finance quiz, units 6 & 7
level perpetuity
the regular payments are the same amount.
Using CAPM: Riskfree rate. 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?
Here, we need to find the risk-free rate, using the CAPM. Substituting the values given, and solving for the risk-free rate, we find: E(Ri) = Rf + [E(RM) - Rf] × βi .1070 = Rf + (.1150 - Rf)(.91) .1070 = Rf + .10465 - .91Rf Rf = .0261, or 2.61%
Dianne diversifies her portfolio. The expected return of her investment is an expected return of .1195 and a variance of .0342, for NDS it is er of .1325 and variance of .0235. Dianne has $1000 to invest and is going to invest half, $500 in each. Dianne now wants to know the return this equally divided portfolio will return.
The expected return of Dianne's portfolio is (.5)(.1195) + (.5)(.1325) = .126. Her portfolio now has an expected return of 12.6%
discrete probability distribution
contains a discrete, or countable, number of observations.
simple interest
interest earned on the original principal. each period the interest rate is applied, but the principal remains the same.
future value
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
nondiversifiable risk
risk which is not reduced by diversification
periodic interest rate
the interest per period, such as the semiannual rate for bond interest payments and the quarterly rate paid via dividends.
Using CAPM: Market rate of return. 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?
Here we need to find the expected return of the market, using the CAPM. Substituting the values given, and solving for the expected return of the market, we find: E(Ri) = Rf + [E(RM) - Rf] × βi .109 = .028 + [E(RM) - .028](.85) E(RM) = .1233, or 12.33%
A Family Loan Example. Christina has asked Dianne for a $1,000 loan. Dianne is happy to loan her the money, but she wants this loan to be a fair deal for her. She is concerned about inflation eating away at the true value of the loan's interest payments, and wants the loan adjusted for the rate of inflation. The loan terms are that the interest payments are set at 6%, but after the first year they are indexed to inflation, which is expected to be 2% per annum. Christina will repay the principle at the conclusion of the loan period. The discount rate is 4%.
The first period interest is computed using the stated rate on the loan. $1,000(.06)= $60 The interest paid in each subsequent period is 2% larger than the previous payment. 2nd payment: $60(1.02) = $61.20 3rd payment $61.20(1.02) = $62.42 and so on. Dianne may be a biotch but she's getting use out of the time value of her money!!!
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.
Calculating present values. Imprudential, Inc., has an unfunded pension liability of $730 million that must be paid in 25 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 5.5 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
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. If the required return on this investment is 5 percent, how much will you pay for the policy?
To find the PV of a perpetuity, we use the equation: PV = C / r PV = $30,000 / .05 PV = $600,000.00
annuity
a series of regular payments at regular intervals for a defined period of time.
market risk
the risk that affects all companies in the economy. Unique risk, varying from company to company, is the risk that can be diversified. Market risk, because it affects all market participants, cannot be diversified away and thus the risk that is relevant for determining the opportunity cost.
unique risk
the risk that is unique to an individual company.
If you deposit $5,000 at the end of each year for the next 20 years into an account paying 9.6 percent interest, how much money will you have in the account in 20 years? How much will you have if you make deposits for 40 years?
Here we need to find the FVA = C{[(1 + r)^t - 1] / r} FVA for 20 years = $5,000[(1.096^20 - 1) / .096] FVA for 20 years = $273,685.74 FVA for 40 years = $5,000[(1.096^40 - 1) / .096] FVA for 40 years = $1,985,526.07 Notice that because of compound interest, doubling the number of periods does not merely double the FVA, it increases it by over ($1,985,526 - $273,686)/$273,686 = 6.25 or 600%--an example of how compounding produces exponential growth!
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
covariance
a statistical measure of the degree to which two rates of return move relative to each other.
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.
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.
Calculating asset expected returns and standard deviations. Based on the following information, calculate the expected return and standard deviation for the two stocks.
For example if probability of boom is 40%, rate of return of stock a is 15% and stock b is 31%. another scenarios is probability of normal is 50%, rate of return is 10% for stock a and rate of return is 18% for stock b. final scenario is probability of recession is 10%, rate of return on stock a is 2% and stock b is -30%. Expected return. The expected return of an asset is the sum of the probability of each state occurring times the rate of return if that state occurs. So, the expected return of each asset is: E(RA) = .10(.02) + .50(.10) + .40(.15) = .1120, or 11.20% E(RB) = .10(-.30) + .50(.18) + .40(.31) = .1840, or 18.40% Variance and standard deviation. To calculate the standard deviation, we first need 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 and standard deviation of each stock is: VarA = .10(.02 - .1120)^2 + .50(.10 - .1120)^2 + .40(.15 - .1120)^2 = .00150 SDA = .001501/2 = .0387, or 3.87% VarB = .10(-.30 - .1840)^2 + .50(.18 - .1840)^2 + .40(.31 - .1840)^2 = .02978 SDB = .029781/2 = .1726, or 17.26%
portfolio variance
Portfolio variance = Variance of FF x (Percent invested in FF)2 + Variance of NDS x (Percent invested in NDS)2 + 2 x (Percent invested in FF) x (Percent invested in NDS) x Covariance of FF and NDS
Calculating present values. You have just received notification that you have won the $1 million first prize in the Centennial 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
probability distribution
a formula or table of information that gives the potential outcomes and the likelihood of those outcomes. There are two types of probability distributions: discrete and continuous
mutual fund
a type of financial vehicle made up of a pool of money collected from many investors to invest in securities such as stocks, bonds, money market instruments, and other assets. Mutual funds are operated by professional money managers, who allocate the fund's assets and attempt to produce capital gains or income for the fund's investors. A mutual fund's portfolio is structured and maintained to match the investment objectives stated in its prospectus."
