FIN 485 Exam 3: Module 3
Multi-factor models
- Arbitrage and APT - Fama-French 3 Factor Model
Duration/Price Relationship
- Price change is proportional to duration and NOT to maturity - D* represents interest rate elasticity of a bond's price
As yields rise, prices fall:
A property of bond prices is convexity: - increase in interest rates results in a price decline that is smaller than price gain from a decrease of equal magnitude in the interest rate
alpha (ai)
abnormal rate of return on security in excess of that predicted by equilibrium model (CAPM) - the excess return will be proportional to beta
Debenture
bond not backed by specific collateral
Interest rate sensitivity
bond prices and interest rates (or yields) are inversely related
premium
bonds selling above par value
discount
bonds selling below par value
Passive investment strategy
buying well-diversified portfolio without attempting to find misplaced securities
puttable bonds
holder may choose to exchange for par value or to extend for given number of years; often multiple sell dates throughout the life of a putable bond - bond ----> cash (whenever the holder wants)
A bond with an annual coupon rate of 5.7% sells for $950. What is the bond's current yield?
$1,000 x 0.057 = $57 Current Yield = Annual Coupon Payment/Price = $57/$950 = 0.06 = 6%
YTM vs. Expected Yield
- Stated Yield/YTM assumes bond won't default - Expected Yield takes default risk into account - If Ford is about to go bankrupt, would you pay more or less for their bonds? Less, and as price goes down, yields go up.
Yield Spread
- The difference between the corporate bond yield and the treasury yield - includes state tax premium, illiquidity premium, and default risk premium
Price-Yield Relationship
- inverse relationship - as rates decrease, prices go up - bond will be called as it approaches the callable price
CAPM assumptions
- perfect markets - investors are homogeneous
Duration Example #1: You have an investment that in today's dollars returns: - 15% of your investment in year 1 - 12% in year 2 - 9% in year 3 - remainder in year 4 What is the duration of this investment?
0.15 x 1 = 0.15 0.12 x 2 = 0.24 0.09 x 3 = 0.27 0.64 x 4 = 2.56 = 3.22
Duration is the term for the effective maturity of a bond:
1. Time value of money tells us we must calculate the present value of each of the five zero-coupon bonds to construct an average. 2. We then need to take the present value of each zero and divide it by the price of the coupon bond. This tells us what percentage of our money we get back each year. 3. We can now construct the weighted average of the times until each payment is received.
Nominal Return
= Interest + Price Appreciation/Initial Price
Real Return
= [(1 + Nominal)/(1 + Inflation)] -1
Accrued Interest Example: What is the accrued interest on a 10% (annual rate) semiannual coupon bond 30 days after its last coupon payment? Assume a 364 day year.
Accrued interest = (annual coupon payment/coupon payments per year) x (days since last coupon/days between coupons) (0.10 x $1,000)/2 = $50 364/2 = 182 days $50 x 30/182 = $8.24
Some individuals contend that professional managers are incapable of continually outperforming the market. Others believe that professional managers are able to continually outperform the market. What are the assumptions in support of passive portfolio management?
Assumptions supporting passive management are: 1) informational efficiency and 2) primacy of diversification motives. Active management is supported by the opposite assumptions, in particular, that pockets of market inefficiency exist.
Yield to Call Example: What is the yield to call for a 20 year annual coupon bond (rate = 6%), with a par value of $1,000, callable in 10 years at $1,100, and currently selling at $1,055.84?
N = 20 I/Y = ? PV = -1,055.84 PMT = 60 FV = 1000 I/Y = 5.53% for YTM N = 10 I/Y = ? PV = -1,055.84 PMT = 60 FV = 1100 I/Y = 6% for YTC
Here are data on two companies. The T-bill rate is 4.4% and the market risk premium is 8.6%. Company: $1 Discount Store: Everything $5 Forecast Return: 15%: 14% Standard Deviation of Returns: 21%: 23% Beta: 1.40: 1.00 What would be the fair return for each company, according to the capital asset pricing model (CAPM)?
Company: Expected Return $1 Discount Store: 16.44% Everything $5: 13.00% E(r) = rf + B[E(rm) - rf] rf = 4.4% [E(rm) - rf] = 8.6% $1 Discount Store: E(r) = 0.044 + 1.40(0.086) = 0.1644 Everything $5: E(r) = 0.044 + 1.00(0.086) = 0.1300
Zero-Coupon Bond Example: What is the price of a 30-year zero-coupon bond yielding 15%?
N = 30 I/Y = 15 PV = ? PMT = 0 FV = 1000 PV = $15.10
Determinants of bond safety:
Coverage Ratios: - company earnings to fixed costs Leverage Ratio: - debt to equity Liquidity Ratios: - current: current assets to current liabilities - quick: assets excluding inventories to liabilities Profitability Ratios: - measures of ROR on assets or equity (ROA, ROE) Cash Flow-to-Debt Ratio
Security Alpha Example: Suppose the return on the market is expected to be 13%. We are analyzing a stock with a beta of 1.3, and the risk-free rate (T-bill) rate is 4%. The SML predicts an expected return of what for this stock?
E(rm) = 13%; B = 1.3; rf = 4% E(ri) = rf + Bi[E(rm) - rf] E(ri) = 0.04 + 1.3(0.13 - 0.04) E(ri) = 0.04 + 1.3(0.09) E(ri) = 0.04 + 0.117 E(ri) = 0.157 = 15.70%
What must be the beta of a portfolio with E(rp) = 11.80%, if rf = 6.00%, and E(rm) = 10.00%?
E(r) = rf + B[E(rm) - rf] 0.1180 = 0.06 + B(0.10 - 0.06) 0.1180 = 0.06 + B(0.04) 0.058 = B(0.04) B = 1.45
CAPM Expected Return Example: Let's suppose we want to invest in MSFT. Assume the risk premium of the market portfolio (E(rm) - rf) = 8% and the risk-free rate (rf) = 3%. We look on Yahoo! Finance and see Bmsft = 0.73. What is the expected rate of return on MSFT?
E(r) = rf + B[E(rm) - rf] E(r) = 0.03 + 0.73(0.08) E(r) = 0.03 + 0.73(0.08) E(r) = 0.03 + 0.0584 E(r) = 0.0884 = 8.84%
T/F Securities that reflect highly variable prices suggest that the market does not know how to price the security accurately.
False In the short-term, markets reflect a random pattern. Information is constantly flowing in the economy and investors each have different expectations that vary constantly. A fluctuating market accurately reflects this logic. Furthermore, while increased variability may be the result of an increase in unknown variables, this merely increases risk and the price is adjusted downward as a result.
Credit Default Swaps
Insurance policy on default risk of corporate bond or loan. - it's really insurance, but if they say "swap" then it is unregulated Designed to allow lenders to buy protection against losses on large loans. - later used to speculate on financial health of companies
Maturity is a major determinant of bond price sensitivity to interest rate changes, but:
It is not the only factor; in particular, the coupon rates and current yield to maturity are also major determinants.
The longer-term bond is more sensitive to a given change in the discount rate. This will always be the case. Mathematically, there are more terms in the equation for the longer-term bond that are influenced by the discount rate:
Practically speaking, your money is tied up longer with the longer-term bond so you will experience greater capital losses when interest rates increase and greater capital gains when interest rates decrease.
