FIN 320F Duvic WB Test 3 Spring 2022

What is the expected return of a stock with: Recession- 30% probability & -11% rate of return Boom- 70% probability & 21% rate of return

Expected Return = (Prob of Return1)(Return1) + (Prob of Return2)(Return2) + ... Expected Return = (.30)(-0.11) + (.70)(-0.21) = -0.033 + 0.147 = 0.114 = 11.4%

expected return formula

Expected Return = (Prob of Return1)(Return1) + (Prob of Return2)(Return2) +...

covariance equation

CovFF,NDS=∑Pi(RFF,RNDS){(RFF,i−E(RFF))(RNDS,i−E(RNDS))}

$10000 inheritance. Wants to make a down payment on a home she hopes to purchase in 7 years. Her account pays 5% per year & down payment must be 10% of house value. How much house can she afford?

$14071 use financial calculator

Eulis Co. has identified an investment project with the following cash flows. If the discount rate is 10 percent, what is the present value of these cash flows?

$2,547.97 = $680 / 1.10 + $490 / 1.10^2 + $975 / 1.10^3 + $1,160 / 1.10^4

Booker, Inc., has identified an investment project with the following cash flows. If the discount rate is 8 percent, what is the future value of these cash flows in Year 4.

$5,519.84 = $985(1.08)^3 + $1,160(1.08)^2 + $1,325(1.08) + $1,495

level perpetuity equation

(per - period cash flow) / the interest rate = C/r

The comparison of 2 expected returns tells us whether the investment is:

- Fairly priced: We are being fairly compensated for the risk of the project. o RRR = IRR - Overpriced: The investment is not a good deal. It is not providing an acceptable risk-return relationship. o RRR > IRR - Underpriced: The investment is a good deal. It provides a superior risk-return relationship. o RRR < IRR

2 Cash Flow streams:

1) Annuity 2) Perpetuity

2 types of probability distributions

1) Discrete Probability Distribution 2) Continuous Probability Distribution

2 returns investors face

1) Expected return 2) Realized return

4 variables for developing a house purchase plan:

1) The present amount that she will deposit. 2) The future amount that she expects to get 3) The Interest rate, which gives the rate of change in the value of her deposit. 4) The maturity, or time period, which determines the number of periods that interest rates are applied.

2 summary measures for distributions of stock returns:

1) expected return 2) variance

2 types of perpetuities:

1) level perpetuity 2) growing perpetuity

2 ways to apply interest rates to cash flows:

1) simple interest 2) compound interest

Portfolio variance has 2 types of risk:

1) unique risk 2) market risk

The Beta of the market = ?

1, - because the market return is perfectly correlated with itself.

4 steps of CAPM

1. Determine the IRR of the asset. 2. Set up these Security Market Line, which gives the current market's risk/return relationship. 3. Calculate the opportunity cost: the rate of return that assets of equivalent risk are earning. 4. Decide

An investment decision involves the comparison of 2 expected returns:

1. Required Rate Of Return (RRR) 2. Internal Rate Of Return (IRR)

3 interest rates involved w the existence of compounding periods of less than one year makes time value calculations more complex

1. Stated annual interest rate 2. Periodic interest rate 3. Effective annual interest rate (EAR)

3 elements of portfolio variance

1. The variability that FF contributes to the portfolio, which consists of the variance of FF and the proportion of the portfolio invested in FF, xFF. As the variance is a squared term, we must also square the weights. 2. The variability that NDS contributes to the portfolio, which consists of the variance of NDS and the proportion of the portfolio invested in NDS, xNDF. As the variance is a squared term, we must also square the weights. 3. The covariance of FF and NDS returns. As the covariance is not a squared term, we do not have to square the weights.

Elements of probability distribution

1. economic activity 2. likelihoods of growth 3. analysis of stock price

2. likelihoods of growth

12% chance of a "boom", 15% "high growth", 25% normal, 30% low growth, 18% recession

statistics

A branch of mathematics dealing with the collection, analysis, interpretation and presentation of masses of numerical data.

Capital Asset Pricing Model (CAPM)

A financial model that uses market/systematic risk as measured by beta to calculate the risk premium in the opportunity cost/expected rate of return. helps us evaluate if a price is good in comparison w the future cash flows & their risk Beta used here

hedge fund

A private investment organization that employs risky strategies that often made huge profits for investors

What happens to the future value of an annuity if you increase the rate, r? What happens to the present value?

Annuities are a series of regular cash flows. Each of the cash flows in the annuity acts in the same way as individual cash flows. Just as increasing the interest rate increases the future value of a single cash flow and reduces the present value of an individual cash flow, assuming positive cash flows and a positive interest rate, the future value of the annuity will rise and the present value of an annuity will fall.

A portfolio has 165 shares of Stock A that sell for $69 per share and 125 shares of Stock B that sell for $44 per share? What is the weight/percentage invested in Stock A?