growing annuity
as 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.
hedge fund
basically a fancy name for an investment partnership. It's the marriage of a professional fund managers, who can often be known as the general partner, and the investors, sometimes known as the limited partners, who pool their money together into the fund . . . Hedge funds they are generally considered to be more aggressive, risky and exclusive than mutual funds"
Calculating Annuity Cash Flows. For each of the following annuities, calculate the annual cash flow. (using future value, years, and interest rate)
for example, future value = $30,000, years = 8, interest rate = 5% FVA = C{[(1 + r)^t - 1] / r} $30,000 = C[(1.05^8 - 1) / .05] C = $30,000 / 9.54911 C = $3,141.65 FVA = C{[(1 + r)^t - 1] / r} $1,200,000 = C[(1.07^40 - 1) / .07] C = $1,200,000 / 199.63511 C = $6,010.97 FVA = C{[(1 + r)^t - 1] / r} $625,000 = C[(1.08^25 - 1) / .08]C = $625,000 / 73.10594 C = $8,549.24 FVA = C{[(1 + r)^t - 1] / r} $125,000 = C[(1.04^13 - 1) / .04] C = $125,000 / 16.62684 C = $7,517.9 not that there are no cash flows at time 0 so the annuity for this calculation has a present value of 0.
solve for the unknown number of years in each of the following.
for instance when present value = $195, interest rate = 9%, and future value = $873 V = $873 = $195 (1.09)t t = ln($873 / $195) / ln 1.09 t = 17.39 years FV = $3,500 = $2,105(1.07)t t = ln($3,500 / $2,105) / ln 1.07 t = 7.51 years FV = $326,500 = $47,800(1.12)t t = ln($326,500 / $47,800) / ln1.12 t = 16.95 years FV = $213,380 = $38,650(1.19)t t = ln($213,380 / $38,650) / ln 1.19 t = 9.82 years
diversification
is the process of reducing the risk of a portfolio by holding assets whose returns are not perfectly correlated. In a general sense, diversification 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 tend to cancel each other out, thus reducing the overall risk (variability) of the portfolio's overall expected return.
correlation coefficient
measures not only the direction of covariability (positive or negative) but also the strength of the relationship. the largest a correlation coeffecient can be is +1 which means the two stocks move exactly together. -1 means they move exactly opposite. correlation coeffecient = Covariance / (SD1)(SD2). as the correlation coefficient declines, the variance of the risk drops. correlation is not necessarily the problem, but market risk is!
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 of return on this sculpture?
r = (FV / PV)1 / t - 1 = ($10,311,500 / $12,377,500)1/4 - 1 = -.0446, or -4.46%
joint probability distribution,
shows how the returns of investment a and investment b vary over the possible states of the economy. Given the expected returns of these two companies and Dianne's decision on how much to invest in each company, she can determine the expected return of her portfolio, which is the wealth-weighted average of the returns expected from the assets held in the portfolio.
calculating rates of return. In 2014, an 1874 $20 double eagle sold for $15,000. What was the rate of return on this investment?
solving for r, we get: r = (FV / PV)1 / t - 1 r = ($15,000 / $20)1/140 - 1 r = .0484, or 4.84%
expected return
the average of the possible realized returns weighted by their probability of occurring.
compounding
the exponential increase in the value of an investment because interest 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. compounding allows a decision maker to restate a cash flow to the future.
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 increase in the winner's check over this period? If the winner's prize increases at the same rate, what will it be in 2045?
(using the future value formula to solve) Solving for r, we get: r = (FV / PV)1 / t - 1 = ($1,620,000 / $150)1/119 - 1 = .08117, or 8.117% Given the interest rate of 10.23%, 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
some stuff
- uncertainty exists because the realized return from an investment may be different from the estimated expected return - probability distributions estimate the expected return and the variance - portfolios are groups of assets. the portfolio expected return is the wealth weighted sum of the individual asset expected returns. - the variance-risk- of the portfolio is affected by the degree to which the individual asset returns are correlated - diversification is the reduction of portfolio risk by combining imperfectly correlated assets - diversification reduces, but does not eliminate risk - investors should be concerned about the market risk that they cannot diversity away - market risk determines the risk premium that is part of the opportunity cost = risk free rate + risk premium
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 flows.