Realized Compound Returns vs. Yield to Maturity
Realized compound return: - compound rate of return on bond with all coupons reinvested until maturity Horizon analysis: - analysis of bond returns over multi-year horizon, based on forecasts of bond's yield to maturity and investment options Reinvestment rate risk: - uncertainty surrounding cumulative future value of reinvested coupon payments - YTM/IRR assumes reinvestment of coupons, at the same rate
Weak-form tests: patterns in stock returns
Returns over Short Horizons: - momentum effect: tendency of poorly or well-performing stocks to continue abnormal performance in following periods Returns over Long Horizons: - reversal effect: tendency of poorly or well-performing stocks to experience reversals in following periods
Fama-French Three Factor Model
Ri = ai + Bmi*Rm + Bhml*Rhml + Bsmbi*Rsmb + ei - Rm is the market risk premium (rm - rf) - Rhml is the return of High Book-to-Market stocks minus the return of Low Book-to-Market stocks - Rsmb is the return of Small stocks minus the return of Big stocks
Fama-French Three Factor Model Example: In the last 12 months: - the market has returned 21.43% - the 3-month T-bill yielded 0.01% - small stocks returned 24.69% - large stocks returned 18.17% - value firms returned 26.97% - growth firms returned 15.90% GOOG has the following risk profile: Bm = 1.08, Bsmb = -0.57, and Bhml = -0.70. a) What was the expected return on GOOG this year? b) If the actual return was 16.50%, what was the alpha over the last 12 months? c) Does GOOG behave like a small or large firm? d) Does GOOG behave like a value or growth firm? e) Is GOOG more or less sensitive to the macro economy than an average firm?
Rm = 0.2143 - 0.0001 = 0.2142 Rsmb = 0.2469 - 0.1817 = 0.0652 Rhml = 0.2697 - 0.1590 = 0.1107 a) E(rgoog) = rf + Bmi*Rm + Bhml*Rhml + Bsmbi*Rsmb = 0.0001 + (1.08*0.2142) + (-0.57*0.0652) + (-0.70*0.1107) = 0.0001 + 0.231336 + -0.037164 + -0.07749 = 0.116782 = 11.68% b) If the actual return was 16.50%, then 0.1650 - 0.1168 = 0.0482 = 4.82% = alpha c) Compare Bsmb = -0.57 to 0. It is less than zero, Rsmb = rs - rb, so therefore GOOG behaves like a large firm. d) Compare Bhml = -0.70 to 0. It is less than zero, Rhml = rv - rg, so therefore GOOG behaves like a growth firm. e) The beta of the market (Bm) = 1. Compare Bm = 1.08 to 1. It is greater than one, so it is cyclical and more sensitive than an average firm.
Multi-factor Models Example: Let's assume there are two systematic risk factors that impact GOOGs risk premium: - market risk (like before) - interest rate risk (captured by a T-bond fund) Let Bgm = 1.1 and Bgtb = 0.8 and rf = 2%. Suppose rm = 9% while rtb = 5%. What is the expected return for GOOG?
Rm = rm - rf = 0.09 - 0.02 = 0.07 = 7% Rtb = rtb - rf = 0.05 - 0.02 = 0.03 = 3% E(rgoog) = rf + Bgm*Rm + Bgtb*Rtb = 0.02 + 1.1*0.07 + 0.8*0.03 = 0.02 + 0.077 + 0.024 = 0.121 = 12.1%
An eight-year bond has a yield of 9% and a duration of 7.201 years. If the bond's yield increases by 25 basis points, what is the percentage change in the bond's price as predicted by the duration formula?
The bond's price decreased by 1.65%: Original Yield (y) = 9% Duration (D) = 7.201 years Change in Yield (Δy) = 0.25% = 25/100 The relationship between duration (D), percentage change in price (ΔP/P), and percentage change in yield (Δy/1 +y) can be expressed as follows: ΔP/P = -D x [Δy/(1 + y)] = -7.201 x [0.0025/(1 + 0.09)] = -7.201 x (0.0025/1.09) = -7.201 x 0.002294 = -0.0165 = 1.65% decline
T/F Reid Aerial Systems has never missed or reduced a dividend payment in its 50-year history. If you believe the markets are efficient, this information should have NO effect on your likelihood of adding this stock to your portfolio.
True Reid Aerial Systems is not more attractive as a possible purchase. Any value associated with dividend predictability is already reflected in the stock price.
T/F Syndergaard Industries just announced a decrease in its annual earnings, yet its stock price rose. There is a potentially rational explanation for this phenomenon.
True There is a potentially rational explanation. The market may have anticipated even lower annual earnings. Compared to prior expectations, the announcement was a pleasant surprise.
Bond pricing between coupon dates
Will you sell your bond for the same price the day before and the day after a coupon payment? Dirty Price = Clean Price + Accrued Interest
Suppose you buy a 2 year, 10% coupon bond at par of $1,000. Assume only 1 coupon payment per year for simplicity. YTM assumes that in one year, you reinvest your $100 coupon at the YTM rate. In this example YTM = Coupon Rate only because you bought the bond at par. So, in 2 years, YTM assumes you will have $1,210. What if you can't reinvest the coupon at 10%?
YTM assumes: $1,000 x (1 + 0.10)^2 = $1,210; because r = 10% What happens when you can only invest that Year 1 coupon at 8%? Then, $1,000 x (1 + r)^2 = $1,208, so r = 9.91% So YTM = 10% and RCR = 9.91% Obviously, you can only calculate this after you know what happened.
Convexity Example: A 30-year maturity, 8% annual coupon, duration of 12.1584, convexity of 212.4325, and sells at an initial yield to maturity of 8%. If the bond's yield increases from 8% to 10%, the bond price will fall to $811.46, a decline of 18.85%. a) What is the modified duration? b) What is convexity?
a) Δy = 10% - 8% = 2% -D* Δy D* = D/1 + y -(12.1584/1 + 0.08) x 0.02 = -22.52% b) = -D* Δy + 0.5(convexity)(Δy^2) = -0.2252 + 0.5(212.4325)(0.02^2) = -0.2252 + 0.042487 = -18.27%
convertible bonds
allow bondholder to exchange bond for specified number of common stock shares - bond -----> stock
Immunization
an investment strategy designed to ensure the investor earns the promised YTM
current yield
annual coupon divided by bond price
Bi > 1
cyclical; more sensitivity than market
Indentures
define contracts between issuer and holder
Yield to Maturity
discount rate that makes present value of bond's payments equal to price
Quoted prices (clean prices)
do not include interest accruing between payment dates
key words for bond investors:
duration and convexity
Rank the interest sensitivity of the following from the most sensitive to an interest rate change to the least sensitive: (i) 8% coupon, noncallable 20-year maturity par bond (ii) 9% coupon, currently callable 20-year maturity premium bond (iii) Zero-coupon 30-year maturity bond
iii, i, ii
A higher coupon results:
in lower duration
As maturity increases, sensitivity of bond prices to changes in yields:
increases at a decreasing rate
Default premium
increment to promised yield that compensates investor for default risk = Corporate Yield - Treasury Yield
Sinking fund
indenture calling for issuer to periodically repurchase some proportion of outstanding bonds before maturity - kind of like an amortizing loan (like your car payment) but not exactly
Sensitivity of a bond's price-to-yield change is:
inversely related to current yield to maturity
Interest rate risk is inversely related to a bond's coupon rate:
low coupon bonds are more sensitive to interest rates resulting in higher duration
All else equal, bond price volatility is greater for:
lower coupon rates.