As the portfolio is a combination asset, its value is determined by two elements: 1. The value of the assets in the portfolio 2. The relative investment in these assets. Value of stock investment. Given the price of the stock and the number of shares held, we can calculate the value of the investment in each stock. Stock A: Price per share x number of shares = $69 x 165 = $11,385 Stock B: Price per share x number of shares = $44 x 125 = $5,500 Portfolio value. The portfolio value is the sum of the value of the stocks in the portfolio. Total value = $11,385 + $5,500 = $16,885 Portfolio weights. We now have the value of the investment in each stock and the total value of the portfolio. The portfolio weight for each stock is determined by dividing the asset values by the total portfolio value: Proportion invested in Stock A: XA = $11,385 / $16,885 = .6743 Proportion invested in Stock B: XB = $5,500 / $16,885 = .3257 It's always a good idea to check and make sure that your weights add up to 100% of the portfolio Proportion invested in Stock A + Proportion invested in Stock B = 100% of the portfolio 67.43% + 32.57% = 100%

As you increase the length of time involved, what happens to the present value of an annuity? What happens to the future value?

Assuming positive cash flows and a positive interest rate, both the present and the future value will rise. This makes perfect sense when we realize that annuities are a stream of regular payments at regular intervals. When you add an additional period to an annuity, you're also adding an additional cash flow.

Beta equation

B = Covi,m / Varm

Suppose that when TMCC offered the security for $24,099, the U.S. Treasury had offered an essentially identical security. Do you think it would have had a higher or lower price? Why?

Building on our discussion in the previous problem, probably not! Both alternatives offer a future payment of $100,000 in 30 years. The Treasury security should be evaluated at the risk-free rate of return; the risky security should be evaluated at its appropriate risk-adjusted opportunity cost. Our risky security offers a 4.68% risk adjusted return at a price of $24,099. The treasury security should offer a lower rate of return, as there is no risk premium. To get this rate on an investment that will pay $100,000 with certainty you would accept a lower rate of return and would pay a higher price. If the risk-free rate is 1%, then the present value of the $100,000 would be: $100,000/(1.01)30 = $74,192. You would be willing to accept this much higher price because you would be getting a fair rate of return of 1%, which is appropriate for this risk-less investment. Note that the difference in these rates is magnified by the long time period of the investment! One of the very basic but very important relationships is the inverse relationship between price and rate of return. For a given future cash flow: If price rises the rate of return drops If the price drops the rate of return rises.

Assume the total cost of a college education will be $295,000 when your child enters college in 18 years. You presently have $53,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education?

C0 = C1 / (1+r)^n 53000 = 295000 / (1+x)^18 53000 ((1+x)^18) = 295000 (1+x)^18 = 5.566037736 1 + x = 1.100067237 x = 0.100067237 = 10%

What is annual cash flow? PV = 24500 Years = 6 Interest Rate = 11%

Calculator: PMT = $5791.23

What is compounding? What is discounting?

Compounding is the exponential increase in the value of an investment because interest earned is added tothe principal, which produces an increased interest payment in the subsequent period. It is also the process of determining the future value of an investment. Discounting is the process of determining the value today of an amount to be received in the future. As many financial decisions are made today, many problems involve taking present values and the rate used in time value calculations is often referred to as the "discount rate" whether or not you're taking present values or future values.

correlation coefficient equation

CorFF,NDS=CovFF,NDS / (SDFF)(SDNDS)

What is the EAR for quarterly compounding if APR = 10%

EAR = (1+ r/m)^m - 1 EAR = (1+ .10/4)^4 - 1 = 0.1038128906 = 10.38%

riskless investment = has/doesn't have variability in returns

DOES NOT HAVE the expected return will be the realized return.

In broad terms, why is some risk diversifiable? Why are some risks nondiversifiable? Does it follow that an investor can control the level of unsystematic risk in a portfolio, but not the level of systematic risk?

Diversifiable risk: Things happen to companies: a great new product is introduced, a new CEO with an effective strategy is appointed, a restaurant chain suffers a series of well-publicized occurrences of food poisoning, etc. These events are unsystematic, in that they occur in a random pattern unconnected to the economy. Good events and bad events have a major impact on the companies involved; however, for an investor holding a large portfolio these events tend to cancel each other out. The increased return of the company with the new product is balanced by the decreased return from the restaurant chain. With larger and larger portfolios these nonsystematic risks are reduced by diversification. By investing in a variety of assets, this unsystematic portion of the total risk can be eliminated at little cost. Nondiversifiable risk: Some events are systematic, in that they affect the entire economy. A rise in the price of oil, an increase in interest rates by the Federal Reserve, a major increase in tariffs as part of a "trade war" between countries will have a general impact on economic activity and most companies. For example, an increase in interest rates will increase the opportunity cost for most companies and reduce the desirability of their projects (their NPVs) As this impacts many companies, even a well-diversified portfolio will suffer a decline in its expected return. Investors can control the level of unsystematic risk in their portfolios by holding larger portfolios, which will reduce total volatility at low cost. They cannot diversify away systematic risk and will thus require a risk premium appropriate for the amount of systematic—nondiversifiable risk—in their portfolios

Calculate the opportunity cost: the rate of return that assets of equivalent risk are earning.

