3 - applications

2B. ...

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Calculating APR Given an EAR of 14% and semiannual compounding, what is the APR? (6-d #3)

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Calculating EAR (6-d #4)

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Calculating EAR (6-d #5)

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Simple Interest versus Compound Interest First City Bank pays 6% simple interest on its savings account balances, whereas Second City Bank pays 6% 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?

1,545.87

Calculating Rates of Return 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?

10.01

3A. Calculating expected return of an asset Based on the following information, calculate the asset's expected return. State of economy: Probability: Rate of Return in given state: Recession 30% -11% Boom 70% 21%

11.4

Calculating the Number of Periods At 4.7% interest, how long does it take to double your money?

15.09

Calculating the Number of Periods You invest $195 today and receive $873 at the conclusion of the investment, earning a 9% rate of return. How long did you have to wait for your payoff?

17.39

Calculating Present Values You make an investment that earns 7% annual interest and pays you $17,328 in 16 years. How much did you invest today?

5,869.59

Calculating Interest Rates You invest $715 today and receive $1,381 in 11 years. What interest rate have you earned?

6.17

Calculating Future Value You invest $3,150 for 7 years at 13%. What is the future value?

7,410.71

Calculating the Number of Periods You are trying to save to buy a new $150,000 Ferrari. You have $35,000 today that can be invested at your bank. The bank pays 2.1% annual interest on its accounts. How long will it be before you have enough to buy the car?

70.02

A tale of two distributions 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.

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 the variance--how the historic realized returns differed from the average return. 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 their occurring. From the probability distribution we can calculate the expected return the probability-weighed return that we might expect to get, and the variance which measures how the realized return might vary from the expected return. The historic frequency distribution does contain useful information, including the finding that asset classes with more volatility earn a higher rate of return. From this point of view, using a frequency distribution could be used to forecast the future. However, to do so assumes that future performance will match historic performance. Every decision we make involves looking into the future and we really don't know what is going to happen! While history has many important insights to provide us, the future is likely to be turbulent—remember creative destruction—so, while not ignoring historical returns, making the best estimates with probability distributions is probably the best course of action.

Determining Portfolio Weights A portfolio that has 165 shares of Stock A that sell for $69 per share and 125 shares ofStock B that sell for $44 per share. What proportion of the portfolio is invested in StockA?

A portfolio is a combination assets. Its value is determined by two elements: The value of the assets in the portfolio 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%

Interest Rates 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.

Annuity Period 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.

Measures of diversification What measure gives you more information about a portfolio: the covariance or the correlation coefficient?

Both measures give information concerning the degree to which the rates of return of assets in a portfolio are related. The covariance: measures the degree to which rates of return move together: 𝐶𝑜𝑣𝐴,𝐵 = ∑𝑃𝑖 (𝑅𝐴 , 𝑅𝐵)[(𝑅𝐴𝑖 ― 𝐸(𝑅𝐴) (𝑅𝐵𝑖 ― 𝐸(𝑅𝐵)] It's important that we understand the thoughts behind the numbers. (𝑅𝐴𝑖 ―𝐸(𝑅𝐴) measures the degree to which the individual return(𝑅𝐴𝑖 )is above or below the assets expected return 𝐸(𝑅𝐴) in a given market state. Given the nature of uncertainty, in some market states (low growth, recession, etc.) the return will be below the expected return and produce a negative difference. In other market states (boom, high growth, etc.) the return will be above the expected return and produce a positive difference. This is the same for the other asset in the portfolio (𝑅𝐵𝑖 ―𝐸(𝑅𝐵). If we multiply these differences for each possible state [(𝑅𝐴𝑖 ― 𝐸(𝑅𝐴) (𝑅𝐵𝑖 ― 𝐸(𝑅𝐵)] we could get a number that is positive or negative: If the numbers for both asset differences are negative—they are both below their respective expected returns--then multiplying the two negative numbers gives a positive result. If the difference for one asset is negative and the other positive, then multiplying them will produce a negative result. If both differences are positive, then their product will also be positive. ∑𝑃𝑖 (𝑅𝐴 , 𝑅𝐵) If we then weight these differences by their likelihoods and sum them, we get the covariance: If the covariance is positive, the two asset returns generally move together. They are generally both above or below their expected values in the possible market states. If the covariance is negative, the two asset returns generally move opposite to each other.When one asset is above its expected value, the other asset return is below its expected value. The covariance thus tells us the degree to which the rates of return "covary" or move together. The correlation coefficient is a standardized covariance. It is produced by taking the covariance and dividing it by the product of the two asset standard deviations. 𝐶𝑜𝑟𝐴,𝐵 = 𝐶𝑜𝑣𝐴,𝐵 / (𝑆𝐷𝐴)(𝑆𝐷𝐵) The correlation coefficient has limits. Assets returns that move exactly together have a correlation coefficient of +1. Asset returns that move exactly in op