Duration increases with:
maturity
A bond's price volatility increases at a decreasing rate as:
maturity increases.
Why do investors like convexity?
more convexity = greater price increases, smaller price decreases when interest rates fluctuate by larger amounts
Index fund
mutual fund which holds shares in proportion to market index representation
Separate Trading of Registered Interest and Principal of Securities (STRIPS)
oversees creation of zero-coupon bonds from coupon bearing notes and bonds - zero-coupon treasuries - synthetic (made my financial intermediaries) - stripped bond (synthetic zero) converges to par at maturity
Term structure of interest rates
relationship between yields to maturity and terms to maturity across bonds
Subordination clause
restrictions on additional borrowing stipulating senior bondholders paid first in event of bankruptcy
Long-term bonds are more price sensitive than short-term bonds:
resulting in higher duration
Given its time to maturity, the duration of a zero-coupon bond is the:
same regardless of the discount rate.
Bi = 1
same risk premium as the market
Increases in a bond's yield to maturity results in:
smaller price change than a yield decrease of equal magnitude
Collateral
specific asset pledged against possible default
Weak-form: Efficient Market Hypothesis
stock prices already reflect all information contained in the history of trading
Semi-strong-form: Efficient Market Hypothesis
stock prices already reflect all public information - already reflects weak-form efficiency plus more
Because of convexity, when interest rates change, the actual bond price will always be higher than:
the bond price predicted by duration.
When the yield to maturity increases:
the duration decreases.
Interest rate risk
the possibility that an investor does not earn the promised YTM because of interest rate changes
An increase in bond's yield to maturity results in a price decline that is smaller than:
the price increase resulting from a decrease in yield of equal magnitude.
support level
unlikely for stock/index to fall below
resistance level
unlikely for stock/index to rise above
You have a 25-year maturity, 9.3% coupon, 9.3% bond yield with a duration of 10 years and a convexity of 134.8. If interest rates fall to 118 basis points, your predicted new price for the bond (including convexity) is?
ΔP/P = -D* Δy + 0.5(Convexity)(Δy^2) D* = D/(1 + y) = (10/1 + 0.093) = (10/1.093) = 9.149131 -D* Δy = -9.149131 x -1.18% = 0.10796 = 10.795% 0.5(Convexity)(Δy^2) = 0.5(134.8)(-1.18%^2) = 0.009385 = 0.9385% -D* Δy + 0.5(Convexity)(Δy^2) = 10.80% + 0.9385% = 11.738% Since the bond's coupon rate was equal to its original yield to maturity, the original price was $1,000. 1.1174 x $1,000 = $1,117.40
Random walk
- notion that stock price changes are random - changes in stock prices occur because of unpredictable new information - this random evolution of prices indicates an efficient market
Implications
- passive investing is efficient - expected excess return will be proportional to beta
Efficient Market Hypothesis
- prices of securities fully reflect available information: this means there should be no mispricing
T/F If financial markets are efficient and all securities are fairly priced, then all securities must offer equal expected rates of return.
False The statement would be true if it were modified to say "expected risk adjusted returns." Securities all have the same risk adjusted expected return if priced fairly; however, actual results can and do vary. Unknown events cause certain securities to outperform others. This is not known in advance, so expectations are set by known information.
Chapter 8
Homework Problems
Strong-form: Efficient Market Hypothesis
stock prices already reflect all relevant information, including inside information - included weak and semi-strong efficiency
Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 7.7%. What is the duration if the yield to maturity is 11.7%?
$1,000 x 0.06 = $60 CF0 = 0 CF1 = 60 x 1 = 60 CF2 = 60 x 2 = 120 CF3 = 1,060 x 3 = 3,180 I = 7.7 NPV = 2,704.71 N = 3 I/Y = 7.7 PV = ? PMT = 60 FV = 1000 PV = 955.95 NPV/PV = D D = 2704.71/955.95 = 2.8293 CF0 = 0 CF1 = 60 x 1 = 60 CF2 = 60 x 2 = 120 CF3 = 1,060 x 3 = 3,180 I = 11.7 NPV = 2,431.64 N = 3 I/Y = 11.7 PV = ? PMT = 60 FV = 1000 PV = 862.39 NPV/PV = D D = 2431.64/862.39 = 2.8197
A bond pays annual interest. Its coupon rate is 7.0%. Its value at maturity is $1,000. It matures in 4 years. Its yield to maturity is 4%. What is the modified duration of this bond?
$1,000 x 0.07 = $70 CF0 = 0 CF1 = 70 x 1 = 70 CF2 = 70 x 2 = 140 CF3 = 70 x 3 = 210 CF4 = 1,070 x 4 = 4,280 I = 4 NPV = 4,041.9967 N = 4 I/Y = 4 PV = ? PMT = 70 FV = 1000 PV = 1,108.8969 NPV/PV = D D = 4,041.9967/1,108.8969 = 3.65 D* = D/(1 + y) = 3.65/(1 + 0.04) = 3.65/1.04 D* = 3.51 years
Find the convexity of a seven-year maturity, 8.0% coupon bond selling at a yield to maturity of 8.5%. The bond pays it coupons annually.
$1,000 x 0.08 = $80 CF0 = 0 CF1 = (1 + 1^2) x 80 = 160 CF2 = (2 + 2^2) x 80 = 480 CF3 = (3 + 3^2) x 80 = 960 CF4 = (4 + 4^2) x 80 = 1,600 CF5 = (5 + 5^2) x 80 = 2,400 CF6 = (6 + 6^2) x 80 = 3,360 CF7 = (7 + 7^2) x 1,080 = 60,480 I = 8.5 NPV = 40,283.6648 N = 7 I/Y = 8.5 PV = ? PMT = 80 FV = 1000 PV = 974.41 Convexity = [1/(P x {1 + y^2})] x NPV = [1/(974.41 x 1.085^2)] x 40,283.6648 = [1/(974.41 x 1.1772)] x 40,283.6648 = (1/1,147.0998) x 40,283.6648 = 0.000872 x 40,283.6648 = 35.1179
A bond pays annual interest. Its coupon rate is 9.2%. Its value at maturity is $1,000. It matures in 4 years. Its yield to maturity is currently 6.2%. The duration of this bond is?
$1,000 x 0.092 = $92 CF0 = 0 CF1 = 92 x 1 = 92 CF2 = 92 x 2 = 184 CF3 = 92 x 3 = 276 CF4 = 1,092 x 4 = 4,368 I = 6.2 NPV = 3,914.0759 N = 4 I/Y = 6.2 PV = ? PMT = 92 FV = 1000 PV = 1,103.4789 NPV/PV = D D = 3,914.0759/1,103.4789 D = 3.55 years
A 30-year maturity bond making annual coupon payments with a coupon rate of 16.3% has duration of 10.54 years and a convexity of 161.2. The bond currently sells at a yield to maturity of 9%. a) Find the price of the bond if its yield to maturity falls to 8%. b) What price would be predicted by the duration rule? c) What price would be predicted by the duration-with-convexity rule? d-1) What is the percent error for each rule? d-2) What do you conclude about the accuracy of the two rules? e-1) Find the price of the bond if it's yield to maturity rises to 10%. e-2) What price would be predicted by the duration rule? e-3) What price would be predicted by the duration-with-convexity rule? e-4) What is the percent error for each rule? e-5) Are your conclusions about the accuracy of the two rules consistent with parts (a) - (d)?
