# HW #5: Financial Econ Basics

## 1) What is a return?

A **stock return** is the percent change (new - old / old) in wealth from holding the stock over a period of time. It usually comes from two places: the stock price changing and (sometimes) a dividend paid to shareholders. Returns are the basic unit we use to measure performance in financial economics.

Suppose that you buy a share for \$50 and sell it one month later for \$55. There is no dividend.  
What is your **simple monthly return** as a percentage?  

Answer: ____

To check your answer: if your answer is X, is $50 \times (1 + X) = 55$?

## 2) Including dividends in returns

If a company pays a dividend, that's cash you receive as long as you still hold the stock. Shareholder return includes both the price change and dividends. This is why "price return" (ignoring dividends) can understate the total return.

You buy a share for \$100. One month later the price is \$102, plus you received a \$1 dividend.  
What is your **monthly return** as a percentage?

Answer: ____

## 3) Simple vs log returns

Simple returns are what most people think of as "percent change." Log returns are based on logarithms and are especially convenient when you add up returns over time. For small returns (like a few percent), simple and log returns are very close. The formula for log returns is $\ln(P_t / P_{t-1})$, or equivalently, $\ln(P_t) - \ln( P_{t-1})$

A stock goes from \$200 to \$204 in one month (no dividend).  
1) Compute the **simple return**.  
2) Compute the **log return**. You can use the R function `log`.

Answer: ____

## 4) Compounding across time

A central idea in finance is that returns compound. If you earn a return this month, your wealth base changes, which affects next month’s dollar gains. This means you **multiply** growth factors across time, rather than adding percent changes.

A stock earns **+10%** in Month 1 and **+20%** in Month 2. Starting from \$100, what is the ending value after two months? What is the **total two-month return**?

Answer: ____

Here's what I mean by "multiplying growth factors":
$$100 \times 1.10 \times 1.20 = 100 \times 1.32 = 132$$

## 5) Average return (arithmetic mean)

When we summarize returns, the simplest measure is the arithmetic mean (just the average). This is often used to estimate expected monthly return from data. But it’s not always the best measure of multi-period growth.

Suppose monthly returns are: $5\%, -3\%, 2\%$.  
Compute the **average monthly return** (arithmetic mean).

Answer: ____

## 6) Geometric mean (growth rate)

The **geometric mean return** tells you the constant return that would produce the same total growth over time. This is tied to compounding and long-run wealth. It’s often lower than the arithmetic mean when returns are volatile.

A stock returns **+50%** in Month 1 and **-50%** in Month 2.  
a) Starting from \$100, what is your ending value ($100 \times 1.5 \times 0.5$)?  
b) What is the **two-month total return**?  
c) What is the **arithmetic average** monthly return?

Answers: ____

## 7) Risk-free rate and excess returns

In finance we often compare risky returns to a "safe" alternative. The risk-free rate is the return on an investment that is treated as essentially default-free over that horizon. **Excess return** is the risky return minus the risk-free rate.

A stock earns **1.2%** this month. The risk-free rate is **0.3%** this month. What is the stock’s **excess return**?

Answer: ____

## 8) Price vs value (present value intuition)

A stock’s **price** is what it trades for today. Its **value** is what you think it’s worth based on future cash flows, discounted back to the present. Discounting reflects the idea that future money is worth less than money today.

You will receive \$4200 one year from now. The interest rate is 5%. What is the **present value** today?

Answer: ____

## 9) Law of one price (no-arbitrage logic)

The law of one price says that two assets with the same payoffs should sell for the same price. If they don’t, someone can buy the cheap one and sell the expensive one for riskless profit (an arbitrage). In efficient markets, strong arbitrage opportunities should not last.

Asset A and Asset B will both pay exactly \$100 for sure in one year.  
The interest rate is 5%.  
a) What should their price be today?  
b) If Asset A sells for \$92 and Asset B sells for \$95, which one is underpriced?

Answers: ____

## 10) A simple stock valuation (discounted dividends)

One way to think about owning a stock is that it gives you **cash flows in the future**. These cash flows can come from:
1) **Dividends** you receive while holding the stock, and  
2) The **price you can sell the stock for later**.

To decide what the stock is worth today, we take the future cash you expect to receive and "discount" it back to the present. The discount rate is like an interest rate: it reflects the idea that \$1 in the future is worth less than \$1 today, because money today can earn a return.

Suppose you buy a stock today and hold it for one year, and you expect to receive:
- A **\$2 dividend** in one year, and
- **\$50 from selling the stock** in one year.

So your total cash flow in 1 year will be \$52. Discount that back to the present at a rate of 10%. What should the stock's price today be?

Answer: ____

## 11) Indexes and portfolio returns

A **market index** is a portfolio of stocks, usually weighted by market value. A portfolio return is just the weighted average of the returns of its holdings. Equal-weight portfolios weight all stocks the same, while value-weight portfolios put more weight on larger companies.

You hold a portfolio with:

- Stock A weight 60%, return 4%  
- Stock B weight 40%, return -1%

What is the portfolio return?

Answer: ____

## 12) Why "abnormal returns" require a benchmark

A raw return isn't always informative because the whole market might be up or down. In financial economics, we often compare performance to a benchmark like the market index. One simple version of "abnormal return" is: stock return minus market return.

A stock return this month is **3%**. The market index return is **1%**. What is the stock's **market-adjusted abnormal return**?

Answer: ____