Compounding What is compounding? What is discounting?

Compounding is the exponential increase in the value of an investment because interest earned is added to the 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

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

Discounting is the inverse of compounding Future values: Future values are positively related to the length of time of the investment: with each additional period interest is earned and added to the principal so the future value increases. This can be seen in the Future Value Factor = (1 + r)T. The present value is multiplied by this FVF to get the future value. The longer the period, the greater 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. As the time period increases, the PVF decreases the future value.

Diversifiable and Non diversifiable Risks. In broad terms, why is some risk diversifiable? Why are some risks non diversifiable? 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. Non diversifiable 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—non diversifiable risk—in their portfolios.

Portfolio with equal investments You want to create a two-asset portfolio of equal proportions of the stock of Calvin Clothing (CC) and Perfect Pets (PP). What is the variance of your portfolio? Economy: Probability: ROR Boom: 80%: 18% (CC) 2% )PP) Bust: 20%: 12% (CC): 5%: PP

First must calculate ER and variances of each stock: RCC = .8(18) + .2(12) = 16.8 RPP = .8(2) + .2(5) = 2.6 Variance: CC: 5.76 PP: 1.44 ER on portfolio is: 5(16.8) + .5(2.6) = 9.7 .5 is the weight of each portfolio, which we know is of equal proportions (50%) To find portfolio variance: first calculate covariance: = .8[18 - 16.8] [2 - 2.6] + .2[12 - 16.8] [5 - 2.6] = -0.576 - 2.304 = -2.88 then find variance: .52(5.76) + .52(1.44) + 2(.5)(.5)(-2.88) = .36

Calculating EAR You're considering a loan with a 10% stated annual rate and quarterly compounding. Whatis the effective annual rate?

For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)]m - 1 APR of 10%, compounded quarterly: EAR = [1 + (.10 / 4)]4 - 1 = .1038, or 10.38% Solving for the Effective Annual Rate is quite easy with our financial calculator. For an APR of 10%compounded quarterly take the following steps. To easily change the number of compounding periods: Function 11: Change the number of compounding periods per year 1. Enter the number of periods 2. Press the down function key. 3. Press the PMT/P/YR key The compounding periods have been changed. You might want to do Function 10 to make sure your change was made. Function 10: Check the number of periods per year 1. Press the down function key. 2. Press the clear keyThe number of periods in a year is displayed. 1. Set calculator for quarterly payments 2. Enter 10 3. Press down shift key 4. Press down shift key 4. Press I/YR/NOM% Press down shift key 5. Press PV/EFF% The effective annual interest rate of a nominal rate of 10% per year compounded quarterly is 10.38

Calculating Annuity Cash Flows.You plan to spend six years in college and obtain both your bachelors and master'sdegrees. Your uncle is gifting you a six-year ordinary annuity to help you pay for youreducation. The annuity has a present value of $24,500, with interest rates set at 11%. How much will you have each year to support your education?

Here we're given four variables and asked to solve for the annual payments. Using the PVA equation and solving for the payment in each case, we find: First annuity PVA = C({1 - [1 / (1 + r)t]} / r)) $24,500 = $C{[1 - (1 / 1.11)6] / .11} C = $24,500 / 4.23054 C = $5,791.23 You would have $5,791 to support your education. Start reading up on student loans. Calculator: To solve for the annuity's payments. 1. Enter 6 2. Press N 3. Enter 11 4. Press I/YR 5. Enter -24500 6. Press PV 7. Enter 0 8. Press FV 9. Solve by pressing PMT, which gives a present value of $5,791.23

Calculating Annuity Future Values You make annual end-of-year deposits $1,900 in your savings account that earns 8%. How much will your investment be worth in 10 years?