The accounts payable decision
Diane is negotiating a new contract with TMI's web developers. TMI can pay $1,000 to the developers in two months or $1,100 in three months for their services. Diane knows the interest rate for this decision is 6%, but should she use discounting or compounding values? First, we determine the present value of each payment using: PV = FV/(1+r)^n the $1,000 payment has a present value of $890 and the $1,100 payment has a present value of $924. Then, we figure out the future value of each payment using: PV*(1+r)^n=FV. (Remember, the interest rate is 6%) The $1,000 payment in two months has a future value of $1,124 and the $1,100 payment in three months has a future value of $1,166. The present value calculation and the future value calculation suggest that paying $1,000 in two months is the better deal. $1,000 payment: $890 present value, $1,124 future value. $1,100 payment: $924 present value, $1,166 future value.
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 City Bank: The simple interest per year is: $8,100 × .06 = $486 So, after 10 years, you will have: $486 × 10 = $4,860 in interest. The total balance will be $8,100 + 4,860 = $12,960 With compound interest, we use the future value formula to get the future value, which includes getting your deposit back plus the accumulated interest on interest. FV = PV(1 +r)t FV = $8,100(1.06)10 FV = $14,505.87
Using the IRR rule we can identify:
Good projects where the rate of return earned (IRR) exceeds what we'd accept (the opportunity cost) Acceptable projects, where the rate of return earned (IRR) equals the rate we'd accept (the opportunity cost) Poor projects, where the rate of return earned (IRR) is less than the rate we'd accept (the opportunity cost)
Calculating Perpetuity Discount Rate. In the previous problem, suppose Curly's told you the policy costs $645,000. At what interest rate would this be a fair deal?
In this problem we're given all of the cash flows and asked to calculate the interest rate. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation we can now solve for the interest rate as follows:PV = C / r$645,000 = $30,000 / rr = $30,000 / $645,000 = .0465, or 4.65%
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. This is not a trivial question, as it does require an understanding of what an average return r
Returns and Standard Deviations. Consider the following information: Probability of boom = 60%, in that wase rate of return for stock a = 15%, stock b = 2%, and stock c = 34%. probability of bust = 40%, in that case rate of return on stock a = 3%, stock b = 16%, and stock 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% Another way to calculate these returns is to multiply each security's expected return by the weight of each asset (1/3, or .3333) in the portfolio. Boom: Rp = 1/3(.15) + 1/3(.02) + 1/3(.34) = .17, or 17% Bust: Rp = 1/3(.03) + 1/3(.16) + 1/3(-.08) = .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% 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
calculating the number of periods. You're 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 percent annual interest on its accounts. How long will it be before you have enough to buy the
Solving for t, we get: t = ln(FV / PV) / ln(1 + r) FV = $150,000 = $35,000(1.021)t t = ln($150,000 / $35,000) / ln 1.021 t = 70.02 years
capital asset pricing model (CAPM)
The expected return on the asset is what we're looking for--the opportunity cost of an asset--that the CAPM estimates. The riskfree rate of return: the basic time value interest rate for riskless assets. The average expected rate of return earned by an asset in the market. The market risk premium. This is the risk premium earned by the average asset in the market. The beta of the asset being evaluated. The risk premium for the asset being evaluated. The average risk of the market is adjusted by the risk of the asset. This is the risk premium that we've been using since Unit 5! While this looks a bit complex, it has an simple logic. An asset with average volatility will have a risk premium equal to that of the market. An asset with above average volatility will have a risk premium higher than that of the market as a whole. An asset with below average volatility will have a risk premium lower than that of the market as a whole. CAPM is a pricing model which helps us to evaluate a price to see if it is a good price in comparison with the future cash flows and their risk.
Calculating future values. Your coin collection contains fifty 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2063, assuming they appreciate at an annual rate of 4.3 percent?
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.
Calculating Annuity Present Values. 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? What would the value be if the payments occurred for 40 years? For 75 years? Forever?