$1,000 x 0.163 N = 30 I/Y = 9 PV = ? PMT = 163 FV = 1000 PV = 1,749.9767 a) N = 30 I/Y = 8 PV = ? PMT = 163 FV = 1000 PV = 1,934.396 b) Using duration, predicted price change is: = -D/(1 + y) x Δy x Po = [-10.54/(1 + 0.09) x -0.01] x 1,749.9767 = 0.096697 x 1,749.9767 = 169.22 1,749.9767 + 169.22 = $1,919.1947 c) Using duration-with-convexity, predicted price change is: = [-D/(1 + y) x Δy] + 0.5(Convexity)(Δy^2) x Po = [-10.54/(1 + 0.09) x -0.01] + 0.5(161.2)(-0.01^2) x 1,749.9767 = 0.096697 + 0.00806 x 1,749.9767 = 0.104757 x 1,749.9767 = 183.32 1,749.9767 + 183.32 = $1,933.2995 d-1) Percent Errors Duration: 1,934.40 - 1,1919.19/1,934.40 = 0.00786 = 0.79% Duration-with-convexity: 1,934.40 - 1,933.30/1,934.40 = 0.000569 = 0.057% d-2) The duration-with-convexity rule provides more accurate approximations to the actual change in price. e-1) N = 30 I/Y = 10 PV = ? PMT = 163 FV = 1000 PV = 1,593.896 e-2) Using duration, predicted price change is: = -D/(1 + y) x Δy x Po = [-10.54/(1 + 0.09) x 0.01] x 1,749.9767 = -0.096697 x 1,749.9767 = -169.2179 1,749.9767 + (169.2179) = $1,580.759 e-3) Using duration-with-convexity, predicted price change is: = [-D/(1 + y) x Δy] + 0.5(Convexity)(Δy^2) x Po = [-10.54/(1 + 0.09) x 0.01] + 0.5(161.2)(0.01^2) x 1,749.9767 = -0.096697 + 0.00806 x 1,749.9767 = -0.088637 x 1,749.9767 = -155.113 1,749.9767 + (155.113) = $1,594.864 e-4) Percent Errors Duration: 1,593.90 - 1,580.76/1,593.90 = 0.008244 = 0.82% Duration-with-convexity: 1,593.90 - 1,594.87/1,593.90 = -0.000609 = -0.061% e-5) Yes, similar to the decrease in YTM, using the duration-with-convexity rule for the increase in YTM predicts a more fine-tuned estimate of price than the duration rule alone.
Market Risk Premium (Rm) Example: Assume we calculated the equilibrium value of the market risk premium and found it = 5.20%. We know the T-bill yield is 5%. Therefore the expected rate of return on the market must be?
(rm - rf) = 0.0520 rf = 0.05 E(r) = rf + B[E(rm) - rf] E(r) = 0.05 + 1(0.052) E(r) = 0.1020 = 10.20%
Consider the following 5 year 10% coupon annual payment corporate bond: 1: $100 2: $100 3: $100 4: $100 5: $1,100
- Because the bond pays cash prior to the maturity, it has an "effective" maturity less than 5 years. - We can think of this bond as a portfolio of 5 zero coupon bonds with the given maturities. - The average maturity of the five zeros would be the coupon bond's effective maturity.
But the IRS doesn't see it that way....
- Built in price appreciation on original issue discount bonds constitutes implicit interest payment to holder - IRS calculates price appreciation schedule to determine taxable interest income for built in appreciation
CAPM and the real world
- CAPM is false based on the validity of its assumptions because they are not realistic - but the ultimate thought is that it could be a useful predictor of expected returns - principles are still valid: investors should diversify, systematic risk is the risk that matters, and well-diversified risky portfolio can be suitable for wide range of investors
Lower coupon rate < market rate
- interest income is worse than what is offered by the market - built-in capital gain ensures that the bond investors do receive the competitive market rate
Arbitrage Portfolio
- IF we find a well-diversified portfolio with a positive alpha, we can make risk-free money (a zero net investment, risk-free portfolio, with a positive return) - How do we find a portfolio with an alpha? Common traits like size and book-to-market - "No such thing as a free lunch." If I find a positive alpha portfolio, someone else will find it, and the price will rise until there is no more alpha. - So, the return of a security is based on these "alpha-generating" factors - market risk premium, size, and book-to-market
Would you invest in projects or securities above the SML? Below?
- SML provides "hurdle rates" for investments - expected return should either be on the line or above it for it to be a good investment for us because we want positive returns (below the line means negative returns) - above the line means it is underpriced - below means it is overpriced, which is not good and we do not want that
Yield to Maturity vs. Holding Period Return (HPR)
- Yield to maturity measures average ROR if investment held until bond matures - HPR is ROR over particular investment period; depends on market price at the end of the period
Capital Asset Pricing Model (CAPM)
- a model that relates the required rate of return on a security to its market risk, systematic risk, as measured by beta - developed before the index model (index model is based on CAPM) - has a lot of problems in reality, but it works really well theoretically, so parts of it are still widely used (beta and alpha)
Portfolio management in efficient market
- active management assumes market inefficiency - passive management is consistent with semi-strong efficiency - inefficient market pricing leads to inefficient resource allocation
Theoretical Equilibrium if markets are perfect and all investors are the same
- all investors choose to hold market portfolio - market portfolio is on efficient frontier, optimal risky portfolio - Market portfolio: the portfolio that includes all securities held in proportion to their market value
Stock market analysts
- analysts are overly positive about firm propsects
Arbitrage Pricing Theory (APT)
- arbitrage is the creation of riskless profits made possible by relative mispricing among securities - APT is a theory of risk-return relationships from no-arbitrage considerations in large capital markets - basically, if security prices are affected by any factors, traders will buy/sell accordingly, and the return of the security will become a function of those factors - if arbitrage investments are risky, then they aren't really arbitrage because there's firm-specific risk we haven't accounted for (eP) - How do we get rid of eP? Diversify! We need a portfolio that has an alpha, either positive or negative. We need a well-diversified portfolio that has negligible nonsystematic risk which means that eP = 0 approximately.
Predicting betas
- betas are mean reversion, which means betas move towards the mean over time - to predict future betas, we need to remember to adjust estimates from historical data to account for regression towards 1.0 (i.e. if beta > 1, we should expect to see it lower in the future as it moves forward; if beta < 1, we should see that the beta grows and is larger in the future as it gets closer to 1) - raw beta from bloomberg is calculated using 2 years of monthly data and adjusted beta is (2*RAWBeta+1)/3
Preferred stock
- bond without a maturity: firm can miss payment without bankruptcy - commonly pays fixed dividend: floating-rate preferred stock becoming more popular - dividends not normally tax-deductible: corporations that purchase other corporations' preferred stock are taxed on only 30% of dividends received
Yield to Call
- calculated like yield to maturity - time until call replaces time until maturity; call price replaces par value - premium bonds are more likely to be called than discount bonds
What is a bond?