E(R) = Rf + B (E(Rm)-Rf))

Opportunity Cost equation (Expected return equation CAPM)

E(R) = Rf + B (E(Rm)-Rf))

A stock has an expected return of 10.7 percent and a beta of .91, and the expected return on the market is 11.5 percent. What must the risk-free rate be?

E(R) = Rf + B (E(Rm)-Rf)) .107 = Rf + .91(.115 - Rf) .107 = Rf + 0.10465 - .91Rf 0.00235 = .08Rf 0.029375 = Rf = 2.94%

A stock has an expected return of 10.9 percent, its beta is .85, and the risk-free rate is 2.8 percent. What must the expected return on the market be?

E(R) = Rf + B (E(Rm)-Rf)) .109 = .028 + .85(E(Rm) -.028) .081 = .85(E(Rm) -.028) 0.09529411765 = E(Rm) -.028 0.1232941177 = E(Rm) = 12.33%

portfolio expected return equation

E(Rp) = Xa * E(Ra) + Xb * E(Rb)

You own a portfolio that is 15 percent invested in Stock X, 40 percent in Stock Y, and 45 percent in Stock Z. The expected returns on these three stocks are 10 percent, 13 percent, and 15 percent, respectively. What is the expected return on the portfolio?

E(Rp) = Xa * E(Ra) + Xb * E(Rb) E(Rp) = (.15 * .10) + (.4 * .13) + (.45 * .15) = 0.1345 = 13.45%

EAR equation

EAR = (1 + r/m)^m - 1

What is the APR for semiannual compounding if EAR = 14%

EAR = (1+ r/m)^m - 1 .14 = (1+ r/2)^2 -1 1.14 = (1+ r/2)^2 1.067707825 = 1+ r/2 0.067707825 = r/2 0.13541565 = r = APR = 13.54%

What is the future value if: PV = 3150 Years = 7 Interest rate = 13%

FV: C0 (1 + r)^n = C1 3150 (1+.13)^7 = $7410.71

Future Value calculation

FV: C1 = C0 (1+r)^n

First City Bank pays 6 percent simple interest on its savings account balances, whereas Second City Bank pays 6 percent interest compounded annually. If you made a deposit of $8,100 in each bank, how much more money would you earn from your Second City Bank account at the end of 10 years?

First: PV*(1 + rn) = FV 8100(1+0.06(10) = $12960 Second: PV*(1 + r)^n = FV 8100(1+0.06)^10 = $14505.87 $14505.87 - $12960 = $1544.87

What happens to a future value if you increase the rate, r? What happens to a present value?

Future values are positively related to interest rates, as the higher the interest rate, the higher the amount of interest earned in each period. Present values are inversely related to interest rates, as calculating present values involves dividing (1 + r)T. Remember: future values increase with longer time periods and higher interest rates; present values decrease with longer time periods and higher interest rates.

As you increase the length of time involved, what happens to future values? What happens to present values?

Future values: Future values are positively related to the length of time of the investment, as each additional period additional interest is earned. This can be seen in the Future Value Factor = (1 + r)T, which is multiplied by the present value to get the future value. Present values: Present values are inversely related to the length of time of the investment. The Present Value Factor = 1/(1 + r)T is the inverse of the Future Value Factor. This shows that discounting is the inverse of compounding.

high risk investment = high/low variability in returns

HIGH iThe realized return may be far below or above the expected return.

sales increase/decrease when the economy is doing well & increase/decrease when it suffers

INCREASE; DECREASE

The Diversifiable risk is eliminated as the number of assets in the portfolio increases/decreases

INCREASES

Determine the IRR of the asset.

IRR = Ending Value - Beg Value / Beg Value

1. Simple interest

Interest earned on the original principal. Each period the interest rate is applied, but the principal remains the same.

Would you be willing to pay $24,099 today in exchange for $100,000 in 30 years? What would you consider in answering yes or no? Would your answer depend on who is making the promise to repay?

In the previous question TMCC received $24,099. Now let's look at this exchange from the investor's point of view. If you are the investor, would you give TMCC the $24,099? Consideration: Your key considerations would be based on opportunity cost. Is the rate of return implicit in the offer attractive relative to other, similar risk investments? The rates offered on investments reflect the perceived risk of the investments: i.e., how certain are we that we will actually get the $100,000 in 30 years? Few investments are riskless, where you will get exactly what you're promised. As we saw in Unit 5, the opportunity cost consists of the risk-free rate plus an appropriate risk premium based on the likelihood that the future payment will show up: Opportunity cost = risk free rate + risk premium. Decision: How would you decide? Calculate what you'd earn: the IRR. Investing $24,099 today and receiving $100,000 in thirty years produces an IRR of 4.86%. If other alternatives offer you less than 4.86% then take this investment. If another alternative offers you more than this investment's 4.86%, then take the alternative. The higher the risk premium the larger the interest rate and the lower the present values, as you don't value a highly risky future payment as much as an investment of lower risk.Final thought: The rate of return earned by the creditor is often less than the rate earned by the company that invests the borrowed funds in its capital budgeting projects, as the risk to the creditor is less than the risk of the company.

