Finance Ch. 7-10

If you invest $2,000 per year for the next 15 years at a 6% annual interest rate, beginning 1 year from today compounded annually, how much are you going to have at the end of the 15th year?

$2,000 X [((1+.06)15 - 1)/.06] = $2,000 X 23.27596989 = $46,551.94

You deposit $2,000 in an account that pays 8% annual interest, compounded annually. How long will it take to triple your money?

$2,000 x 1.08n = $6,000 1.08n = 3 n = ln(3)/ln(1.08) = 14.2749

The Apple stock you purchased 12 years ago for $23 a share is now worth $535.86. What is the compounded annual rate of return you have earned on this investment?

$23.00 = $535.86 X [PVFk%, 12 years ] .0429 = PVFk%, 12 years; k = 30.00%

Dolph Starbeam will deposit $5,000 at the beginning of each year for 20 years into an account that has an annual interest rate of 4%. How much will Dolph have to withdraw in 20 years?

$5,000 X [(1.0420 - 1)/0.04] X 1.04 = $154,846.01

If your required annual rate of return is 6%, how much will an investment that pays $50 a year at the beginning of each of the next 40 years be worth to you today?

$50 X [(1-1/1.0640)/.06] X 1.06 = $797.45

Two methods to determine cost of common equity

-Dividend growth model -Capital asset pricing model

Factors that affect the cost of capital

-General economic conditions (affect interest rates) -Market conditions (affect risk premiums) -Operating decisions (affect business risk) -Financial decisions (affect financial risk) -Amount of financing (affects flotation costs and market price of security)

Cost of internal common equity

-Management should retain earnings only if they earn as much as stockholders' next best investment opportunity of the same risk -Cost of internal equity = opportunity cost of common stockholder's funds

present value of a single amount

-PV= FV/(1+k)^n -value today of an amount to be received or paid in the future

Two types of common equity financing

-Retained earnings (internal common equity) -Issuing new shares of common stock (external common equity)

market-related risk

-Risk due to overall market conditions -measures how individual stock returns are affected by this market -measured by beta

diversification

-Spreading out investments to reduce risk -if investors hold stock in many companies, the firm-specific risk will be cancelled out

Marginal Cost of Capital

-WACC of the next dollar of capital raised -when graphing, breakpoint (or point where new equity will need to be issued) = available retained earnings/percentage of total

standard deviation

-a measure of uncertainty of future returns -square root of variance -standard deviation = SQRT[sum(P(V-mean)^2]

Perpetuity

-a series of equal payments at equal time intervals (an annuity) that will be received into infinity -PVP = PMT/k

Crowdfunding

-companies raise capital from a large number of different people who have a connection with the company -usually done through a website

Continuous Compounding

-compounding frequency is infinitely large; compounding period is infinitely small -FV = PV x e^kn

Internal Rate of Return (IRR)

-discount rate that forces the NPV to equal zero -rate of return on project given its initial investment and future cash flows -rate earned only if all CFs are reinvested at IRR rate -accept the project if IRR >= to required rate of return (k) and reject otherwise

Modified Internal Rate of Return (MIRR)

-discount rate which causes project's PV of outflows to equal project's TV (terminal value) of inflows -PV outflow = TV inflows/(1 + MIRR)^n -assumes cash inflows are reinvested at k, the cost of capital

Non-simple projects

-have one or more negative future cash flows occurring after the initial investment -could have as many IRRs as there are sign changes -if a project has more than one IRR, use the NPV method for project accept/reject decisions

Capital Asset Pricing Model (CAPM)

-measures required rate of return for investments, given the degree of market risk measured by beta -kj = kRF + bj(kM - kRF) where kj = required rate of return on the jth security, kRF = risk free rate of return, kM = required rate of return on the market, and bj = beta for the jth security

Net present value

-present value of all costs and benefits (measured in terms of incremental cash flows) of a project -NPV = PV of inflows - initial investment = CF1/(1 + k)^1 + CF2/(1+k)^2 + ... + CFn/(1 + k)^n - initial investment -if NPV > 0, then we accept it

systemic risk

-the failure of one company can lead to the failure of others -aka contagion -cannot be diversified away -sometimes different groups of assets go up and down together in value