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 $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 of the 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, we use the equation: PV = C / rPV = $5,430 /.08 PV = $67,875
Calculating Annuity Values. For each of the following annuities, calculate the present value. (using annuity payments, years, and interest rate)
for example annuity payments = $2,100, years = 7, and interest rate = 5% PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $2,100{[1 - (1 / 1.05)^7] / .05} PVA = $12,151.3 PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $1,095{[1 - (1 / 1.10)^9 ] / .10} PVA = $6,306.13 PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $11,000{[1 - (1 / 1.08)^18 ] / .08} PVA = $103,090.7 PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $30,000{[1 - (1 / 1.14)^28] / .14} PVA = $208,819.8
Calculating EAR. Find the EAR in each of the following cases. (using stated rate (APR), compounding period)
for example, if stated rate = 10% and compounding is quarterly. 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% APR of 17%, compounded monthly EAR = [1 + (.17 / 12)]^12 - 1 = .1839, or 18.39% APR of 13%, compounded daily EAR = [1 + (.13 / 365)]^365 - 1 = .1388, or 13.88% APR of 9%, compounded semiannually EAR = [1 + (.09 / 2)]^2 - 1 = .0920, or 9.20%
required rate of return v internal rate of return
if RRR = IRR we are being fairly compensated for risk. if RRR>IRR the investment is not a good deal. if RRR<IRR the investment is a good deal. remember RRR is the return we should receive on the investment given its risk- this is the opportunity cost. Internal rate of return (IRR) is the actual return on the investment
interest rate
interests rates are not in our control, they are affected by market conditions and represent opportunity costs, the downside of time value is that interest rates drop so do the future value of our investment.
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 $.10. What was the annual increase in the value of this comic book?
r = (FV / PV)1 / t - 1 = ($3,207,852 / $.10)1/76 - 1 = .2554, or 25.54%
diversifiable risk
a risk that is reduced as assets with imperfectly correlated rates of return are added to the portfolio
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? Explai
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.
The Award: An Ordinary Annuity Example. The financial payment will be made over four months at $900 per payment. The monthly interest rate is 1%, so how much is Amanda's reward worth today?
There are three ways that Dianne can value Amanda's bonus. 1) take the present values of the cash flows individually. 2) Use a formula 3) Use a calculator Calculating the present value of each of the five payments individually using the present value equation, we get 891.09, 882.27, 873.53, 864.88 to total out to $3,511.77 for the present value of her $3600 bonus. apparently there is also a PVA equation which looks really complicated but if there's not a long period you can individually calculate.
statistics
a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data
variance
a measure of how the possible realized returns might vary from the expected return calculated above.
perpetuity
a series of regular payments for regular periods that go on indefinitely. two types, level and growing
what happens to a future value if you increase the rate, r?
future values increase with longer time periods and higher interest rates, r. future value is positively related to interest rates. present values decrease with longer time periods and higher interest rates, r. present value is inversely related to interest rates.
level annuity
has the same cash flow for each period of time. Level annuities are further classified as to when the payments are made in each period.
annuity due
occur at the beginning of each period. Your landlord expects the rent at the beginning of each month. (a type of level annuity apparently)
ordinary annuity
occur at the end of each period, such as the salary a worker gets paid at the end of each month. (a type of level annuity apparently)
frequency distribution
shows the historic relationship between return and risk and the frequency with which each rate of return is earned.
growing perpetuity
the amounts grow at a constant rate. each cash flow is a growing percentage larger than the previous cash flow.
effective annual interest rate (EAR)
the annual interest rate that reflects the impact of intra-year compounding. For the effective annual interest rate, you apply the periodic interest rate for the number of compounding periods within a year, so it is effectively a combination of the stated annual interest rate and the periodic rate. EAR = [1+r/m]^m - 1
four steps in deciding on investment
1 determine the IRR of the asset IRR = ending value-beginning value/beginning value 2 set up the security market line, which gives the current markets risk/return relationship risk premium for one unit of market risk is 11%-7%=4% risk of S&P index minus government security 3 calculate the opportunity cost: the rate of return that assets of equivalent risk are earning. E(R) = Rf + Beta[E(Rm)-Rf] E(R) = .07 + 1.3 [.11-.07] E(R) =.112, assets with 1.3 B should earn 12.2%, this is the opportunity cost/discount rate to evaluate Epson service stock price. 4 decide because epson is producing 10% only we will decline the stock.
the home purchase example
4 variables involved: 1) the present amount of the deposit 2) the future amount we expect 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 If Diane gets a $10,000 inheritance and wants to use the money as a down payment on a home in 7 years, and her account pays 5% per year, and the down payment must be 10% of the house value, how much house can she afford? PV*(1+r)^n=FV $10,000 * (1.05)^7=$14,071 If the down payment is 10% of the houses total value, then Diane can afford a $140,710 house. Obviously, increasing the initial amount saved or increasing the amount of years interest compounds increases the expected return and ultimate home deposit.
Focusing on cash flows and NPV analysis:
A project that earns more than its opportunity cost will have a positive NPV and should be acceptable. A project that earns its opportunity cost will have a zero NPV. This is an acceptable project in that you are earning a rate of return sufficient to compensate you for the estimated risk of the project. A project that earns less than its opportunity cost will have a negative NPV and should be rejected, as it is expected to reduce wealth.
the legal settlement: calculating present value
As a part of a settlement a company, TMI will receive a cash payment in four years of $60,000. If TMI wanted the money right now and they ended into a contract to get paid out today, how much should TMI receive today in exchange for the $60,000 it will receive 4 years from now, with the annual interest rate being 4%? We must compute the present value of a future cash flow. Co=C4/(1+r)^n (This equation is. simple extension of computing a present value of a cash flow received at the end of the time period.) $51,288=$60,000/(1.04)^4 Thus, even though it is lower, $51,288 is the present value of the $60,000 which will be received in 4 years. If we delayed the payment for 2 years, the payment would be $55,473. If the interest rate rises (from 4% to 8% for example) the current payment would decrease
What happens to the future value of an annuity if you increase the rate, r? What happens to the present value?
