Time Value of Money Bond Valuation Risk and Return Stock Valuation WEB CHAPTER 28 Basic Financial Tools: A Review

Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

Time line for a $100 lump sum due at the end of Year Year i%

Time line for an ordinary annuity of $100 for 3 years i%

What’s the FV of an initial $100 after 1, 2, and 3 years if i = 10%? FV = ? % Finding FVs (moving to the right on a time line) is called compounding. 100FV = ?

After 1 year: FV 1 = PV + INT 1 = PV + PV (i) = PV(1 + i) = $100(1.10) = $ After 2 years: FV 2 = PV(1 + i) 2 = $100(1.10) 2 = $

After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 = $ In general, FV n = PV(1 + i) n.

What’s the FV in 3 years of $100 received in Year 2 at 10%? % 110

What’s the FV of a 3-year ordinary annuity of $100 at 10%? % FV= 331

NI/YRPVPMTFV Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT

% What’s the PV of $100 due in 2 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding PV = ?

Solve FV n = PV(1 + i ) n for PV:   PV= $ = $100PVIF = $ = $ i,n       2

What’s the PV of this ordinary annuity? % = PV

Have payments but no lump sum FV, so enter 0 for future value NI/YRPVPMTFV INPUTS OUTPUT

How much do you need to save each month for 30 years in order to retire on $145,000 a year for 20 years, i = 10%? PMT months before retirementyears after retirement -145k...

How much must you have in your account on the day you retire if i = 10%? How much do you need on this date? years after retirement -145k... 0

You need the present value of a 20- year 145k annuity--or $1,234, NI/YRPVFV PMT 1,234,467 INPUTS OUTPUT

How much do you need to save each month for 30 years in order to have the $1,234,467 in your account? You need $1,234,467 on this date PMT... months before retirement...

You need a payment such that the future value of a 360-period annuity earning 10%/12 per period is $1,234, / NI/YRPVFV PMT INPUTS OUTPUT It will take an investment of $ per month to fund your retirement.

Key Features of a Bond 1.Par value: Face amount; paid at maturity. Assume $1, Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed. (More…)

Maturity: Years until bond must be repaid. Declines. 4.Issue date: Date when bond was issued.

NI/YR PV PMTFV -1,000 The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10: $ $1, PV annuity PV maturity value PV annuity ====== INPUTS OUTPUT

NI/YR PV PMTFV When r d rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount. What would happen if expected inflation rose by 3%, causing r = 13%? INPUTS OUTPUT

What would happen if inflation fell, and r d declined to 7%? NI/YR PV PMTFV -1, If coupon rate > r d, price rises above par, and bond sells at a premium. INPUTS OUTPUT

The bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

M Bond Value ($) Years remaining to Maturity 1,372 1,211 1, r d = 7%. r d = 13%. r d = 10%.

At maturity, the value of any bond must equal its par value. The value of a premium bond would decrease to $1,000. The value of a discount bond would increase to $1,000. A par bond stays at $1,000 if r d remains constant.

Assume the Following Investment Alternatives Economy Prob.T-BillHTCollUSRMP Recession %-22.0% 28.0% 10.0%-13.0% Below avg Average Above avg Boom

What is unique about the T-bill return? The T-bill will return 8% regardless of the state of the economy. Is the T-bill riskless? Explain.

Do the returns of HT and Collections move with or counter to the economy? HT moves with the economy, so it is positively correlated with the economy. This is the typical situation. Collections moves counter to the economy. Such negative correlation is unusual.

Calculate the expected rate of return on each alternative. r = expected rate of return. r HT = 0.10(-22%) (-2%) (20%) (35%) (50%) = 17.4%. ^ ^

r HT17.40% Market15.00 USR13.80 T-bill8.00 Collections1.74 ^ HT has the highest rate of return. Does that make it best?

What is the standard deviation of returns for each alternative? = Standard deviation..

 T-bills = 0.0%.  HT = 20.0%.  Coll =13.4%.  USR =18.8%.  M =15.3%.. = (( ) ( ) ( ) ( ) ( ) ) 1/2 = 20.0%. HT:

The coefficient of variation (CV) is calculated as follows: CV HT = 20.0%/17.4% = 1.15  1.2. CV T-bills = 0.0%/8.0% = 0. CV Coll = 13.4%/1.74% = 7.7. CV USR = 18.8%/13.8% = 1.36  1.4. CV M = 15.3%/15.0% = 1.0.  /r. ^

Prob. Rate of Return (%) T-bill US R HT

Standard deviation measures the stand-alone risk of an investment. The larger the standard deviation, the higher the probability that returns will be far below the expected return. Coefficient of variation is an alternative measure of stand-alone risk.

Expected Return versus Risk Security Expected return Risk,  HT 17.4% 20.0%1.2 Market USR T-bills Collections Which alternative is best? CV

Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections. Calculate r p and  p. ^

Portfolio Return, r p r p is a weighted average: r p = 0.5(17.4%) + 0.5(1.74%) = 9.6%. r p is between r HT and r Coll. ^ ^ ^ ^ ^^ ^^ r p =   w i r i  n i = 1

Alternative Method r p = (3.0%) (6.4%) (10.0%) (12.5%) (15.0%)0.10 = 9.6%. ^ Estimated Return EconomyProb.HTColl.Port. Recession % 28.0% 3.0% Below avg Average Above avg Boom (More...)

 p = (( ) ( ) ( ) ( ) ( ) ) 1/2 = 3.3%.  p is much lower than: either stock (20% and 13.4%). average of HT and Coll (16.7%). The portfolio provides average return but much lower risk. The key here is negative correlation.