Here, we need to find the future value of an annuity. Using the FVA equation, we find: FVA = C{[(1 + r)t - 1] / r} FVA = $1,900[(1.0810 - 1) / .08] FVA = $27,524.47 To solve for the annuity's future value we just enter the four variables given and solve for the FV: 1. Enter 10 2. Press N 3. Enter 8 4. Press I/YR 5. Enter -1900 6. Press PMT 7. Enter 0 8. Press PV 9. Solve by pressing FV, which gives a present value of $27,524.47

3B. Calculating the variance and standard deviation of an assetBased on the following information, calculate the variance and standard deviation of an asset. Please enter your answer for the standard deviation. State of economy: Probability: Rate of Return in given state: Recession 30% -11% Boom 70% 21%

In Problem 3 we determined that the expected return is 11.4% .E(R) = .30(-.11) + .70(.21) = .1140, or 11.40% Variance: The variance measures how the realized returns may differ from the expected return. 𝑉𝑎𝑟𝐴= Σ𝑃𝑖 (𝑅𝑖−𝐸(𝑅𝐴))2 We are given two possible realized returns. In a recession we would likely receive a negative return of -11%, which is substantially below the expected return of 11.4% .-0.11 - 0.1140 = -0.224 In a boom we would enjoy a 21% return, which is nicely above the expected return of 11.4%. 0.21 - 0.114 = 0.096 We are looking for variably of the return, and just adding the differences would, to an extent, just cancel out the differences, so these differences are squared. Recession: (-0.11 - 0.114)2 = 0.0502 Boom: (0.21- 0.114)2 = 0.0092 These differences are not equally likely to occur, so we must weight them by their probabilities to get the variance. Var = 0.30(0.11 - 0.114)2 + 0.70(0.21- 0.114)2 = 0.30(0.0502) + 0.70(0.0092) = 0.02150 Standard deviation: The variance, being a squared term, cannot be directly compared to the expected return, which is not squared. We can thus take the square root of the variance to get the standard deviation. SD = 𝑆𝐷 = 𝑉𝑎𝑟 = 0.2150 = 0.1466 So, you need to calculate the variance to get the standard deviation. In practical work the standard deviation, which is in percent, is more often used when working with the expected return, as we saw with the confidence interval example in question 2.

Values and interest rates What happens to a future value if you increase the rate, r? What happens to a present value?

Major determinants of value are the amount of time and interest rates. 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 the future value by (1 + r)T. future values increase with longer time periods and higher interest rates; present values decrease with longer time periods and higher interest rates

Systematic versus Unsystematic Risk. Classify the following events as mostly systematic or mostly unsystematic. Is the distinction clear in every case?

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.

Corporate Downsizing In recent years, it has been common for companies to experience significant stock price changes in reaction to announcements of massive layoffs. Critics charge that such events encourage companies to fire longtime employees and that Wall Street is cheering them on. Do you agree or disagree?

Such layoffs generally occur in the context of corporate restructurings. To the extent that the market views a restructuring as value-creating, stock prices will rise. So, it's not the layoffs per se that are being cheered on but the cost savings associated with the layoffs. Nonetheless, Wall Street does encourage corporations to take actions to create value, even if such actions involve layoffs. These layoffs may also be the result of creative destruction, which is increasingly affecting higher level jobs. The impact AI and applications such as ChatGPT may have a major impact on many types of jobs, producing an increase in profits, but a reduction in many types of jobs and an increase in economic inequality.

1B. Time value of money: From the company's point of view. 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 of4.86%. 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 onTMCC'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. This question focuses on issues we'll see in our capital budgeting decisions.

Calculating Portfolio Betas You own a stock portfolio invested 15 percent in Stock Q, 25 percent in Stock R, 40 percentin Stock S, and 20 percent in Stock T. The betas for these four stocks are .75, .87, 1.26, and1.76, respectively. What is the portfolio beta?