As explained in the Key Concepts, 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.
multiple cash flows: which payment plan is better for TMI? (interest rate remains at 6%) Option 1: Cash Payment- TMI would pay SNAP $28,000 today. Option 2: Payment as Work Progresses- TMI would pay SNAP six payments totaling $31,000 as the system work is accomplished. (2,000 now, then 3,000 year 1, 4,000 year 2, 7,000 year 3, 7,000 year 4 and finally 8,000 year 5.)
Option 1: nominal contract payment today: $28,000 Option 2: Nominal contract value: $31,000 Economic total value of contract payments today $25,790. (ie. present value of 2,000 is 2,000 because were getting it today, 3,000 is 2,830, 4,000 is 3,650, 7,000 is 5,877, next 7,000 is 5,545, and 8,000 is 5,978. these numbers are based on the interest rate of 6%, the time of the payment, ie of year 5 its 8,000.) Diane like oh shit bitch the payment plan is better for us because we end up giving up less rn (the present value of the payment plan is $25,790 which is lower than the $28,000 payment today.
calculating the 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? calculating the number of periods. At 4.7 percent interest, how long does it take to double your money? To quadruple it?
SO basically we are solving for (r) the rate of return, in this example were gonna use the future value equation to start out. FV = PV(1 + r)t r = (FV / PV)1 / t - 1 = ($295,000 / $53,000)1/18 - 1 = .1001, or 10.01% 2ND PART To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. 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, double your money: FV = $2 = $1(1.047)t t = ln 2 / ln 1.047 t = 15.09 years FV = $4 = $1(1.047)t t = ln 4 / ln 1.047 t = 30.18 years
TMI's preferred stock example. TMI is considering financing an acquisition by issuing preferred stock. Nathan estimates that TMI would pay $2 in preferred dividends every period. Nathan also estimates the appropriate discount rate to be 4%. If TMI issues this preferred stock, the what is the economic value of the stock?
The issue of preferred stock would be a level perpetuity, where a set cash flow is paid each period into infinity. TMI would pay $2 in preferred dividends. The appropriate discount rate is 4%. Co = C/r = $2/.04. Given these factors, investors should be willing to pay $50 for this security. But how can an infinite amount have a finite value? This calculation is not added up multiples of $2, it is summing their present values as of today. Given the nature of discounting, the further out a cash flow is, the less its worth today. note that PV = C/r is the present value of a perpetuity while Co = C1/(1+r) is the present value of a single period cash flow!
compounding period
The length of time that passes before interest is recognized and added to the principle. 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.
Present Value. If you were an athlete negotiating a contract with a signing bonus of $1 million, 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. (because 1,000,000 divided by 10 is 100,00 each time). 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 paidThe 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 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 APR. Vandermark Credit Corp. wants to earn an effective annual return on its consumer loans of 14.2 percent per year. The bank uses daily compounding on its loans. What interest rate is the bank required by law to report to potential borrowers? Explain why this rate is misleading to an uninformed borrower.
The reported rate is the APR, so we need to convert the EAR to an APR as follows: Algebraically convert the EAR equation to solve for the APR EAR = [1 + (APR / m)]^m - 1 APR = m[(1 + EAR)^1/m - 1] = 365[(1.142)^1/365 - 1] = .1328, or 13.28% This is deceptive because the borrower is actually paying annualized interest of 14.2 percent per year, not the 13.28 percent reported on the loan contract. If you were borrowing $2,000: You think, given the APR that you're paying $2,000 x 13.28% = $265. You're actually paying $2,000 x 14.2% = $284.
The Award: An Annuity Due Example. Amanda has just been informed that her cash award, an ordinary annuity, is now an annuity due. Amanda is confused. Is she getting less money or more money because of this change?
We already knew that the present value of Amanda's ordinary annuity is $3,511.77. With the annuity due there are still four payments, but each payment is discounted for one less period. So, payment one comes at 0 instead of one on our timeline, otherwise known as right now. As each cash flow is closer to today, it has a larger present value which means that the annuity is due, which still has four payments over four periods, has a higher present value than its equivalent ordinary annuity. You have to adjust the ordinary annuity formula to make it work. Fortunately, this adjustment factor is (1+r). in the end, this increases Amandas bonus by $95. INSERT ANNUITY DUE FORMULA CHANGE SCREENCAP HERE
Using CAPM: Beta. 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?