Suppose you deposit a large sum in an account that earns a low interest rate and simultaneously deposit a small sum in an account with a high interest rate. Which account will have the larger future value?

It depends on the length of time involved. The large deposit will have a larger future value for some period, but after time, the smaller deposit with the larger interest rate will eventually become larger due to the effect of compound interest. The length of time for the smaller deposit to overtake the larger deposit depends on the amount deposited in each account and the interest rates.

What does a covariance of 0.0282 tell Diane about her portfolio?

It tells her that the returns on her securities tend to move together. When FF's return is higher than its expected return, NDS's return is also higher than its expected return

The lower the interest rate, the _________ the annual (compounded) present value.

LARGER

The shorter the time period the ________ the present value

LARGER

The higher the interest rate, the _______the future value

LARGER because the rate you earn in each period will compound into larger future wealth.

The longer the time period, the ________ the future value

LARGER the longer you save the more periods you'll earn compound interest.

More frequent compounding period = larger/smaller FV

LARGER bc its a loan

A homogeneous portfolio of the same type of stocks (automobiles, for example) would likely be more/less diversified and have a higher/lower variance

LESS; HIGHER

low risk investment = high/low variability in returns

LOW you don't expect the realized return to be substantially different from the expected return

to increase wealth you must earn more/less than the opportunity cost

MORE

Suppose the government announces that, based on a just-completed survey, the growth rate in the economy is likely to be 2 percent in the coming year, as compared to 5 percent for the year just completed. Will security prices increase, decrease, or stay the same following this announcement? Does it make any difference whether or not the 2 percent figure was anticipated by the market? Explain.

Market prices reflect information. Investors will look for information that will impact stock prices. If new favorable information on a stock reaches the market investors will want to buy the stock and the price will increase. If investors receive unfavorable information on the stock they will sell the stock and the price will drop. Information impacting the market as a whole—systematic information—will affect all securities. In this case, if the market expected the economy's growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate different than 2 percent and the expectation was incorporated into security prices, then the government's announcement would most likely cause security prices in general to change; prices would typically drop if the anticipated growth rate had been more than 2 percent, and prices would typically rise if the anticipated growth rate had been less than 2 percent.

Use the opportunity cost provided by the CAPM to evaluate _________

NPVs As these projects are independent and have normal cash flow patterns, the NPV and IRR rules should give us the same accept/reject decisions. The proper calculation of the opportunity cost is crucial for all economic decision rules.

If a portfolio has a positive investment in every asset, can the expected return on the portfolio be greater than that on every asset in the portfolio? Can it be less than that on every asset in the portfolio? If you answer yes to one or both of these questions, give an example to support your answer.

No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return.

Does diversification eliminate risk?

No, only reduces risk

opportunity cost equation

Opportunity cost = Risk-free rate + Risk premium

Risk and expected return are positively/negatively correlated

POSITIVELY

growing perpetuity equation

PV = C1 / r-g

Curly's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $30,000 per year forever. If the required return on this investment is 5 percent, how much will you pay for the policy?

PV of Perpetuity: PV = C/r PV = 30000 / .05 = 600000

What is the present value if: FV = 17328 Years = 15 Interest rate = 7%

PV: C0 = C1 / (1+r)^n C0 = 17328 / (1+.07)^15 = $6280.46

Present Value calculation

PV: C0 = C1 / (1+r)^n

What is the # of years if: PV = 195 FV = 873 Interest rate = 9%

PV: C0 = C1 / (1+r)^n 195 = 873 / 1.09 ^n 195 (1.09 ^n) = 873 1.09 ^n = 4.476923077 n = log 4.476923077 / log 1.09 = 17.39354917 = abt 17 years

What is the interest rate if: PV = 715 FV = 1381 Years = 11

PV: C0 = C1 / (1+r)^n 715 = 1381 / (1 + r)^11 715 ((1 + r)^11) = 1381 (1 + r)^11 = 1.931468531 1 + r = 1.061670586 r = 0.061670586 = 6.17%

Boom: prob = 60%, Stock A rate of return = 15%, B = 2%, C = 34% Bust: prob = 40%, Stock A rate of return = 3%, B = 16%, C = -8% What is the expected return on an equally weighted portfolio of these three stocks? What is the variance of a portfolio invested 20 percent each in A and B and 60 percent in C?

Portfolio expected return with equal investment in the three assets: To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. In this case all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is: Boom: Rp = (.15 + .02 + .34) / 3 = .17, or 17% Bust: Rp = (.03 + .16 - .08) / 3 = .0367, or 3.67% The expected return of the portfolio is calculated by multiplying the returns in Boom and Bust states by the likelihood of those states occurring. E(Rp) = .60(.17) + .40(.0367) = .1167, or 11.67% Portfolio variance with unequal investment in the three assets: If the asset weights are different the portfolio return in Boom and Bust must be computed by multiplying the returns of each security by its proportional weight in the portfolio. To do this, multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: Rp = .20(.15) +.20(.02) + .60(.34) = .2380, or 23.80% (the weights are listed in problem) Bust: Rp = .20(.03) +.20(.16) + .60(-.08) = -.0100, or -1.00% The expected return of the portfolio is: E(Rp) = .60(.2380) + .40(-.0100) = .1388, or 13.88% With the expected return of the portfolio determined, the next step is to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then sum. The result is the variance. So, the variance of the portfolio is: Varp = .60(.2380 - .1388)2 + .40(-.0100 - .1388)2 = .01476

portfolio variance equation

Portfolio variance = (Xa)2VARa + (Xb)2VARb + 2(Xa)(Xb)COVa, b

You have two distributions available to you: a frequency distribution and a probability distribution. Please define each and identify which one you'd use to make your investment decision.