The beta of a portfolio is the sum of the weight of each asset in the portfolio times the beta of each asset. So, the beta of the portfolio is: = .15(.75) + .25(.87) + .40(1.26) + .20(1.76) = 1.19 Note that this calculation involves only market risk: there is no reduction in risk due to diversification because there is no unique risk in the asset betas, so the portfolio beta is just sum of the betas of the three assets in the portfolio. The risk of the portfolio is influenced in how wealth is allocated among the three assets.Increasing the relative investment in the low beta asset (p = .75) by reducing the investment in the high beta asset (p = 1.76) will reduce the portfolio beta.

Portfolio Expected Return 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%

Diversification: How does diversification work?

The future is uncertain. In business and investments, this takes the form of an uncertain rate of return. A probability distribution measures the variability of possible asset realized rates of return around the asset's expected return. The risk is that investors expect to earn a rate of return, but the return they eventually get is likely to be less or greater than what was expected. Investors don't generally consider getting a larger rate of return than expected a "risk" to avoid. The risk they are concerned about is getting a smaller return than they bargained for. In many cases, investors can reduce this downside risk by holding multiple assets in a portfolio. Uncertain events may affect the portfolio's assets to different degrees. Some assets will end up earning more than expected, others less than expected. These differences tend to cancel out, making the overall return on the portfolio less volatile, thus reducing downside risk. The assets held in the portfolio determine the amount of risk reduction. The more diverse the assets, the greater the potential risk reduction.

Annuity Present Value Tri-State Megabucks Lottery advertises a $10 million grand prize. The winner receives a20-year annuity due of $500,000 per year. A lump sum option of $5 million payableimmediately is also available. If your opportunity cost is 12% which would you prefer?

The lottery is just one example of the many annuities we see in daily life. The headline doesn't really specify the fact that it's an annuity, but the payments fit the definition of an annuity due. In this example we'll see how to use a calculator to evaluate these alternatives. Lottery face values are the total payments, in this case $10,000,000. This is a nominal value and not a cash flow. The fine print specifies that the payment will be split into equal annual payments, which give $10,000,000/20 = $500,000. You would receive a cash payment of $500,000 every year. These payments do indeed add up to $10,000,000, so the lottery statement is not deceptive; however, it does not take time value into account. Your choices are: Take annual payments of $500,000 Take the lump sum of $5,000,000 today These cash flows-- receiving the first payment today and nineteen subsequent payments at the end of each year--fits the definition of a 20-period annuity due. In setting these payments, the lottery assumed a discount rate. You can use the calculator to determine this rate and thus make your decision. To solve: Step 1: Change from End Mode (ordinary annuity) to Begin Mode(annuity due).1. Press down function key 2. Press MAR/Beg/EndNote: The calculator normally operates in the End mode, and treats all payments as end-of-period. When you shift to Begin mode you'll see "Beg" on the bottom of your screen. Nothing changes as far as how you enter numbers Step 2: 1. Enter 20 2. Press N 3. Enter -5,000,000 4. Press PV 5. Enter 500,000 6. Press PMT 7. Enter 0 8. Press FV 9. Press I/YR 10. This gives you 8.92% The lottery thus assumes an interest rate of 8.92. In effect if you "invest" the lump sum of $5,000,000 and receive the 20 annual payments of $500,000 you would earn 8.92%. If your own opportunity cost--the rate you'd earn in your own investments--differs from 8.92%, then you would prefer one or the other alternative. Your opportunity cost is 4%. If you were to take the lump sum of $5,000,000 today and invest it to obtain a 20-year annuity due, you would receive annual payments of only $353,758. You'd get a higher return from taking the annual lottery payments. Your opportunity cost is 12%. If you were to take the lump su

Present Value If you were an athlete negotiating a contract with a signing bonus of $1million, would you want the signing bonus payable immediately, or divided into smaller payments over the duration of your contract? How about looking at it from the team's perspective?

The nominal value of the bonus is $1 million. The economic value of the contract depends on how the cash is actually paid. To put some numbers on this, assume that the contract is for 10 years and the discount rate is 5%. Pay today: The value of the payment today is $1,000,000 Pay over time. The value of the payments is $772,173. The athlete would receive $100,000 each year. $1,000,000/10 = $100,000 Given the placement of the payments and the discount rate, the payments to the athlete have a present value of $772,173. This example fleshes out the impact of how money is actually paid. The athlete would like the total bonus paid today. Spreading out the payments would make the economic value of the bonus smaller. This result would be just fine to the team owner! Given a fixed amount, splitting it up into future payments means that the economic value of the payments to the athlete is smaller. These differences in perspectives makes sense when we know that a dollar received today is more valuable than a dollar received in the future.