We are given the values for the CAPM except for the β of the stock. We need to substitute these values into the CAPM, and solve for the β of the stock. One important thing we need to realize is that we are given the market risk premium. There are two ways to incorporate the market rate of return into the CAPM. The market rate of return is the expected return of the economy/market. As we cannot invest in the entire economy the CAPM uses a large market index as a proxy for the economy. When the market index—Dow Jones Industrials, S&P 500, MSCI or Wilshire 5000—rises and falls it reflects changes in the economy. The market risk premium is the expected return of the market minus the risk-free rate. This is the risk premium for bearing one unit of market risk: βi = 1, and is represented in the CAPM by: Market risk premium = [E(RM) - Rf] 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] × βi .114 = .037 + .071βi The beta is βi = 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 Using the standard equation for the CAPM we can use the market rate of return to get the same beta. E(Ri) = Rf + [E(RM) - Rf] × βi .114 = .037 + [.108 - .037] βi The beta is βi = 1.085 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.
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?
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.
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.
As seen in Unit 5, risk involves two returns: The expected return is future return, generally uncertain, that one expects to get from an investment. This is the return that is used in making most business decisions. 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 periods 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. 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. What are the portfolio weights for a portfolio that has 165 shares of Stock A that sell for $69 per share and 125 shares of Stock B that sell for $44 per share?
As the 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 = .6743 $16,885 Proportion invested in Stock B: xB = $5,500 = .3257 $16,885 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%
Portfolio Expected Return. 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?
Many portfolios have more than two assets. The expected return of a portfolio is the sum of the expected return of each asset weighted by its proportion in the portfolio. So, the expected return of the portfolio is: E(Rp) = .15(.10) + .40(.13) + .45(.15) = .1345, or 13.45%
diversifiable vs. nondiversifiable 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. 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.
Compounding periods: Christinas consumer loan. Christinas loan was for one year at 18%, compounded annually. When the loan matured Christina paid back the principal of $1,000 and the accumulated interest of $180.
FV = PV (1+r) = $1,000(1.18) = $1,180.00 The interest rate stated On an annual basis is 18%. With annual compounding, this is the rate that Christina paid on this loan. IRR = Ending value - Beginning value / beginning value 18% = $1,180 - $1,000 / $1,000 What if the FCC wanted to do quarterly compounding on her next loan? with compounding periods of less than a year, the period and interest rate must be consistent. number of periods = m, periodic interest rate for m periods = r/m. With quarterly compounding, there are 4 periods. For an annual rate of 18%, the periodic interest rate, the quarterly rate, is 4.5%. (18%/4=4.5%. FV=PV(1+r^m = $1,000(1.045)^4 = $1,000(1.1925)=$1,192.52. While this looks like only a minor change, Christinas yearly interest payment goes from $180 to $192, an increase of almost 7% !!!#$)@#$#@$(@#)()@
Calculating EAR. First National Bank charges 10.1 percent compounded monthly on its business loans. First United Bank charges 10.3 percent compounded semiannually. As a potential borrower, which bank would you go to for a new loan?
For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)]^m - 1 First National Bank First National Bank charges an APR of 10.1%, but compounds monthly. EAR = [1 + (.101 / 12)]^12 - 1 = .1058, or 10.58% First United Bank First United Bank charges a higher APR of 10.3% EAR = [1 + (.103 / 2)]^2 - 1 = .1057, or 10.57% For a borrower, First United would be preferred since the EAR of the loan is lower. Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR. A difference of 10.58% - 10.57% = 0.01% seems like a small amount. However, if your company was borrowing $500,000, you'd be paying $5,000 more in interest. 0.01% is called a point. In markets this is not a trivial amount!
Calculating APR. Find the APR, or stated rate, in each of the following cases. (using compounding period and EAR)
Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m)]^m - 1 We can now solve for the APR. Doing so, we get: APR = m[(1 + EAR)^1/m - 1] for example if compounding period is semiannually and EAR is 14%. EAR of 14%, compounded semiannually EAR = .14 = [1 + (APR / 2)]^2 - 1 APR = 2[(1.14)^1/2 - 1] = .1354, or 13.54% EAR of 9%, compounded monthly EAR = .09 = [1 + (APR / 12)]^12 - 1 APR = 12[(1.09)^1/12 - 1] = .0865, or 8.65% EAR of 8%, compounded weekly EAR = .08 = [1 + (APR / 52)]^52 - 1 APR = 52[(1.08)^1/52 - 1] = .0770, or 7.70% EAR of 13%, compounded daily EAR = .13 = [1 + (APR / 365)]^365 - 1 APR = 365[(1.13)^1/365 - 1] = .1222, or 12.22%
Calculating Annuity Cash Flows. For each of the following annuities, calculate the annual cash flow. using present value, years, and interest rate)
Here we're given four variables and asked to solve for the annual payments. This is the same process used when you're taking out a loan to purchase a car: given what you've borrowed, how long your note will be, and what rate will you be charged, what are your monthly payments? Using the PVA equation and solving for the payment in each case, we find: for example present value = $24,500, years =6, and interest rate = 11% 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 Second annuity PVA = C({1 - [1 / (1 + r)^t]} / r )) $19,700 = $C{[1 - (1 / 1.07)^8] / .07} C = $19,700 / 5.97130 C = $3,299.11 Third annuity PVA = C({1 - [1 / (1 + r)^t]} / r )) $136,400 = $C{[1 - (1 / 1.08)^15] / .08} C = $136,400 / 8.55948 C = $15,935.55 Fourth annuity PVA = C({1 - [1 / (1 + r)^t]} / r )) $286,650 = $C{[1 - (1 / 1 .06)^20] / .06} C = $286,650 / 11.46992 C = $24,904.2
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.