Expected return

-the mean value of the probability distribution of possible returns -expected return = sum(Vi x Pi) where Vi = possible value of return during period i and Pi = probability of V occurring during period i

Capital budgeting

-the process of evaluating proposed investment projects for a firm -managers must determine which projects are acceptable and must rank mutually exclusive projects by order of desirability to the firm

Beta

-the slope of the regression (characteristic) line -if beta=1, there is average risk for the company -if beta<1, low risk company and return on stock will be less affected by the market than average -if beta>1, high market risk company and stock return will be more affected by the market than average

Jay Lo Enterprises finances its assets with 60% debt, 10% preferred stock, and 30% common stock. Jay Lo's after-tax cost of debt is 5%, its cost of preferred stock is 8%, and its cost of common equity financing is 12%. Given these conditions, what is Jay Lo's WACC?

0.60(0.05) + 0.10(0.08) + 0.30(0.12) = 0.074 = 7.4%

Four Methods of Capital Budgeting Analysis

1. payback period (years to recoup the initial investment) 2. net present value (NPV) (change in value of firm if project is undertaken) 3. internal rate of return (IRR) (projected percent rate of return project will earn) 4. modified internal rate of return (MIRR)

Steps for calculating cost of capital model

1.) Compute cost of each source of capital (after tax cost)(ATkd = before tax cost(1-tax rate)) 2.) Determine percentage of each source of capital in the optimal capital structure (kp = Dividend (Dp)/Market Price (Pp) - F) 3.) Calculate Weighted Average Cost of Capital (WACC)

Amortized loans

A loan that is paid off in equal amounts that include principal as well as interest

Alvin C. York, the founder of York Corporation, thinks that the optimal capital structure of his company is 30% debt, 15% preferred stock, and the rest common equity. If the company has a 25% interest subsidy tax rate, compute its weighted average cost of capital given that: -YTM of its debt is 10% -New preferred stock will have a market value of $31, a dividend of $2 per share, and flotation costs of $1 per share. -Price of common stock is currently $100 per share, and new common stock can be issued at the same price with flotation costs of $4 per share. The expected dividend in one year is $4 per share and the growth rate is 6%. Assume the addition to retained earnings for the current period is zero.

AT kd = 0.10(1-0.25) = 7.5% kp = $2/($31 - $1) = 6.67% kn = $4/($100 -$4) + .06 = 10.17% ka = (0.3)(6) + (0.15)(6.67) + (0.55)(10.17) = 8.394%

Amy Jolly is the treasurer of her company. She expects the company will grow at 4% in the future and debt securities (YTM = 14%, tax rate = 30%) will always be a cheaper option to finance the growth. The current market price per share of its common stock is $39 and the expected dividend in one year is $1.50 per share. Calculate the cost of the company's retained earnings and check if Amy's assumption is correct.

AT kd = 0.14(1-.30) = 9.80% ks = ($1.50/$39.00) + 0.04 = 7.85% The cost of the company's retained earnings is lower. This would lead you to reevaluate the estimated numbers, or question the applicability of the valuation models used here, since ks cannot be lower than AT kd for a given company.

future value of a single amount

FV = PV (1 + k)^n

What is the present value of a $2,000 10-year annual ordinary annuity at a 4% annual discount rate?

PVA = $2,000 X [(1-1/1.0410)/0.04] = $16,221.79

Beginning a year from now, Clancy Wiggum will receive $80,000 a year from his pension fund. There will be 15 of these annual payments. What is the present value of these payments if a 3% annual interest rate is applied as the discount rate?

PVOA = 80,000 X [(1-1/1.0315)/0.03] = $955,034.81

Upon reading your most recent credit card statement, you are shocked to learn that the balance owed on your purchases is $4,000. Resolving to get out of debt once and for all, you decide not to charge any more purchases and to make regular monthly payments until the balance is 0. Assuming that the bank's credit card annual interest rate is 19.5% and the most you can afford to pay each month is $350, how long will it take you to pay off your debt?

PVOA = PMT X PVFOA k,n $4,000 = $350 X PVFOA k=.195/12, n=? 11.4286 = PVFOA k=.01625, n=? n = 12.75 months

firm-specific risk

Risk due to factors within the firm

Annuity due

The cash payments occur at the BEGINNING of each time period

Ordinary annuity

The cash payments occur at the END of each time period

What is the expected rate of return on a portfolio that has $4,000 invested in Stock A and $6,000 invested in Stock B? The expected rates of return on these two stocks are 13% and 9%, respectively.