Calculating the Rate of Return In 2014, an 1874 $20 double eagle sold for $15,000. What was the rate of return on this investment over its 140 year life?

The time line for this golden investment from when it was issued until 2014 is: 0 yrs = -$20 140 yrs = $15,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)1 / t - 1 r = ($15,000 / $20)1/140 - 1 r = .0484, or 4.84% Again, you have a present value, a future value and a time period, so solve for I/YT To solve: 1. enter 140 2. press N 3. enter -20 4. press PV 5. enter 0 6. press PMT 7. enter 15,000 8. press FV 9. solve by pressing 1/YR, which gives the rate of return of 4.84% 4.84

Two measures of volatility Why the heck 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 and given in confidence intervals.

Calculating Annuity Present Value You're a consultant for a non-profit and have a contract that pays you $2,100 annually for seven years. If your opportunity cost is 5%, what is the present value of this contract?

There are only five variables that most of the problems in These Applications use. Here we need to find the present value of an annuity. Using the PVA equation, we find: First annuity: PVA = C({1 - [1 / (1 + r)t]} / r) PVA = $2,100{[1 - (1 / 1.05)7] / .05} PVA = $12,151.38 To solve for the annuity's present value we just enter the numbers: 1. Enter 7 2. Press N 3. Enter 5 4. Press I/YR 5. Enter 2100 6. Press PMT 7. Press PV, which gives a present value of -$12,151.38 Note that the calculator requires an outflow and an inflow to compute these values, so if your annual inflows are positive, the calculator will give you a negative present value. This doesn't mean that you're losing money, it's just how the calculator works. Receiving $2,100 per year for 7 years has a value of$12,151.38 today.

APR and EAR Should lending laws be changed to require lenders to report EARs instead of APRs? Why or why not?

There are three rates involved in compounding: The stated annual interest rate is the interest rate stated on an annual basis. The periodic interest rate is the interest per period. The effective annual interest rate (EAR) is the annual interest rate that reflects the impact of intra-year compounding. With annual compounding these rates are the same; however, for compounding more frequently than annually, they are different. The answer to our question is yes, they should. APRs generally don't provide the relevant rate. The only advantage is that they are easier to compute, but, with modern computing equipment, that advantage is not very important. Also, the APR's on debt generally look more attractive than the true rate charged.You'll find that our course's discussion of compounding will save you money on anything—clothing car,house—that you buy on credit!

2A. Calculating Perpetuity Values 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. The payments would be made at the end of each year.If the required return on this investment is 5 percent, how much would you pay for the policy today?

This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:PV = C / rPV = $30,000 / .05PV = $600,000.00 Perpetuities are similar to annuities but have no end. The present value of this infinite payment stream is finite because we are not adding the payments, but rather adding the present value of the payments. To solve: 1. Enter 500 2. Press N 3. Enter 5 4. Press I/YR 5. Enter 30000 6. Press PMT 7. Press PV to get the present value of -$600,000. This is the amount you would pay today for this stream of payments that will make your heirs very happy! Why enter 500? With the perpetuity we are not adding infinite amounts of $30,000: we are adding the present values of these $30,000 payments. As we look at more distant cash flows, their present value declines. The present value of the payment made in year 10 is: $30,000/(1.05)10 = $18,417.40 The present value of the payment made in year 50 is: $30,000/(1.05)50 = $2,616.11 The present value of the payment made in year 500 is: $30,000/(1.05)500 = $0.00500 is not a set number, but it generally means that we've taken into account any future cash flow of our perpetuity that has a positive present value

Calculating Rates of Return Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, Sotheby's sold the Edgar Degas bronze sculpture "Petite Danseuse de Quatorze Ans" at auction for a price of $10,311,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,377,500. What was his annual rate ofreturn on this sculpture?