Annuity Present Values. Tri-State Megabucks Lottery advertises a $10 million grand prize. The winner receives $500,000 today and 19 annual payments of $500,000. A lump sum option of $5 million payable immediately is also available. Is this deceptive advertising?
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. The problem, in stating that you'll receive 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 y
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.
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.
what is the difference between standard deviation and variance?
Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance. The variance measures the average degree to which each point differs from the mean—the average of all data points. The two concepts are useful and significant for traders, who use them to measure market volatility
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 of Covid-19 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.
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 payed 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.
Calculating Portfolio Betas. 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?
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: βp = .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 (βp = .75) by reducing the investment in the high beta asset (βp = 1.76) will reduce the portfolio beta.
Systematic versus Unsystematic Risk. 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. 3. 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.
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.
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: %. 𝑅̅ = ∑(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.
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 that 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!
Using the CAPM: The opportunity cost. Jackne Business Services (JBS) provides IT, procurement, shipping and financial services to small retail companies in Central Texas. Their controller wants to calculate the discount rate to use in their capital budgeting projects. She's determined that the riskfree rate is 0.41%, the market rate of return is 7.25, and JBS' beta is 1.2. What is JBS' discount rate?
There are two steps to this problem: First, set up the Security Market Line. Second, determine JBS' discount rate. Set up the Security Market Line (SML). The SML is a visual representation of the CAPM. We'll build the SML on a step-by-step basis. First, set up the axes. Second, enter the market information where 7.25 is at the top and .41 is at the bottom. there is a 6.84% difference Third, draw the Security Market Line. The SML is a visual representation of the risk-return relationship in the market. It is a straight-line connecting the riskfree rate and the market rate of return on the Y-axis with their appropriate betas--0 for the riskfree rate and 1 for the market portfolio—on the X-axis. The risk premium for bearing one unit of market risk is E(RM) - RF. For this problem this market risk premium would be 7.25% -0 0.41% = 6.84%. Given the risk-return relationship in the market we can calculate the discount rate for any asset—all we need is the asset's beta. Assets that are less risky than the market have a beta less than one and will thus have a discount rate less than the market return of 7.25%. Assets that are riskier than the market have a beta greater than one and will thus have a discount rate greater than the market return of 7.25%.
TMI's common stock example. TMI's stock's next annual dividend is $3. Nathan expects the company's profits to increase, and knows that those profits go to the shareholders through regular payments of dividends. These dividend payments are expected to grow at a rate of 5% a year. If the discount rate for the stock is 15%, how much is the stock worth- What is its economic value? (as common stock is riskier than the preferred stock, it has a higher discount rate)
There is a pattern to the dividend cash flows: each dividend is 5% larger than the previous dividend. second dividend is $3(1.05)=$3.15, third period dividend $3.15(1.05 = $3.31... as this pattern extends indefinitely, the dividend payments fit the definition of a growing perpetuity. Co = $3 / .15-.05 = $30 Given its cash flows, their timing and risk investors should be willing to pay $30 for TMI's stock. Even though both common and preferred stock begin with the same period 1 cash flow of $3, the common stock dividends grows over time. But the cash flows of the common stock are more uncertain, and investors thus require a higher rate of return.
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? At 24 percent?
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 / 1.10^2 + $975 / 1.10^3 + $1,160 / 1.10^4 Present value at 18%: $2,119.91 = $680 / 1.18 + $490 / 1.18^2 + $975 / 1.18^3 + $1,160 / 1.18^4 Present value at 24%: $1,869.09 = $680 / 1.24 + $490 / 1.24^2 + $975 / 1.24^3 + $1,160 / 1.24^4
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? What is the future value at an interest rate of 11 percent? At 24 percent?
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 payments are as follows: $985, $1,160, $1,325, $1,495 at years 1, 2, 3, and 4 respectively. Future value at 8% $5,519.84 = $985(1.08)^3 + $1,160(1.08)^2 + $1, 325(1.08) + $1,495 Future value at 11% $5,742.10 = $985(1.11)^3 + $1,160(1.11)^2 + $1,325(1.11) + $1,495 Future value at 24% $6,799.64 = $985(1.24)^3 + $1,160(1.24)^2 + $1,325(1.24) + $1,495 Notice, since we are finding the value at Year 4, the cash flow at Year 4 is added to the FV of the other cash flows. In other words, we do not need to compound this cash flow. As with an individual cash flow, the future value of a series of cash flows increases as the interest rate increases.
