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Future value of Multiple Cash Flows(differing amounts)

You deposit $1000 now, $1500 in one year, $2000 in two years and $2500 in three years in an account, paying 10% interest per year. How much do you have in the account at the end of the third year? You can solve by either:

- compounding the accumulated balance forward one year at a time; or - calculate the future value of each cash flow first and then total them. Solution 1

End of year 1: ($1 000 × 1.10) + $1 500 =$2 600 End of year 2: ($2 600 × 1.10) + $2 000 =$4 860 End of year 3: ($4 860 × 1.10) + $2 500 =$7 846 Solution 2

$1000 × (1.10) 3 = $1331 $1500 × (1.10) 2 = $1815 $2000 × (1.10) 1 = $2200 $2500 × (1.10) 0 = $2500

______________________________ Total = $7846

Present value of Multiple Cash Flows(differing amounts)

The cash flows are $1500 in one year, $2000 in two years and $2500 in three years. The interest rate is 10% per year. What is the present value of these cash flows? $2500 × (1.10) -3 = $1878 $2000 × (1.10) -2 = $1653 $1500 × (1.10) -1 = $1364

______________________________ Total = $4895

Suppose you are 20 years old and just start your career. You plan to retire at your 50 years old birthday. If you prepare to receive a $10000 annually annuity in the first 20 years after your retirement, you should deposit from now. Suppose you would receive the annuity at the beginning of each year. The interest rate is 10%, and the deposit will start at your 21st birthday and will end when you are 50. How much should you deposit each year?

Example

Suppose we can invest $50 today and receive $60 in one year. Should we accept the project given a 10% expected return?

60NPV=-50+=$4.55

1.10Example

Suppose we can invest $50 today and receive $60 in one year. Should we accept the project given a 10% expected return?

profit60-50?Return ==0.20 or 20%

investment50

Internal Rate of Return Example

You can make an investment for $4,000. The investment will generate $2,000 and $4,000 in cash flows for two years, respectively. What is the IRR on this investment?

20004000??0NPV=-4000+(1?IRR)1(1?IRR)2

IRR?28.08%

Profitability Index Example

We only have $300,000 to invest. Which do we select? Project NPV Investment PI A 230,000 200,000 1.15 B 141,250 125,000 1.13 C 194,250 175,000 1.11 D 162,000 150,000 1.08 Select projects with highest Weighted Average PI WAPI (BD) = 1.01 WAPI (A) = 0.77 WAPI (BC) = 1.12

CP8-10

? At the beginning of the year, the stock was selling for $37 per share. If you had bought 100 shares. Suppose, over the year, the stock paid a dividend of $1.85 per share. Also, the value of the stock has risen to $40.33 per share by the end of the year. By the end of the year, you would have received: ? Dividend = $1.85 × 100 = $185

? Capital gain = ($40.33 – $37) × 100 = $333 On the other hand, if the price had dropped to$34.78 Capital loss = ($34.78 - 37) × 100 = -$222

Total dollar return = Dividend income + Capital gain (or loss)

Review of the Capital Asset Pricing Model

1. Investors like high expected return and low standard deviation. Common stock portfolios that offer the highest expected return for a given standard deviation are known as efficient portfolios.

2. If the investor can lend or borrow at the risk-free rate of interest, one efficient portfolio is better than all the others: the portfolio that offers the highest ratio of risk premium to standard deviation. A risk-averse investor will put part of his money in this efficient portfolio and part in the risk-free asset. A risk-tolerant investor may put all her money in this portfolio or she may borrow and put in even more.

3. The composition of this best efficient portfolio depends on the investor’s assessments of expected returns, standard deviations, and correlations. But suppose everybody has the same information and the same assessments. If there is no superior information, each investor should hold the same portfolio as everybody else; in other words, everyone should hold the market portfolio

4. Don’t look at the risk of a stock in isolation but at its contribution to portfolio risk. This contribution depends on the stock’s sensitivity to changes in the value of the portfolio.

5. A stock’s sensitivity to changes in the value of the market portfolio is known as beta. Beta, therefore, measures the marginal contribution of a stock to the risk of the market portfolio.

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