to 1.
• Historical annual returns are independent observations.
• For the t-test, annual returns are approximately normally distributed or the test is used as a reasonable small-sample approximation.