- debt security that obligates issuer to make payments to holder over time - face value, par value: payment to bondholder at maturity of bond - coupon rate: bond's annual interest payment per dollar of par value - zero-coupon bond: pays no coupons, sells at discount, provides only payment of par value at maturity
risk premium of market portfolio
- demand drives prices, lowers expected rate of return/risk premiums - when premiums fall, investors move funds into risk-free assets - equilibrium risk premium of market portfolio positively proportional to risk of market and risk aversion of average investor - investors are only compensated for bearing market risk (not firm-specific) - expected returns on individual securities should have a direct relationship between beta and expected return
eurobonds
- denominated in currency (usually that of issuing country) different than that of market - going into a different country and using currency that is not theirs - German firm issuing bonds in the U.S. but in euros
Index Model
- derived from the CAPM - E(ri) = rf + Bi[E(rm) - rf] E(ri) - rf = ai + Bi[(E(rm) - rf] ri - rf = ai + Bi[rm - rf] + ei - Estimate CAPM using the index model
The Need for Convexity
- duration is only an approximation - duration asserts that the percentage price change is linearly related to the change in the bond's yield - underestimates the increase in bond prices when yields fall - overestimates the decline in price when the yields rise
Are Markets Efficient?
- efficiency is relative, not binary - selection bias issue: investors who find successful investment schemes are less inclined to share findings; observable outcomes preselected in favor of failed attempts - lucky event issue: lucky investments receive disproportionate attention - semi-strong for most investors - enough that only differentially superior information will earn money - professional manager's margin of superiority likely too slight for statistical significance
investors are homogeneous (the same)
- except for initial wealth, risk aversion - investors plan for single-period horizon; they are rational, mean-variance optimizers - use same inputs, consider identical portfolio opportunity sets
Higher coupon rate > market rate
- interest income is better than what is offered by the market - built-in capital loss ensures that the bond investors only receive the competitive market rate
Recent innovations
- inverse floaters: coupon rate falls when interest rates rise - asset-backed bonds: income from specified assets used to service debt - pay-in-kind bonds: issuers can pay interest in cash or additional bonds - catastrophe bonds: bonds with natural disaster-linked payments & higher coupon rates to investors for taking on risk - indexed bonds: payments tied to a price index, or price of a commodity & TIPS - par value of the bond increases with CPI
Competition as Source of Efficiency
- investor competition should imply stock prices reflect available information - investors exploit available profit opportunities - competitive advantage can verge on insider trading - investors reveal information when they buy/sell securities
Capital Market Line (CML)
- is a capital allocation line that contains the entire market - the point where the efficient frontier is just barely tangent to the CML, which means that's the highest slope or the highest Sharpe ratio we can get, and that is going to be the market portfolio - CML is tangent to the efficient frontier
foreign bonds
- issued by borrower in a different country from where the bond is sold, denominated in currency of market country - issuer goes to a different country and issues a bond in their currency - Brazilian firm issuing bonds in the U.S. in dollars
Multi-factor models
- models of security returns that respond to several systematic factors by adding them into the model for better descriptions of security returns - right now, we assume that only market risk matters - we add more systematic risk factors, which is technically the multi-factor security market line - to make it easier, from now on: Rx = [E(rx) - rf]
Perfect markets
- no investor is wealthy enough to individually affect prices - all information publicly available; all securities public - no taxes on returns, no transaction costs - unlimited borrowing/lending at risk-free rate
Bond Pricing
- prices fall as market interest rates rise - if you only remember one thing in this chapter, remember that interest rate fluctuations are the primary source of bond market risk - bonds with longer maturities are more sensitive to fluctuations in interest rates
Security Market Line (SML)
- represents expected return-beta relationship of CAPM - graphs individual asset risk premiums as function of asset risk - it is NOT the capital market line (CML) - the slope of the SML is beta
Fundamental Analysis
- research on determinants of stock value, i.e. earnings, dividend prospects, future interest rate expectations, and firm risk - assumes stock price equal to discounted value of expected future cash flow
Technical Analysis
- research on recurrent/predictable price patterns and on proxies for buy/sell pressure in the market
Predictors of broad market performance
- return on aggregate stock market tends to be higher when dividend yield is low - earnings yield can predict market returns - bond market data (spread between yields) can predict market returns
CAPM implications
- risk premium on market portfolio is proportional to variance of market portfolio and investor's risk aversion - risk premium on individual assets: positively proportional to risk premium on market portfolio and positively proportional to beta coefficient of security on market portfolio - passive strategy is efficient - mutual fund theorem: all investors desire same portfolio of risky assets, can be satisfied by single mutual fund composed of that portfolio - if passive strategy is costless and efficient, why follow active strategy? active traders are what ensure market efficiency
Duration
- the weighted average maturity for all future bond cash flows - estimates the impact of a 1% change in interest rates could have on a bond's price - duration is a number - higher duration = higher sensitivity
Mutual fund managers
- today's conventional model: Fama French factors plus momentum factor (a portfolio constructed based on prior-year stock return) - Wermers: funds show positive gross alphas before fees. After controlling fees and risk, funds show negative net alphas. - Carhart: minor persistance in relative performance across managers, largely due to expense/transaction costs - Berk and Green: skilled managers with abnormal performance will attract new funds until additional cost and complexity drives alphas to zero - Chen, Ferson, and Peters: On average, bond mutual funds outperform passive bond indexes in gross returns, underperformed once fees subtracted - Kosowki, Timmerman, Wermers, and White: stock pricing ability of minority of managers sufficient to cover costs; performance persists over time - Samuelson: records of most managers show no easy strategies for success
Who issues bonds?
- treasure - corporations - stand and local governments (municipal bonds) - Federal Home Loan Bank Board - farm credit agencies - Ginnie Mae, Fannie Mae, Freddie Mac
Estimating the Index Model:
- we get the Security Characteristic Line (SCL), which is a plot of security's expected excess return over risk-free rate as function of excess return on market - gives us a historical beta
A bond investor faces two types of interest rate risk:
1. price risk: an investor cannot sell the bond for as much as anticipated and an increase in interest rates reduces the sale price. 2. reinvestment risk: the investor will not be able to reinvest the coupons at the promised yield rate and a decrease in interest rates reduces the future value of the reinvested coupons. - The two types of risk are potentially offsetting.
What is the expected rate of return for a stock that has a beta of 1.00 if the expected return on the market is 14.00%?
14% because its expected return is exactly the same as the market return when beta is 1.00.
Factor portfolio
A well-diversified portfolio with a beta of 1.0 on one factor and a beta of 0 on the others.
Consider a firm that unexpectedly announces a large cash dividend to its shareholders. If the market is efficient and there was no information leakage prior to the announcement, one might expect:
An abnormal price change at the announcement. In efficient markets, unexpected announcements that have substantial information regarding the firm should be absorbed instantly.
A coupon bond paying semiannual interest is reported as having an ask price of 122% of its $1,000 par value. If the last interest payment was made one month ago and the coupon rate is 5%, what is the dirty price of the bond? Assume a month has 30 days and a year has 364 days. a) Clean Price? $1,220 b) Accrued Interest? $4.12 c) Dirty Price? $1,224.12
Annual Coupon Payment = Par Value x Coupon Rate = $1,000 x 0.05 = $50 Days since last coupon payment = 30 days or 1 month Semiannual, so coupon payments per year = 2 so; 364/2 = 182 a) Clean Price/Quoted Price/Flat Price = [Par Value x (Ask Price%/100)] or; Par Value x Ask Price = [$1,000 x (122%/100)] = $1,220 or; = $1,000 x 1.22 = $1,220 b) Accrued Interest = (Annual Coupon Payment/Coupon Payments Per Year) x (Days Since Last Coupon Payments/Days Between Coupon Payments) = ($50/2) x (30 days/182 days) = $25 x 0.1648 = $4.121 c) Dirty Price/Invoice Price = Clean Price + Accrued Interest = $1,220 + $4.121 = $1,224.121
Why isn't the yield curve flat?