Risk involves two returns: 1. The expected return is the future return, generally uncertain, that one expects to get from an investment. This is the return that is used in making most business decisions. 2. The realized return is the return that is actually received at the end of the investment period. These distributions help determine the average/expected return and its relationship—variability—with possible realized returns. A frequency distribution presents a measure of how frequently historical returns have occurred over a given time period. From this distribution you can determine the average return earned over a given period and how variable realized returns were from this average. To generalize, using a frequency distribution assumes that the future expected return will be the same as the historic average return. A probability distribution presents an estimate of the future. Possible rates of return are identified along with the likelihood of the returns. From the probability distribution we can calculate the expected return and how the realized return might vary from the expected return. While not ignoring historical returns, making the best estimates with probability distributions is probably the best course of action.

The higher the interest rate, the _______the present value

SMALLER

The longer the time period, the ________ the present value

SMALLER

The lower the interest rate, the _________ the annual (compounded) future value.

SMALLER

The shorter the time period the ________ the future value

SMALLER as you will accumulate fewer compounded interest payments.

3. analysis of stock price

Stock analysts estimate company's expected stock price for each level of econ activity: § Using the current price & Future prices, estimate company's possible rates of return in each state with: Stock Return = (Stock Price - Current Price ) / Current Price

Why would TMCC be willing to accept such a small amount today ($24,099) in exchange for a promise to repay about four times that amount ($100,000) in the future.

TMCC borrows money because it hopes to earn a higher rate of return in its capital budgeting projects than the rate paid to their creditors. If the creditors lend $24,099 and receive $100,000 in thirty years they would earn an IRR of 4.86%. If TMCC takes the $24,099 and invests it wisely in projects that produce desirable products for its customers it would earn more than the 4.86% they pay to the creditors. If rate of return on TMCC's projects was 6%, the borrowed $24,099 would grow to an inflow $138,412 In thirty years, TMCC's investment would be worth $138,412. After paying off the debt, they would have $38,412 ($138,412 - $100,000) that they would not otherwise have. This ability to use borrowed funds to create wealth is one of the basic rationales for businesses borrowing funds.

If the price drops in IRR equation, the rate of return increases/decreases

increases

You own a portfolio that has $2,750 invested in Stock A and $3,900 invested in Stock B. If the expected returns on these stocks are 9 percent and 14 percent, respectively, what is the expected return on the portfolio?

The expected return of a portfolio is the sum of the expected returns of the assets comprising the portfolio weighted by the relative weight of each asset in the portfolio. Total value. The total value of the portfolio is: Total value = $2,750 + 3,900 = $6,650 Portfolio weights. We now have the value of the investment in each stock and the total value of the portfolio. The portfolio weight for each stock is determined by dividing the asset values by the total portfolio value: Proportion invested in Stock A: XA = $2,750 / $6,650 = 0.4135 Proportion invested in Stock B: XB = $3,900 / $6,650 = 0.5865 Our check: 41.35% + 58.65% = 100% Expected return. The expected return of this portfolio is: Portfolio expected return = Expected return on A x Proportion invested in A + Expected return on B x Proportion invested in B E(Rp) = 0.09 x 0.4135 + 0.14 x 0.5865 = .1193, or 11.93%

Security Market Line (SML)

The graph of the Capital Asset Pricing Model (CAPM)

compunding period

The length of time that passes before interest is recognized and added to the principle.

1. Required Rate Of Return (RRR)

The return we should receive on the investment, given its risk. This is the opportunity cost.

Why do we have two measures of volatility?

The variance measures how much the realized returns might vary from the expected return. Given that the expected return is an average of the possible returns, it's likely that if we just add up the possible returns they'd sum to near zero: the returns above the expected value and the returns below the expected value would cancel each other out—not very useful! To eliminate this canceling the differences are squared, eliminating negative signs and emphasizing larger differences between the realized and expected returns. Thus, the variance formula: (𝑅) = ∑[𝑝(𝑟𝑒𝑡𝑢𝑟𝑛)𝑥(𝑅 − 𝐸(𝑅) 2 ] While very useful, the variance is measured in squared percents: %2. We have a problem here, in that the expected value is measured in percents: %. 𝑅 = ∑(𝑝(𝑟𝑒𝑡𝑢𝑟𝑛)𝑥 𝑟𝑒𝑡𝑢rn Just as you can't add feet and square feet when measuring a room for a carpet, you can't directly combine expected return and variance. So, we take the square root of the variance to get the standard deviation. The SD, like the expected return, is measured in percents: %. We can thus add and subtract the SD from the expected value to give an indication of the amount of dispersion of realized returns around the expected return.