To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)1 / t - 1 = ($10,311,500 / $12,377,500)1/4 - 1 = -.0446, or -4.46% So, collectibles are not always a sure thing! Unless you're a math major who just loves formulas, a financial calculator is really useful when solving for interest rates! 1. Enter 4 2. Press N 3. Enter -$12,377,500 4. Press PV 5. Enter $10,311,500' 6. Press FV 7. Enter 0 8. Press PMT 9. Press I/YR to see the unpleasant interest rate of -4.46%.

Calculating the Growth Rates and Future Values In 1895, the first U.S. Open Golf Championship was held. The winner's prize money was$150. In 2014, the winner's check was $1,620,000. What was the annual percentage increasein the winner's check over this period?

To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)1 / t - 1 = ($1,620,000 / $150)1/119 - 1 = .08117, or 8.117% Using the calculator is a sure winner! 1. Enter -$150 2. Press PV 3. Enter $1,620,000 4. Press FV 5. Enter 119 6. Press N 7. Press I/YR find the interest rate is 8.117% Given this rate, solve for the future value in a future year—2045. Given the interest rate of 8.117%, we can find the FV of the future first prize in 2045, we use: FV = PV(1 + r)t FV = $1,620,000(1.08117)31 = $18,206,589 To solve for the first future value. 1. Enter -$1,620,000 2. Press PV 3. Enter 31 4. Press N 5. Enter 8.117 6. Press I/YR 7. Solve by pressing FV, which gives a future value of $18,206,589.

Calculating the Rates of Return In 2014, an Action Comics No. 1, featuring the first appearance of Superman, was sold at auction for $3,207,852. The comic book was originally sold in 1938 for $0.10. What was the annual increase in the value of this comic book?

To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1+ r)t Solving for r, we get:r = (FV / PV)1 / t - 1 = ($3,207,852 / $.10)1/76 - 1 = .2554, or 25.54% Not a bad rate of return? Unfortunately, as more and more people see such examples, comic books, toys and other collectibles are increasingly being saved, which reduces the rate collectors can expect to receive in the future. As always, the calculator makes it easy, as long as you can visualize what you're given and what you need to calculate. 1. Enter 76 2. Press N 3. Enter -0.10 4. Press PV 5. Enter 0 6. Press PMT 7. Enter 3207852 8. Press FV 9. Solve by pressing I/YR, which gives a Super rate of return of 25.54%.

Calculating Future Values. Your coin collection contains fifty 1952 silver dollars that your grandparents obtained at their face value when they were new. How much will your collection be worth when you retire in 2063, assuming they have appreciated at an annual rate of 4.3 percent since they were issued?

To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $50(1.043)111 = $5,352.15 You have PV, N, and I/YR. Solve for FV. To solve for the future value: 1. Enter $50 2. Press PV 3. Enter 111 4. Press N 5. Enter 4.3 6. Press I/YR 7. Solve by pressing FV, which gives a future value of $5,352.15

Calculating Present Values Imprudential, Inc., has an unfunded pension liability of $730 million that must be paid in 25 years. To assess the impact of this liability on the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is5.5 percent, what is the present value of this liability?

To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV = $730,000,000 / (1.055)25 PV = $191,430,603.85 This should be getting routine by now!!1. 1. Enter $730,000,000 2. Press FV 3. Enter 25 4. Press N 5. Enter 5.5 6. Press I/YR 7. Solve by pressing PV, which gives a present value of $191,430,603.85

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

To find the Present value of Annuity (PVA), we use the equation: PVA = C({1 - [1/(1 + r)t]} / r) $5,430 for 15 years PVA = $5,430{[1 - (1/1.08)15] / .08} = $46,478 Using our calculator: 1. Enter 15 2. Press N 3. Enter 8 4. Press I/YR 5. Enter 5430 6. Press PMT 7. Enter 0 8. Press FV 9. Solve by pressing PV, which gives a present value of $46,478 Some additional calculations: While the problem asks for fifteen years, we'll also showing the results for longer periods: $5,430 for 40 years: PVA = $5,430{[1 - (1/1.08)40] / .08} = $64,751 $5,430 for 75 years: PVA = $5,430{[1 - (1/1.08)75] / .08} = $67,664 As with individual payments, the longer the payment stream the higher the future value. As the length ofthe annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. A perpetuity is a stream of payments that goes on (technically) forever. To find the PV of a perpetuity, weuse the equation: PV = C / r PV = $5,430 /.08 PV = $67,875 Note that the difference between the present value of the 75-year annuity and the present value of the perpetuity is only $67,875 - $67,664 = $211.