Expectations Hypothesis: - yields to maturity determined solely by expectations of future short-term interest rates Liquidity Preference Hypothesis: - investors demand a risk premium on long-term bonds
T/F The Efficient Market Hypothesis implies that technical analysis has added value to security analysis.
False Technical analysis involves the search for recurrent and predictable patterns in stock prices in order to enhance returns. The EMH implies that technical analysis is without value. If past prices contain no useful information for predicting future prices, there is no point in following any technical trading rule.
Are the following bonds more or less valuable than a standard fixed-coupon bond, all else equal?
Feature: Value vs. Plain Vanilla Bond Callable: less Puttable: more Convertible: more Floating-rate: closer to par
Yield Curve Example: (1 + Yn)^n = (1 + Yn-1)^n-1 x (1 + Fn)
Forward Rate - inferred short-term ROI for a future period - makes expected total return of long-term bond equal to that of rolling over short-term bonds
Chapter 10
Homework Problems
Chapter 11
Homework Problems
Chapter 7
Homework Problems
Which of the following statements would provide support against the semi-strong form of the efficient market theory?
Low P/E stocks tend to have positive abnormal returns over long-run. The P/E ratio is public information so this observation would provide evidence against the semi-strong form of the efficient market theory.
What would happen to market efficiency if all investors attempted to follow a passive strategy?
Market efficiency will decrease. If all investors follow a passive strategy, sooner or later prices will fail to reflect new information. At this point there are profit opportunities for active investors who uncover mispriced securities. As they buy and sell these assets, prices again will be driven to fair (and accurate) levels.
Capital Asset Pricing and Arbitrage Pricing Theory
Module 3 PowerPoint
Efficient Market Hypothesis
Module 3 PowerPoint
Managing Bond Portfolios
Module 3 PowerPoint
Semi-Annual Bond Pricing Example: If yields on similar bonds are 10%, what would the price of a 10-year, 8% (annual rate) semiannual coupon be?
N = 10 x 2 I/Y = 10/2 PV = ? PMT = (1000 x 0.08)/2 FV = 1000 PV = $875.38
Dirty Price Example: What is the dirty price of the above bond, which is 15 years to maturity and yielding 9%?
N = 15 x 2 I/Y = 9/2 PV = ? PMT = 50 FV = 1000 PV = 1081.44 1081.44 + 8.24 = $1,089.68
According to the efficient market hypothesis:
Positive alphas on stocks will quickly disappear. Stocks producing abnormal excess returns will increase in price to eliminate the positive alpha.
Market Anomalies
Semi-strong Tests: Market Anomalies: - anomalies: patterns of returns contradicting efficient market hypothesis P/E Effect: - portfolios of low P/E stocks exhibit higher average risk-adjusted returns than high P/E stocks Small-firm effect: - stocks of small firms can earn abnormal returns, primarily in January Neglected-firm effect: - stock of little-known firms can generate abnormal returns Book-to-market effect: - shares of high book-to-market firms can generate abnormal returns Post-earnings announcement price drift: - sluggish response of stock price to firm's earnings announcement - abnormal return on announcement day, momentum continues past market price Bubbles and market efficiency: - speculative bubbles can raise prices above intrinsic value - even if prices are inaccurate, it can be difficult to take advantage of them Risk premiums or inefficiencies? - Fama and French: market phenomena can be explained as manifestations of risk premium - Lakonishok, Shleifer, and Vishny: market phenomena are evidence of inefficient markets
A share of stock is now selling for $145. It will pay a dividend of $10 per share at the end of the year. Its beta is 1.0. What do investors expect the stock to sell for at the end of the year? Assume the risk-free rate is 7% and the expected rate of return on the market is 18%.
Since the stock's beta is equal to 1.0, its expected rate of return should be equal to that of the market, that is, 18%: E(r) = D + P1 - Po/Po 0.18 = 10 + P1 - 145/145 0.18(145) = 10 + P1 - 145 26.10 = -135 + P1 161.10 = P1 Expected selling price = $161.10
Current Yield & Yield to Maturity Example: If a bond with 5 years until maturity, paying an annual 7% coupon sells for $1,086.59, what is the bond's current yield? What is the bond's yield to maturity?
The current yield should be < 7% Current Yield = Annual Coupon/Price Today = $70/$1086.59 = 6.44% The yield to maturity: N = 5 I/Y = ? PV = -1,086.59 PMT = 70 FV = 1000 I/Y = 5% YTM
You own a fixed-income asset with a duration of four years. If the level of interest rates, which is 8.1%, goes down by 20 basis points, how much do you expect the price of the asset to go up (in percentage terms)?
The price of the asset increases by 0.74%: The relationship between duration (D), percentage change in price (ΔP/P), and percentage change in yield (Δy/1 +y) can be expressed as follows: ΔP/P = -D x [Δy/(1 + y)] = -4 x [-0.20%/(1 + 0.081)] = -4 x [-0.0020/(1.081)] = -4 x -0.00185 = 0.007401 = 0.74% increase
You have invested in a Treasure Inflation Protected Security (TIPS) that has a par value of $1,000 and a coupon rate of 2.68%. You paid par value for the security, and it matures in two years. Assume that the inflation rate for the next year is 2.10% and for the year after is 1.22%. Time: 0 Inflation in Year Just Ended: - Par Value: $1,000 Coupon Payment: - Principal Repayment: - Total Payment: - Nominal Return: - Real Return: - Time: 1 Inflation in Year Just Ended: 2.10% Par Value: $1,021 Coupon Payment: $27.36 Principal Repayment: $0.00 Total Payment: $27.36 Nominal Return: $4.84% Real Return: $2.68% Time: 2 Inflation in Year Just Ended: 1.22% Par Value: $1,033.46 Coupon Payment: $27.70 Principal Repayment: $1,033.46 Total Payment: $1,061.15 Nominal Return: 3.93% Real Return: 2.68%
Time 1: Inflation = Given in the problem of 2.10% Par Value = [Time 0 Par x (1 + Inflation Rate of Time 1)] = $1,000 x (1 + 0.0210) = $1,021 Coupon Payment = Par Value x Coupon Rate = $1,021 x 0.0268 = $27.36 Principal Repayment = $0.00 Total Payment = Coupon Payment + Principal Payment = $27.36 + $0.00 = $27.36 Nominal Return = [(Par Value + Coupon Payment)/Par Value at Time 0] - 1 = [($1,021 + $27.36)/$1,000] - 1 = 0.04836 Real Return = [(1 + Nominal)/(1 + Inflation)] -1 = [(1 + 0.04836)/(1 + 0.0210)] - 1 = 0.026797 Time 2: Inflation = Given in the problem of 1.22% Par Value = [Time 1 Par x (1 + Inflation Rate of Time 2)] = $1,021 x (1 + 0.0122) = $1,033.456 Coupon Payment = Par Value x Coupon Rate = $1,033.456 x 0.0268 = $27.696 Principal Repayment = $1,033.456 Total Payment = Coupon Payment + Principal Payment = $27.696 + $1,033.456 = $1,061.152 Nominal Return = [(Par Value + Coupon Payment)/Par Value at Time 1] - 1 = [($1,033.456 + $27.696)/$1,021] - 1 = 0.039327 Real Return = [(1 + Nominal)/(1 + Inflation)] -1 = [(1 + 0.039327)/(1 + 0.0122)] - 1 = 0.026800 Principal repayment occurs at maturity and is based on the par value at that time.