2. Internal Rate Of Return (IRR)

This is the actual return on the investment.

True or false: The most important characteristic in determining the expected return of a well-diversified portfolio is the variances of the individual assets in the portfolio. Explain.

This statement is false. Variance is a measure of total risk: the sum of the differences between the portfolio returns at different states relative to the portfolio's expected return. We first recall the equation of the portfolio variance—the total volatility (risk) of the portfolio. Portfolio variance = Var A x (Percent invested A)2 + Var of B x (Percent invested in B)2 + Covariance of A and B We then recall that this total risk is the combination of two types of risk. Variance = Unsystematic risk + Systematic risk Our minds then recall our discussion of diversification as assets are added to a portfolio to reduce the risk of the portfolio. Unsystematic risk can be diversified away, but systematic risk remains. Variance = Diversifiable risk + Nondiversifiable risk The variance and expected return on a well-diversified portfolio are functions of systematic risk only, as the diversifiable risk is reduced by diversification.

Which 2 variables determine relationship btwn present & future variables

interest rate & time period

The TMCC security is traded on an exchange. If you looked at the price today, do you think the price would exceed the $24,099 original price? Why? Holding all economic and risk factors constant, if you looked in 2028, do you think the price would be higher or lower than today's price? Why?

Today: We would say that the price in the market would be the same as the economic value calculated. $24,099. This because the U.S. capital markets are reasonably efficient. For now we'll just accept this statement, but will examine market efficiency in Unit 8. In ten years: We're making some major assumptions that the economy and the company will not change over the next ten years!!! We cannot be sure that interest rates will remain constant or that TMCC's financial position will not change—or for that matter if they will still be in existence! However, given these assumption, the price would be higher because as time passes the price of the security will tend to rise toward $100,000. This increase is just a reflection of the time value of money. As time passes, the time until receipt of the $100,000 grows shorter, and the present value rises.

An investment offers $5,430 per year for 15 years, with the first payment occurring one year from now. If the required return is 8 percent, what is the value of the investment?

Using calculator, and 0 for FV, PV= $-46477.97

A stock has an expected return of 11.4 percent, the risk-free rate is 3.7 percent, and the market risk premium is 7.1 percent. What must the beta of this stock be?

Using the market risk premium. In this problem we are given the market risk premium, not the market rate of return. Using the CAPM, we find: E(Ri) = Rf + [Market risk premium] × ßi .114 = .037 + .071ßi The beta is ßi = 1.085 Using the market rate of return. In this problem we are not given the market rate of return, but we can calculate it using the above relationship. Market risk premium = [E(RM) - Rf] 0.071 = [E(RM) - 0.037] E(RM) = 0.071 + 0.037 = 0.108 Why two methods? Our course primarily uses the market rate of return in the CAPM. In practice, analysts often use the market risk premium, which reduces the impact of volatility in the risk-free rate of return and allows a more direct focus on market risk. .071 is the market risk PREMIUM, not the market rate of RETURN, which is E(Rm)

Variance equation

Var1 = SUM OF Pi(R1,R2)(R1-E(R1))2

Is it possible that a risky asset could have a negative beta? What does the CAPM predict about the expected return on such an asset? Can you give an explanation for your answer?

Yes, it is possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument, so adding it to a portfolio would actually reduce portfolio beta. One example of a negative beta would be gold or other countercyclical asset.

portfolio

a collection of assets such as stocks and bonds, a holding of real estate properties, even an investment in collectibles such as baseball cards.

probability distribution

a formula or table that shows possible outcomes & their likelihood of occurring integrates expected returns, realized returns, variances and standard deviations.

variance

a measure of how the possible realized returns might vary from the expected return calculated above

Beta

a measure of the market/systematic risk of an asset. (measure of covarability)

portfolio variance

estimates the risk of her portfolio: how much her realized return might vary from the expected return

Joint probability distribution

a probability distribution containing two assets rather than one.

annuity

a series of regular payments at regular intervals for a defined period of time. Rent, salary, pensions, and car payments are all examples of annuities.

2) Perpetuity

a series of regular payments for regular periods that go on indefinitely; regular payments that never end

Covariance

a statistical measure of the degree to which two rates of return move relative to each other gives the direction of the relationship

annuity due

a type of level annuity payments occur at the beginning of each period. Your landlord expects the rent at the beginning of each month.

ordinary annuity

a type of level annuity payments occur at the end of each period, such as the salary a worker gets paid at the end of each month.

Set up these Security Market Line, which gives the current market's risk/return relationship.

a. Determine the general tradeoff between risk and return in the current market. Do this by determining the riskless rate of return and the expected return on the market portfolio. b. The current riskless rate of return on U.S. government securities is 7%. The current return on the S&P 500 index is 11%.