Future Value and Multiple Cash Flows 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?

To solve this problem, we must find the future values of each cash flow and then sum them at N = 4.To find the future value of a lump sum, we use: FV = PV(1 + r)t Future value at 8%: $5,519.84 = $985(1.08)3 + $1,160(1.08)2 + $1, 325(1.08) + $1,495 Notice, since we are finding the value at Year 4, the cash flow at Year 4 is already at year 4, so we do not need to compound this cash flow. We can use our financial calculator. If you need to, go back to problem 2 to review the CFj andPRC/NPV buttons. To solve for the future value at the 8% discount rate, first enter the cash flows. 1. Enter 0. This entry tells the calculator that there is no time 0 cash flow. 2. Press CFj Note: You will see CF 0 flash and then disappear 3. Enter 985 4. Press CFj Note: You will see CF 1 flash and then disappear 5. Enter 1160 6. Press CFj 7. Enter 1325 8. Press CFj. 9. Enter 1495 10. Press CFj Second, enter the discount rate. 11. Enter 8 12. Press I/YR Third, solve for NPV. 13. Press down function key 14. Press PRC/NPV to give the answer that NPV = -$4,057.25 We can convert this value at N = 0 to its future value at N = 4. 1. Press down function key 2. Press k/SWAP This gives us the future value of $5,519.84 You could also take the present value of $4,057 and take its future value in four years using the 8%discount rate. $5,519.84 = $4,057.25(1.08)4 As always, we can also use our calculator: 1. Enter 4 2. Press N 3. Enter 8 4. Press I/YR 5. Enter -$4,057 6. Press PV 7. Solve by pressing FV, which gives a future value of$5,519.84

Comparing Annuities Investment X offers to pay you $3,400 per year for nine years, whereas Investment Y offers to pay you $5,200 per year for five years. All payments are made at the end of the year. If the discount rate is 6 percent, what is the present value of the investment with the higher present value?

While the first investment has more payments over a longer period of time, the second investment has larger payments. To find the PVA, we use the equation: PVA = C({1 - [1/(1 + r)t]} / r ) At an interest rate of 6 percent: Present value of X: $23,125.75 = $3,400{[1 - (1/1.06)9] / .06 } Present value of Y $21,904.29 = $5,200{[1 - (1/1.06)5] / .06} You would prefer Investment X, which has a higher present value. While Investment Y has higher annual cash flows, Investment X has more cash flows. This question deals with annuities, so we can use the Calculator Guide to solve for present value of an ordinary annuity. Here we go back to what I call our five favorite buttons. To solve for the present value of the first annuity at 6%: 1. Enter 9 2. Press N 3. Enter 6 4. Press I/YR 5. Enter 3400 6. Press PMT 7. There is no terminal payment, so enter 0 for FV 8. Press PV to get the present value of $23,125.75. Some additional comments: We also want to show the present values if interest rates were 22% and make an important point. At an interest rate of 22 percent: Present value of X: $12,873.37 = $3,400{[1 - (1/1.22)9] / .22} Present value of Y: $14,890.93 = $5,200{[1 - (1/1.22)5] / .22} Notice that: At 6% discount rate the PV of Investment X is greater than the present value of Investment Y The reason is that X has a greater number of total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At 22% discount rate the PV of Investment Y is greater than the present value of Investment X At a higher interest rate, Y is more valuable since it has larger annual payments. At a higher interest rate, getting these payments early are more important since the cost of waiting (the interest rate) is so much greater.

Present Value and Multiple Cash Flows 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? What is the present value at 18 percent?