Total Portfolio = $10,000 Weights: Stock A: 4,000/10,000 = .4 Stock B: 6,000/10,000 = .6 .4(13) + .6(9) = 10.6% Expected Rate of Return = 10.6%

Weighted Cost of Capital

WACC = ka = (WTd x ATkd) + (WTp x kp) + (WTs x ks)

Calculate the after-tax cost of debt for loans with the following effective annual interest rates and interest subsidy tax rates. a.) Interest rate, 10%; interest subsidy tax rate, 0% b.) Interest rate, 10%; interest subsidy tax rate, 22% c.) Interest rate, 10%; interest subsidy tax rate, 25%

a ) AT kd = 10% x (1-.00) = 10.0% b ) AT kd = 10% x (1-.22) = 7.8% c ) AT kd = 10% x (1-.25) = 7.5%

A firm is issuing new bonds that pay 8% annual interest. The market required annual rate of return on these bonds is 13%. The firm has an interest subsidy rate of 25%. a.) What is the before-tax cost of debt? b.) What is the after-tax cost of debt?

a ) kd = 13% b ) AT kd = 13% x (1-.25) = 9.75%

African Queen River Tours, Inc. has capitalized on the renewed interest in riverboat travel. Charlie Allnut, the lone financial analyst, estimates the firm's earnings, dividends, and stock price will continue to grow at the historical 5% rate. AQRT's common stock is currently selling for $30 per share. The dividend just paid was $2. They pay dividends every year. The rate of return expected on the overall stock market is 12%. a.) What is AQRT's cost of equity? b.) If they issue new common stock today and pay flotation costs of $2 per share, what is the cost of new common equity? c.) If AQRT has a risk-free rate of 3% and a beta of 1.4, what will be AQRT's cost of equity using the CAPM approach?

a ) ks = ($2 X 1.05)/$30 + .05 = 12.0% b ) kn = ($2 X 1.05)/($30-$2) + .05 = 12.5% c ) ks = .03 + (.12 - .03) x 1.4 = 15.6%

Annuity

a series of equal cash flows spaced evenly over time

a.) What would be the after-tax cost of debt for a company with the following yields to maturity for its new bonds, if the applicable interest subsidy tax rate 20%? 1.) YTM = 7% 2.) YTM = 11% 3.) YTM = 13% b.) How would the cost of debt change if the applicable interest subsidy tax rate were 25%?

a) (i) YTM = 7% AT kd = 7% x (1-.20) = 5.6% (ii) YTM = 11% AT kd = 11% x (1-.20) = 8.8% (iii) YTM = 13% AT kd = 13% x (1-.20) = 10.4% b) (i) YTM = 7% AT kd = 7% x (1-.25) = 5.25% (ii) YTM = 11% AT kd = 11% x (1-.25) = 8.25% (iii) YTM = 13% AT kd = 13% x (1-.25) = 9.75%

Calculate the future value of a 12-year, $15,000 annual ordinary annuity, using an annual interest rate of: a.) 0% b.) 5% c.) 10% d.) 15%

a) 0% $15,000 X 12 = $180,000 b) 5% $15,000 X [(1.0512-1)/0.05] = $238,756.90 c) 10% $15,000 X [(1.1012-1)/0.10] = $320,764.26 d) 15% $15,000 X [(1.1512-1)/0.15] = $435,025.01

Calculate the present value of $15,000 to be received 20 years from now at an annual discount rate of: a.) 0% b.) 5% c.) 10% d.) 20%

a) 0% $15,000 X [1/(1+0.00)20] = $15,000.00 b) 5% $15,000 X [1/(1+0.05)20] = $5,653.34 c) 10% $15,000 X [1/(1+0.10)20] = $2,229.65 d) 20% $15,000 X [1/(1+0.20)20] = $391.26

Calculate the future value of $20,000 ten years from now if the annual interest rate is: a.) 0% b.) 2% c.) 5% d.) 10%

a) 0% $20,000 X (1 + 0.00)10 = $20,000.00 b) 5% $20,000 X (1 + 0.02)10 = $24,379.89 c) 10% $20,000 X (1 + 0.05)10 = $32,577.89 d) 20% $20,000 X (1 + 0.10)10 = $51,874.85