Classify the following events as mostly systematic or mostly unsystematic. Is the distinction clear in every case? a. Short-term interest rates increase unexpectedly. b. The interest rate a company pays on its short-term debt borrowing is increased by its bank. c. Oil prices unexpectedly decline. d. An oil tanker ruptures, creating a large oil spill. e. A manufacturer loses a multimillion-dollar product liability suit. f. A Supreme Court decision substantially broadens producer liability for injuries suffered by product users.

a. Short-term interest rates increase unexpectedly. - Systematic: Interest rate changes impact all elements of the economy and thus cannot be diversified away. b. The interest rate a company pays on its short-term debt borrowing is increased by its bank. - Unsystematic: This interest rate change likely reflects a change in the risk of the individual company. As this interest rate change is unique to the company it can be diversified away in a large portfolio. c. Oil prices unexpectedly decline. - Both; probably mostly systematic: Oil prices change the cost of energy in the economy as would thus be systematic; however, not all companies would be equally impacted. d. An oil tanker ruptures, creating a large oil spill. - Unsystematic: This would affect the company, and a lot of fish, but would not impact the entire economy. e. A manufacturer loses a multimillion-dollar product liability suit. - Unsystematic: Again, bad news for the company, but the settlement impacts only the cash flow of this company. f. A Supreme Court decision substantially broadens producer liability for injuries suffered by product users. - Systematic: This would impact many companies and likely have some impact on the economy and security markets.

Indicate whether the following events might cause stocks in general to change price, and whether they might cause Big Widget Corp.'s stock to change price. a. The government announces that inflation unexpectedly jumped by 2 percent last month. b. Big Widget's quarterly earnings report, just issued, generally fell in line with analysts' expectations. c. The government reports that economic growth last year was 3 percent, which generally agreed with most economists' forecasts. d. The directors of Big Widget die in a plane crash. e. Congress approves changes to the tax code that will increase the top marginal corporate tax rate. The legislation had been debated for the previous six months.

a. The government announces that inflation unexpectedly jumped by 2 percent last month. - This is a systematic risk: the increase in inflation will be reflected in interest rates and opportunity costs. Market prices in general will most likely decline and Big Widget will be along for the ride down. b. Big Widget's quarterly earnings report, just issued, generally fell in line with analysts' expectations - This is a firm specific risk; as the report just reflects the expectations of investors, the company price will most likely stay constant. c. The government reports that economic growth last year was 3 percent, which generally agreed with most economists' forecasts. - This is a systematic risk; as with an individual company, if economic activity is as expected market prices in general will most likely stay constant. However, if the growth rate turns out to be different from what was expected market prices would change. d. The directors of Big Widget die in a plane crash. - This is a firm specific risk; the company price will most likely decline. Of course, if the directors were felt to be incompetent, the price could rise. e. Congress approves changes to the tax code that will increase the top marginal corporate tax rate. The legislation had been debated for the previous six months. - This is a systematic risk; market prices in general will most likely stay constant. In this case market participants, following the debate, likely saw that the tax increase would occur and had already adjusted prices to reflect their beliefs.

mutual fund

fund that pools the savings of many individuals and invests this money in a variety of stocks, bonds, and other financial assets

Level annuities

has the same cash flow for each period of time. Further classified as to when the payments are made in each period: ordinary annuity and annuity due

An asset with above average volatility will have a risk premium equal/higher/lower than that of the market as a whole.

higher

For assets with the same amount of risk, investors prefer the asset with the higher/lower return.

higher

Compounding

allows a decision-maker to restate a cash flow to the future.

Discounting

allows a decision-maker to restate a cash flow to the present.

how do we measure market risk

by determining the covariability of the asset's rate of return with that of the economy. The measure of this covariability is called beta, ß.

Time value calculations rely on which interest?

compound interest the base amount for your calculations (the principal) changes with each period as interest is added to principal to make a larger amount for computing interest.

1) Discrete Probability Distribution

contains a discrete, or countable, number of observations.

2) Continuous Probability Distribution

contains an infinite number of observations which are analyzed using quantitative measures based on mathematical relationships and calculus.

Holding more imperfectly correlated assets produces

diversification (risk reduction)

Since all assets are a part of the economy, when the economy goes down, most asset returns go up/down

down

1. economic activity

economists find 5 possible levels of growth: boom, high growth, normal, low growth, recession

An asset with average volatility will have a risk premium equal/higher/lower to that of the average asset in the market.

equal

quarterly compounding

interest recognized at the end of each quarter · at end of first quarter, interest calculated & added to principal o Larger loan balance & higher interest payment for the 2nd month o Continues each quarter until loan is due o With quarterly compounding, total interest payment increases bc ur charged a higher EAR

Both perpetuities and annuities can be _________, where all of the cash flows are the same amount, or ____________, where the cash flows grow at a regular rate.

level; growing

An asset with below average volatility will have a risk premium equal/higher/lower than that of the market as a whole.

lower

For assets with the same return, investors prefer the asset with the higher/lower risk.

lower

undiversifiable risk

market risk

correlation coefficient

measures not only the direction of covariability (positive or negative) but also the strength of the relationship

2. Compound interest

occurs when the Interest earned is applied to the principal each period. With compound interest, the principal grows over time and, if the principal grows, so does the amount of interest paid each period.