While you were asked to solve for the present value at 18%, we're giving you the results at several rates so that you can see how a change in rate impacts present values. To solve this problem, we must find the future values of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r)t Present value at 10%: $2,547.97 = $680 / 1.10 + $490 / 1.102 + $975 / 1.103 + $1,160 / 1.104 Present value at 18%: $2,119.91 = $680 / 1.18 + $490 / 1.182 + $975 / 1.183 + $1,160 / 1.184 Present value at 24%: $1,869.09 = $680 / 1.24 + $490 / 1.242 + $975 / 1.243 + $1,160 / 1.244 Calculator: First enter the cash flows: 1. Enter 0. This entry tells the calculator that there is no time 0 cash flow. 2. Press CFj Note: You will see CF 0 flash and then disappear 3. Enter 680 4. Press CFj Note: You will see CF 1 flash and then disappear 5. Enter 490 6. Press CFj 7. Enter 975 8. Press CFj. 9. Enter 1160 10. Press CFj Second, enter the discount rate 11. Enter 10 12. Press I/YR Third, solve for NPV 13. Press down function key 14. Press PRC/NPV to give the answer that the present value of the timeline is $2,547.97 Once you have worked through this problem at 10%, you need only enter 18% and you'll get its present value. Enter 24% and you'll get its appropriate present value

Beta and CAPM 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. In practice there are very few negative beta assets.

Calculating Present Values You have just received notification that you have won the $1 million first prize in theCentennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you're around to collect), 80 years from now. What is the present value of your windfall if the appropriate discount rate is 7.25 percent?

You are 20 years old and have a promised payment of $1,000,000. Unfortunately, you will receive this in 80 years. To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV = $1,000,000 / (1.0725)80 = $3,700.12 To solve: 1. Enter $1,000,000 2. Press FV 3. Enter 80 4. Press N 5. Enter 7.25 6. Press I/YR 7. Solve by pressing PV, which gives a present value of $3,700.12. Something to lookforward to!

Measures of risk Your friend in this course uses variance as a measure of risk in her other courses and doesn't see why she has to bother with beta, yet another measure of risk she doesn't think is important. Would you agree or disagree with your friend?

You should explain to your friend that she should not ignore beta. You can convince her that beta is important by reviewing the definitions of variance, diversification, beta and opportunity cost. Variance is a summary measure derived from probability distributions. Probability distributions give the possible returns and their likelihood of occurring. Variance measures the dispersion of realized returns from the expected return, and is thus a measure of the total volatility—risk—of the asset. Diversification is the process of reducing the risk of a portfolio by holding assets whose returns are not perfectly correlated. Returns of different assets, which can be in different industries, different countries, have varying quality of management, etc., do not move exactly together through time. The imperfect correlation of returns of assets in a portfolio thus tend to dampen the movement of the portfolio return, making the risk of the portfolio less than the sum of the risk of the assets in the portfolio. Limits of diversification: As more and more assets are added to a portfolio the portfolio variance does drop; however, after a number of assets have been added the variance ceases to drop. This occurs because the variance reflects two types of risk. Unique risk is the risk unique to an asset in the portfolio. It is this risk that is being diversified. Market risk is the risk of the asset that is correlated to the market-the economy. If a major economic event occurs, such as a change in interest rates, expectations about future difficulties in the economy, a trade war, new technological developments, etc., all asset returns will be affected. This economy-wide risk can't be diversified away. Given the two risks, investors will diversify unique risk but must bear market risk. Beta measures the market risk in an asset. As this risk can't be diversified away, investors will demand a risk premium appropriate for the amount of market risk they bear As the markets are controlled by large investors who use diversification, the appropriate risk premium for the opportunity cost is based on beta.

1A. Time value of money: From the investor's point of view. What rate of return would you receive if you pay $24,099 today in exchange for $100,000 in 30 years? Is this an acceptable rate of return?

Your rate of return: Investing $24,099 today to receive $100,000 in thirty years produces an annual IRR of 4.86%. To solve for the IRR, just enter the variables given: Enter 30 Press N Enter -$24,099 Press PV Enter $100,000 Press FV Press 1/YR The IRR is 4.86% Your decision: is 4.86% an acceptable rate of return? Your decision should be based on opportunity cost: the rate of return offered by other equivalent investments of similar risk. The opportunity cost consists of the risk-free rate plus an appropriate risk premium based on the likelihood that the future payment will actually show up. So, your comparison should only be with investments that have the same risk premium. Opportunity cost = risk free rate + risk premium In this decision: If other alternatives offer you less than 4.86% then take the TMCC security. If another alternative—of equivalent risk--offers you more than this investment's 4.86%,then take the alternative

3B. (7-A #5)

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