Assume that coupon interest payments are made semiannually and that par value is $1,000 for both bonds: Bond: Coupon Rate: Maturity: Required Return Bond A: 4.50%: 5 years: 7.03% Bond B: 4.50%: 25 years: 7.03% a) Calculate the values of Bond A and Bond B. b) Recalculate the bonds' values if the required return changes to 8.72%. c) Calculate the increase or decrease in bond value based on the changed in required return.
a) Bond A: N = 5 x 2 I/Y = 7.03/2 PV = ? PMT = 45/2 FV = 1000 PV = $894.874 Bond B: N = 25 x 2 I/Y = 7.03/2 PV = ? PMT = 45/2 FV = 1000 PV = $704.087 b) Bond A: N = 5 x 2 I/Y = 8.72/2 PV = ? PMT = 45/2 FV = 1000 PV = $831.8864 Bond B: N = 25 x 2 I/Y = 8.72/2 PV = ? PMT = 45/2 FV = 1000 PV = $573.3468 c) Bond A: Decrease $894.874 - $831.8864 = $62.988 Bond B: Decrease $704.087 - $573.3468 = $130.74
Assume you have a one-year investment horizon and are trying to choose among three bonds. All have the same degree of default risk and mature in 9 years. The first is a zero-coupon bond that pays $1,000 at maturity. The second has an 8.7% coupon rate and pays the $87 coupon once per year. The third has a 10.7% coupon rate and pays the $107 coupon once per year. Assume all bonds are compounded annually. a) If all three bonds are priced now to yield 8.7% maturity, what are their current prices? b-1) If you expect their yields to maturity to be 8.7% at the beginning of next year, what will their prices be then? b-2) What is your rate of return on each bond during the one-year holding period?
a) Current Prices Zero-Coupon Bond: $471.99 N = 9 I/Y = 8.7 PV = ? PMT = 0 FV = 1000 8.7% Bond: $1,000 N = 9 I/Y = 8.7 PV = ? PMT = 87 FV = 1000 10.7% Bond: $1,121.38 N = 9 I/Y = 8.7 PV = ? PMT = 107 FV = 1000 b-1) Prices One Year From Now Zero-Coupon Bond: $513.05 N = 8 I/Y = 8.7 PV = ? PMT = 0 FV = 1000 8.7% Bond: $1,000 N = 8 I/Y = 8.7 PV = ? PMT = 87 FV = 1000 10.7% Bond: $1,111.94 N = 8 I/Y = 8.7 PV = ? PMT = 107 FV = 1000 b-2) Rate of Return Zero-Coupon Bond: 8.70% $531.05 - $471.99 = $41.06 Price Increase so; Coupon Income + Price Increase = Income $0 + $41.06 = $41.06 Income Rate of Return = Income/Current Price $41.06/$471.99 = 0.086993 8.7% Bond: 8.70% $1,000 - $1,000 = $0 Price Increase so; Coupon Income + Price Increase = Income $87.00 + $0 = $87.00 Income Rate of Return = Income/Current Price $87.00/$1,000 = 0.08700 10.7% Bond: 8.70% $1,111.94 - $1,121.38 = ($9.44) Increase so; Coupon Income + Price Increase = Income $107 + ($9.44) = $97.56 Income Rate of Return = Income/Current Price $97.56/$1,121.38 = 0.08700
The market price of a security is $42. Its expected rate of return is 9%. The risk-free rate is 4%, and the market risk premium is 7%. What will the market price of the security be if its beta doubles (and all other variables remain unchanged)? Assume the stock is expected to pay a constant dividend in perpetuity. a) What is the original beta of the security at a market price of $42? 0.71 b) What is the new beta after it doubles? 1.43 c) What is the new required rate of return for the security given the new beta? 14% d) What is the dividend amount that the firm pays? $3.78 e) What would be the new market price of the security after the beta has doubled? $27.00
a) E(r) = rf + B[E(rm) - rf] 0.09 = 0.04 + B(0.07) 0.05 = B(0.07) B = 0.714 b) 0.714 x 2 = 1.428 If the beta of the security doubles, then so will its risk premium. c) Risk Premium = Expected rate of return - risk-free rate = 9% - 4% = 5% New risk premium = 5% doubled = 5% x 2 = 10% New discount rate = New risk premium + Risk-free rate = 10% + 4% = 14% d) If a stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount Rate $42 = D/0.09 D = $42 x 0.09 D = $3.78 e) At the new discount rate of 14%, the stock would be worth: $3.78/0.14 = $27.00. The increase in stock risk has lowered the value of the stock by 35.71% = ($42-$27)/$42 = 0.35714
You are a consultant to a large manufacturing corporation considering a project with the following net after-tax cash flows (in millions of dollars): Years from Now: After-Tax CF 0: -34 1-9: 14 10: 28 The project's beta is 1.8. Assuming rf = 5% and E(rM) = 15%. a) What is the net present value (NPV) of the project? 20.96 million b) What is the highest possible beta estimate for the project before its NPV becomes positive? 3.54
a) E(r) = rf + B[E(rm) - rf] = 0.05 + 1.8(0.15 - 0.05) = 0.05 + 1.8(0.10) = 0.05 + 0.18 = 0.23 = 23% CF0 = -34,000,000 CF1 = 14,000,000 F1 = 9 CF2 = 28,000,000 I = 23 NPV = 20,956,130.45 IRR = 40.35% b) The internal rate of return on the project is 40.35%. The highest value that beta can take before the hurdle rate exceeds the IRR is determined by: 0.4035 = 0.05 + B(0.15 - 0.05) 0.4035 = 0.05 + B(0.10) 0.3535 = B(0.10) B = 3.54
The current yield curve for default-free-zero bonds is as follows: Maturity(yrs): YTM: Forward Rate Year 1: 10.3%: - Year 2: 11.3%: - Year 3: 12.3%: - a) What are the implied one-year forward rates? Year 1: - Year 2: 12.31% Year 3: 14.33% b) Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will the pure yield curve be next year? shifts upwards c-1) If you purchase a two-year-zero bond now, what is the expected total rate of return over the next year? (HINT: Compute the current and expected future prices.) Ignore taxes. 10.30% c-2) If you purchase a three-year-zero bond now? 10.30% Extra Notes: Maturity(yrs): YTM: Forward Rate: Prices Year 1: 10.3%: -: $906.62 Year 2: 11.3%: 12.31%: $807.25 Year 3: 12.3%: 14.33%: $706.07
a) Forward Rate = (1 + Yn)^n = (1 + Yn-1)^n-1 x (1+ Fn) Year 2 Forward Rate = (1 + 0.113)^2 = (1 + 0.103)^1 x (1 + F2) = (1.113)^2 = (1.103)^1 x (1 + F2) = [(1.113)^2/(1.103)^1)] - 1 = F2 = (1.239/1.103) - 1 = F2 = 1.123 - 1 = F2 = 0.123091 = F2 = 12.31% = F2 Year 3 Forward Rate = (1 + 0.123)^3 = (1 + 0.113)^2 x (1 + F3) = (1.123)^3 = (1.113)^2 x (1 + F3) = [(1.123)^3/(1.113)^2] - 1 = F3 = (1.416/1.239) - 1 = F3 = 1.143 - 1 = F3 = 0.14327 = F3 = 14.33% = F3 b) There will be a shift upwards in next year's curve. c-1:2) $1,000/1.103 = $906.62 $1,000/(1.103 x 1.1231) = $807.25 $1,000/(1.103 x 1.1231 x 1.1433) = $706.07 $1,000/1.1231 = $890.39 $1,000/(1.1231 x 1.1433) = $778.79 Two-Year-Zero Bond: ($890.39/$807.25) - 1 = 0.1030 Three-Year-Zero Bond: ($778.79/$706.07) - 1 = 0.1030
Consider the following information: Portfolio: Expected Return: Beta Risk-free: 6%: 0 Market: 12%: 1.0 A: 10%: 1.5 a) Calculate the return predicted by CAPM for a portfolio with a beta of 1.5. b) What is the alpha of portfolio A? c) If the simple CAPM is valid, is the situation above possible?