Portfolio standard deviation in general  p = Portfolio standard deviation. Where w 1 and w 2 are portfolio weights and r 1,2 is the correlation coefficient between stock 1 and 2.

Two-Stock Portfolios Two stocks can be combined to form a riskless portfolio if r = Risk is not reduced at all if the two stocks have r = In general, stocks have r  0.65, so risk is lowered but not eliminated. Investors typically hold many stocks. What happens when r = 0?

Portfolio beta b p = Portfolio beta Where w 1 and w 2 are portfolio weights, and b 1 and b 2 are stock betas. For our portfolio of 50% HT and 50% Collections, b p = 0.5(1.30) + 0.5(-0.87) =  b p = w 1 b 1 + w 2 b 2

What would happen to the riskiness of an average portfolio as more randomly picked stocks were added?  p would decrease because the added stocks would not be perfectly correlated, but r p would remain relatively constant. ^

Large 0 15 Prob. 2 1  1  35% ;  Large  20%. Return

# Stocks in Portfolio ,000+ Company-Specific (Diversifiable) Risk Market Risk 20 0 Stand-Alone Risk,  p  p (%) 35

Stand-alone Market Diversifiable Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification. Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification. risk risk risk = +.

Conclusions As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.  p falls very slowly after about 40 stocks are included. The lower limit for  p is about 20% =  M. By forming well-diversified portfolios, investors can eliminate about half the riskiness of owning a single stock.

No. Rational investors will minimize risk by holding portfolios. They bear only market risk, so prices and returns reflect this lower risk. The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk. Can an investor holding one stock earn a return commensurate with its risk?

Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio. It is measured by a stock’s beta coefficient, which measures the stock’s volatility relative to the market. What is the relevant risk for a stock held in isolation? How is market risk measured for individual securities?

How are betas calculated? Run a regression with returns on the stock in question plotted on the Y- axis and returns on the market portfolio plotted on the X-axis. The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.

Yearr M r i 115% 18% riri _ rMrM _ Illustration of beta calculation: Regression line: r i = r M. ^^ Beta Illustration

How is beta calculated? The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = Analysts typically use five years’ of monthly returns to establish the regression line.

If b = 1.0, stock has average risk. If b > 1.0, stock is riskier than average. If b < 1.0, stock is less risky than average. Most stocks have betas in the range of 0.5 to 1.5. Can a stock have a negative beta? How is beta interpreted?

HT T-Bills b = 0 riri _ rMrM _ b = 1.30 Collections b = Regression Lines of Three Alternatives

Expected Return versus Market Risk Security Expected return Risk,  b HT17.4%1.30 Market USR T-bills Collections Which of the alternatives is best?

Use the SML to calculate each alternative’s required return. The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM). SML: r i = r RF + (r M - r RF )b i. Assume k RF = 8%; r M = r M = 15%. RP M = r M - r RF = 15% - 8% = 7%. ^

Required Rates of Return r HT = 8.0% + (15.0% - 8.0%)(1.30) = 8.0% + (7%)(1.30) = 8.0% + 9.1%= 17.1%. r M = 8.0% + (7%)(1.00)= 15.0%. r USR = 8.0% + (7%)(0.89)= 14.2%. r T-bill = 8.0% + (7%)(0.00)= 8.0%. r Coll = 8.0% + (7%)(-0.87)= 1.9%.

Expected versus Required Returns ^ r r HT 17.4% 17.1% Undervalued Market Fairly valued USR Overvalued T-bills Fairly valued Coll Overvalued

Coll.. HT T-bills. USR r M = 15 r RF = SML: r i = 8% + (15% - 8%) b i. r i (%) Risk, b i SML and Investment Alternatives Market

What is the required rate of return on the HT/Collections portfolio? r p = Weighted average r = 0.5(17%) + 0.5(2%) = 9.5%. Or use SML: b p = 0.22 (Slide 28-45) r p = r RF + (r M - r RF ) b p = 8.0% + (15.0% - 8.0%)(0.22) = 8.0% + 7%(0.22) = 9.5%.

One whose dividends are expected to grow forever at a constant rate, g. Stock Value = PV of Dividends What is a constant growth stock?.

For a constant growth stock, If g is constant, then:....

$ 0.25 Years (t) 0 If g > r, P 0 = negative

What happens if g > r s ? If r s < g, get negative stock price, which is nonsense. We can’t use model unless (1) g  r s and (2) g is expected to be constant forever. Because g must be a long- term growth rate, it cannot be  r s.

Assume beta = 1.2, r RF = 7%, and r M = 12%. What is the required rate of return on the firm’s stock? r s = r RF + (r M - r RF )b Firm = 7% + (12% - 7%) (1.2) = 13%. Use the SML to calculate r s :

D 0 was $2.00 and g is a constant 6%. Find the expected dividends for the next 3 years, and their PVs. r s = 13% g = 6% D 0 = % 2.12

What’s the stock’s market value? D 0 = 2.00, r s = 13%, g = 6%. Constant growth model: = = = $ $ = D 0 (1 + g) r s - g D 1 r s - g

Rearrange model to rate of return form: Then, r s = $2.12/$ = = 13%. ^

If we have supernormal growth of 30% for 3 yrs, then a long-run constant g = 6%, what is P 0 ? r s is still 13%. Can no longer use constant growth model. However, growth becomes constant after 3 years. ^

Nonconstant growth followed by constant growth: r s =13% = P 0 g = 30% g = 6% D 0 = ^