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. Which of these cash flow streams has the higher present value if the discount rate is 6 percent? If the discount rate is 22 percent?
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} 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.
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. I practice there are very few negative beta assets.
Portfolio Risk. If a portfolio has a positive investment in every asset, can the standard deviation on the portfolio be less than that on every asset in the portfolio?
Yes, the standard deviation of the portfolio can be less than that of every asset in the portfolio. Standard deviation is nothing more that the square root of the variance, and is thus a measure of total risk. Diversification reduces the unique, or diversifiable, risk. With imperfectly correlated returns some of the variability of the individual assets is cancelled out.
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.
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?
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 and 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 that 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.
time value of money
all things being equal, it seems better to have money now rather than later- you can do much more with money you get today because you can earn more interest.
continuous probability distribution
contains an infinite number of observations which are analyzed using quantitative measures based on mathematical relationships and calculus. To allow us to concentrate on the logic of risk rather than having to develop sophisticated math skills we will use discrete probability distributions in our course.
Calculating asset expected return. Based on the following information, calculate the expected return. (using probability of state of the economy and rate of return)
for example if probability is .30 of recession and rate of return is -11% in that case. and if probability of boom is .70 and rate of return is 21%. We are given the following types of information in this probability distribution: There are two possible states of the economy: recession and boom. We expect to earn a higher rate of return when the economy is booming and a lower rate of return when the economy tanks. The expected return of an asset is the sum of the probability of each state occurring times the rate of return if that state occurs. So, the expected return of asset is: E(R) = .30(-.11) + .70(.21) = .1140, or 11.40% Note: developing a probability distribution (at least, one that you'd want to use) takes great insights into economics and how companies would respond given the state of the economy. In a booming economy we'd expect a company to earn a higher rate of return. In a recession we'd likely see a lower rate of return. However, we don't know what the future will be, so we use the expected value to make our decisions. As we will see in Lesson 2, this relationship between the economy as a whole and the returns of assets is a major factor in calculating the opportunity cost.
Solve for the unknown interest rate.
for instance when future value = $1,381, years = 11, and present value = $715 FV = $1,381 = $715(1 + r)11 r = ($1,381 / $715)1/11 - 1 r = .0617, or 6.17% .... FV = $1,718 = $905(1 + r)8 r = ($1,718 / $905)1/8 - 1 r = .0834, or 8.34% .... FV = $141,832 = $15,000(1 + r)23 r = ($141,832 / $15,000)1/23 - 1 r = .1026, or 10.26% .... FV = $312,815 = $70,300(1 + r)16 r = ($312,815 / $70,300)1/16 - 1 r = .0978, or 9.78%
present value
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.
Christina's Consumer Loan: Effective Annual Interest Rate
maturity of loan = 1 year, compounding period = quarterly, annual stated interest rate (APR) r = 18%, periodic interest rate: r/m = 18%/4 = 4.5% The effective annual rate is: EAR = [1+ .18/4]^-1 = [1.045]^4 - 1 = .1925 EAR = 19.25%
beta
measuring market risk, The beta of the market equals 1, because the market return is perfectly correlated with itself. A beta greater than 1 signifies that the asset return is more volatile than the market return. A beta less than 1 signifies that the asset return is less volatile than the market return. beta does not measure absolute risk, but rather measures risk (variability) relative to the market). What the beta calculation shows is that a riskier investment should earn a premium over the risk-free rate. The average market risk beta set at 1. Assets with betas above 1 are more risky, assets with betas less than 1 are less risky. Riskless assets have a beta of 0
annual percentage rate (APR)
the interest rate charged per period multiplied by the number of periods. Federal regulations require this rate to be disclosed to consumers but, as it does not recognize time-value compounding, it is not as useful as the EAR.
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.
discounting
the process of determining the value to day of an amount to be received in the future. Discounting allows a decision maker to restate a cash flow to the present.
probability distribution example
using the probability percentage of economic growth or decline x the stock price we can calculate the expected stock return. for instance, if the probability was 12% and the stock price was $140 = stock return of 41% Total expected return = average (probability of return)(return percentage) (.12)(.41) + (.15)(.31) + etc etc etc .= .1195 On average, Dianne Could expect to earn 11.95% on her investment . But Dianne realizes she will likely earn a return that is higher or lower than her expected return. she needs to know how volatile her investment would be: the level of dispersion she might see in realized returns. Variance = (.12)(.41-.1195)^2 + (.15)(.31-.1195)^2 + etc etc etc. = .0342 variability. the second bubble is the difference of realizes return from expected return. but .0342 is in squared percent while expected return is in percent, %. but we can convert the variability to a measure stated in percent, which is standard deviation by square rooting it. the square root of .0342 is .1849, or the standard deviation of this company stock.