a) Given these data, the SML is: E(r) = 0.06 + B(0.12 - 0.06) E(r) = 0.06 + 1.5(0.12 - 0.06) E(r) = 0.06 + 1.5(0.06) E(r) = 0.06 + 0.09 E(r) = 0.15 = 15% b) The expected return for Portfolio A is 10% so that Portfolio A plots below the SML (i.e. has an alpha of -5%), and hence is an overpriced portfolio. This is inconsistent with the CAPM. c) No, not possible.
I am buying a firm with an expected perpetual cash flow of $1,250 but am unsure of its risk. If I think the beta of the firm is zero when the beta is really 1.0, how much more will I offer for the firm than it is truly worth? Assume the risk-free rate is 5% and the expected rate of return on the market is 20%. a) Present value of the firm if beta equals 0? $25,000 b) Present value of the firm if beta equals 1.00? $6,250 c) Present value difference? $18,750
a) If beta equals 0, then the cash flow should be discounted at the risk-free rate, 5%: PV = $1,250/0.05 = $25,000 b) If, however, beta actually is equal to 1.00, the investment should yield 20%, and the price paid for the firm should be: PV = $1,250/0.20 = $6,250 c) The difference is the amount you will overpay if you assume that beta is zero rather than 1.00. $25,000 - $6,250 = $18,750
You have purchased a convertible bond for $1,093.76. It is convertible into 58 shares of the firm's common stock. The current stock price is $15.60 per share. a) What is the market conversion value of the bond? $904.80 b) What is the conversion premium? $188.96 c) Will you choose to convert the stock now? No
a) Market Conversion Value = #Shares x Share Price = 58 shares x $15.60 = $904.80 b) Conversion Premium = [Price of Bond - (#Shares x Share Price)] = [$1,093.76 - (58 shares x $15.60)] = $1,093.76 - $904.80 = $188.96 c) No, you will not choose to convert the bond now because the value of the stock is less than the value of the bond.
A bond has a par value of $1,000, a time to maturity of 20 years, and a coupon rate of 7.30% with interest paid annually. If the current market price is $730, what will the approximate capital gain be over this next year if its yield to maturity remains unchanged? a) What will be the price of the bond next year if YTM remains unchanged? $734.40 b) Capital Gain? $4.40
a) N = 20 I/Y = ? PV = (730) PMT = 73 FV = 1000 I/Y = 10.6029 N = 19 I/Y = 10.6 PV = ? PMT = 73 FV = 1000 PV = $734.40 b) $734.40 - $730 = $4.40
A zero-coupon bond with face value $1,000 and maturity of 6 years sells for $736.22. a) What is its yield to maturity? 5.24% b) What will the yield to maturity be if price falls to 720? 5.63%
a) N = 6 I/Y = ? PV = (736.22) PMT = 0 FV = 1000 I/Y = 5.2363 b) N = 6 I/Y = ? PV = (720) PMT = 0 FV = 1000 I/Y = 5.62772
Consider the following information: Portfolio: Expected Return: Standard Deviation Risk-free: 5.0%: 0% Market: 13%: 35% A: 11%: 24% a) Calculate the Sharpe ratios for the market portfolio and portfolio A. b) If the simple CAPM is valid, is the above situation possible?
a) Sharpe ratio for Portfolio A: = E(r) - rf/σ = 0.11 - 0.05/0.24 = 0.06/0.24 = 0.25 Sharpe ratio for market portfolio: = E(r) - rf/σ = 0.13 - 0.05/0.35 = 0.08/0.35 = 0.23 This implies that Portfolio A provides a better risk-reward tradeoff than the market portfolio. b) No, because the reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since CAPM predicts that the market portfolio is the most efficient portfolio.
Suppose the yield on short-term government securities (perceived to be risk-free) is about 5%. Suppose also that the expected return required by the market for a portfolio with a beta of 1.00 is 8.0%. According to the capital asset pricing model (CAPM): a) What is the expected return on the market portfolio? 8% b) What would be the expected return on a zero-beta stock? 5% Suppose you consider buying a share of stock at a price of $100. The stock is expected to pay a dividend of $9 next year and to sell then for $103. The stock price has been evaluated at B = -0.5. c-1) Using the SML, calculate the fair rate of return for a stock with a beta = -0.5. 3.5% c-2) Calculate the expected rate of return, using the expected price and dividend for next year. 12% c-3) Is the stock overpriced or underpriced?
a) Since the market portfolio, by definition, has a beta of 1.00, then its expected rate of return is 8.0%. b) A beta = 0 means the stock has no systematic risk. Hence, the portfolio's expected rate of return is the risk-free rate, 5%. c-1) Using the SML, the fair rate of return for a stock with beta = -0.5 is: E(r) = rf + B[E(rm) - rf] = 0.05 + -0.5(0.08 - 0.05) = 0.05 + -0.5(0.03) = 0.05 + (0.015) = 0.035 = 3.5% c-2) The expected rate of return, using the expected price and dividend for next year is: E(r) = [($103 + $9)/$100] - 1 E(r) = ($112/$100) - 1 E(r) = 0.12 = 12% c-3) Underpriced because the expected return exceeds the fair return.
Call provisions on corporate bonds
callable bonds: may be repurchased by issue at specified call price during call period - company can turn the bond into cash whenever they want
Zero-coupon bonds
carry no coupons, provide all return in form of price appreciation
Default risk
chance that the promised payments aren't made by the borrower
floating-rate bond
coupon rates periodically reset according to specified market date - (LIBOR + 150 bp)
Bi < 1
defensive; lower risk premium than the market
Yields on municipal bonds are generally lower than yields on similar corporate bonds because of:
differences in taxation.
Yield curve
graph of yield to maturity as function of term to maturity
All else equal, duration is shorter at:
higher interest rates
Any security that gives an investor more money back sooner (as a % of your investment):
will have lower price volatility when interest rates change
Duration is equal to maturity for:
zero-coupon bonds
Duration is shorter than maturity for all bonds except:
zero-coupon bonds
A bond currently has a price of $1,050. The yield is 6%. If the yield increases by 22 basis points, price will go down to $1,022. The duration of this bond is?
ΔP/P = -D* Δy $1,050 - $1,022 = $28 -28/1050 = -D*(0.22%) -0.0267 = -D*(0.22%) -0.0267/0.0022 = -D* -12.136 = -D* D* = 12.14 D = D*/(1 + y) D = 12.14(1 + 0.0622) D = 12.14(1.0622) D = 12.895 = 12.90 years