Calculate the future value of $40,000 at 5% for the following years: a.) 5 years b.) 10 years c.) 15 years d.) 20 years

a) 5 years $40,000 X (1 + 0.05)5 = $51,051.26 b) 10 years $40,000 X (1 + 0.05)10 = $65,155.79 c) 15 years $40,000 X (1 + 0.05)15 = $83,157.13 d) 20 years $40,000 X (1 + 0.05)20 = $106,131.91

Twister Corporation is expected to pay a dividend of $7 per share one year from now on its common stock, which has a current market price of $143. Twister's dividends are expected to grow at 13%. a.) Calculate the cost of the company's retained earnings. b.) If the flotation cost per share of new common stock is $4, calculate the cost of issuing new common stock.

a) ks = ($7/$143) + 0. 13 = 17.90% b) kn = ($7/($143 - $4) + 0. 13 = 18.04%

Otto Mann has his heart set on a new Miata sports car. He will need to borrow $28,000 to get the car he wants. The bank will loan Otto the $28,000 at an annual interest rate of 3%. a.) How much would Otto's monthly car payments be for a 4-year loan? b.) How much would Otto's monthly car payments be if he obtains a 6-year loan at the same interest rate?

a) n = 4 X 12 = 48 k = .03/12 = .0025 or 0.25% $28,000 = PMT X [(1-1/1.002548)/0.0025] PMT = $28,000/45.17869463 = $619.76 b) n = 6 X 12 = 72 k = .03/12 =0 .0025 or 0.25% $28,000 = PMT X [(1-1/1.002572)/0.0025] PMT = $28,000/65.81685774 = $425.42

Calculate the present values of the following using a 4% annual discount rate at the end of 15 years: a.) $20,000 b.) $60,000 c.) $90,000 d.) $130,000

a)$20,000 $20,000 X [1/(1 + 0.04)15] = $11,105.29 b)$60,000 $60,000 X [1/(1 + 0.04)15] = $33,315.87 c)$90,000 $90,000 X [1/(1 + 0.04)15] = $49,973.81 d)$130,000 $130,000 X [1/(1 + 0.04)15] = $72,184.39

Higher risk, use a ___________ discount rate and lower risk, use a ____________ discount rate

higher; lower

Calculate the expected rates of return for the low-, average-, and high-risk stocks: a.) Risk-free rate = 4.5% b.) Market risk premium = 12.5% c.) Low-risk beta = .5 d.) Average-risk beta = 1.0 e.) High-risk beta = 1.6

kl = 4.5 + .5(12.5) = 10.75% ka = 4.5 + 1.0(12.5) = 17% kh = 4.5 + 1.6(12.5) = 24.5%

Cost of New Common Stock

kn = (D1/P0 - F) + g

A company can sell preferred stock for $26 per share and each share of stock is expected to pay a dividend of $2. If the flotation cost per share is $0.75, what would be the estimate of the cost of capital from this source?

kp = $2/($26 - $0.75) = $2/$25.25 = 7.92%

Dividend growth model

ks = (D1/P0) + g (D = dividend, P = market price, g = expected growth rate)

Free Willy, Inc., has a beta of 1.4. If the rate on U. S. Treasury bills is 4.5% and the expected rate of return on the stock market is 12%, what is Free Willy's cost of common equity financing?

ks = .045 + 1.4(.12 - .045) = 15%

Capital Asset Pricing Model

ks = kRF + beta(kM - kRF) (kRF = risk-free rate, kM = expected market return)

Titania wants to borrow $225,000 from a mortgage banker to purchase a $300,000 house. The mortgage loan is to be repaid in monthly installments over a 30-year period. The annual interest rate is 6%. How much will Titania's monthly mortgage payments be?

n = 30 X 12 = 360 k = .06/12 =0 .005 or 0.5% $225,000 = PMT X [(1-1/1.005360)/0.005] PMT = $225,000/166.7916144 = $1,348.99

Capital Rationing

practice of placing a dollar limit on the total size of the capital budget

risk

the potential for unexpected events to occur