Decide

overvalued = dont buy stock undervalued = bairgain fairly valued = fair deal

growing annuity

payments that grow at a steady rate. The cash flows are not the same, but they grow at a constant rate. An example would be a lease payment that is adjusted for a given rate of inflation.

To make decisions, we convert uncertainty to risk via _________________

probability distributions

correlation

relationship btwn any 2 variables

realized return

return received at end of investment pd; these returns vary

frequency distribution

shows the frequency with which each rate of return is earned summarized the historic relationship between return and risk

the variance is often converted into the

standard deviation iExpected return is in percent (%) and Variance is in squared percent (%^2), so to get the same unit of measurement, you take the square root of variance to get standard deviation so that they both are in % SD = sqrtVariance

Beta is a ____________ measure

standardized, or relative

1) Annuity

stream of periodic payments

2) growing perpetuity

the amounts grow at a constant rate.

3. Effective annual interest rate (EAR)

the annual interest rate that reflects the impact of intra-year compounding.

A Beta less than 1 means

the asset return is less volatile than the market return. Some of these assets are not sensitive to market movements. (ex: grocery stores...people eat regularly)

A Beta greater than 1 means

the asset return is more volatile than the market return. Some assets are very sensitive to market movements. (such as a luxury resort...when economy is down, less ppl will travel)

expected return (summary)

the average of the possible realized returns weighted by their probability of occurring. what the investor expects to get

The variance (risk) of the portfolio is affected by

the degree to which the individual asset returns are correlated

Probability distributions estimate

the expected return & the variance

Positive risk-return trade-off

the expected return offered by the project's cash flows at least matches the project's opportunity cost

2. Periodic interest rate

the interest per period such as the semiannual rate for bond interest payments and the quarterly rate paid via dividends.

Annual Percentage Rate (APR)

the interest rate charged per period multiplied by the number of periods, and also includes any fees or additional costs to the borrower

periodic rate

the interest rate stated on a quarterly basis

1. Stated annual interest rate

the interest rate stated on an annual basis. This is the rate normally used in contracts, including loans such as credit cards, auto loans and mortgages.

volatility

the level of dispersion one might see in realized returns

risk

the possibility of losing something of value. Values (such as physical health, social status, emotional well-being, or financial wealth) can be gained or lost when taking risk resulting from a given action or inaction, foreseen or unforeseen (planned or not planned). Risk can also be defined as the intentional interaction with uncertainty. Uncertainty is a potential, unpredictable, and uncontrollable outcome; risk is an aspect of action taken in spite of uncertainty

Diversification

the process of reducing the risk of a portfolio by holding assets whose returns are not perfectly correlated. occurs if rates of return on the individual assets do not move exactly together, and thus the movement of the rates of return over time tends to cancel each other out, thus reducing the overall risk (variability) of the portfolio's overall expected return.

uncertain results

the rate of return you think you'll receive is often not what you'll actually end up getting

Uncertainty exists bc

the realized return from an investment may be diff from the estoated expected return

1) level perpetuity

the regular payments are the same amount. (no change in cash flows)

A negative covariance means

the returns tend to move in opposite directions.

A positive covariance means

the returns tend to move together.

market risk

the risk that affects all companies in the economy. (systematic) (undiversifiable) Market risk, because it affects all market participants, cannot be diversified away and thus the risk that is relevant for determining the opportunity cost. The variability of all the assets in your portfolio relative to the economy

unique risk

the risk that is unique to an individual company. (unsystematic) (diversifiable)

portfolio expected return

the wealth-weighted average of the expected returns of the assets held in the portfolio. It reflects the returns one might earn in diff companies and how much one invests in each asset.

undiversifiable risk is concerning to investors bc

they will expect some compensation for bearing this risk

expected return (return)

uncertain future return one expects to get from an investment; used in making most business decisions

diversifiable risk aka

unique risk

compund interest effects the ______ of a loan

value

do the treatment of cash flows, time periods, & interest rates have to match

yes

Examples of different compounding periods

· As we will see in Unit 9, bonds have a maturity and interest rates stated in years, but make these stated interest payments semiannually · Stocks (Unit 10) are also compared on an annual basis, such as the annual rate of return earned on stocks compared to the annual rate of return earned in bonds, but stocks pay dividends quarterly. · Investors may also own commercial real estate and determine the annual rate of return earned on these investments, but the leases often require monthly payments.

Multiple cash flows over multiple periods:

· Rather than taking the future value of one cash flow, you take the future values of all of the cash flows and sum them up. · Similarly, rather than taking the present value of one cash flow, you just take the present values for all of the cash flows and then sum them up.

You own a stock portfolio invested 15 percent in Stock Q, 25 percent in Stock R, 40 percent in Stock S, and 20 percent in Stock T. The betas for these four stocks are .75, .87, 1.26, and 1.76, respectively. What is the portfolio beta?

ßp = .15(.75) + .25(.87) + .4(1.26) + .2(1.76) = 1.186