Bonds and Stocks - Class Roadmap Bonds Stocks Conclusion...

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Unformatted text preview: Class Roadmap Bonds Stocks Conclusion Bond Pricing, the Term Structure of Interest Rates and Stock Pricing Ding Ding University of Toronto September 28, 2011 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Class Roadmap Bonds Bonds and Bond Markets Bond Pricing The Yield Curve Forward Rates Stocks Stocks and Equity Financing Valuation D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Characteristics – What is a Bond? A bond is a borrowing arrangement in which the borrower issues (sells) IOU to the investor. Issuer receives a certain amount of money from the Purchaser(Creditor) in return for the bond Issuer is obligated to repay coupons + the principal at the end of the lifetime of the bond (maturity) It specifies Maturity Face/Par Value Coupon Rate (Coupon = CouponRate ×FaceValue No .ofCouponPaymentsPerYear ) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Characteristics – Issuers Governments/Treasuries E.g., T-bills/notes/bonds Municipalities Agencies E.g., CMHC’s Canada Mortgage Bond Program, Ginnie Mae’s Mortgage Bonds, etc Corporations Notes, Debentures, Mortgage-/Asset-backed D.Ding Bonds and Stocks Ca C Class Roadmap Bonds Stocks Conclusion 24.73% Characteristics Bond Pricing quality Very poor Bond Yields Yield to Maturity and Term Structure Imminent default or in default Bond Ratings Equivalent Credit Ratings Credit Risk Moody's Standard & Poor's Fitch IBCA Duff & Phelps INVESTMENT GRADE Highest quality Aaa AAA AAA AAA High quality (very strong) Aa AA AA AA Upper medium grade (strong) A A A A Baa BBB BBB BBB Lower medium grade (somewhat speculative) Ba BB BB BB Low grade (speculative) B B B B Poor quality (may default) Caa CCC CCC CCC Most speculative Ca CC CC CC No interest being paid or bankruptcy petition filed C C C C In default C D D D Medium grade NOT INVESTMENT GRADE Source: The Bond Market Association D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Ratings - Average Default Rates Moody's Rating Average Default Rate Within One Year of Rating (1970-2001) Definition Notes Aaa 0.00% Highest Rating Available Investment grade bonds. Aa 0.02% Very High Quality A 0.01% High Quality Baa 0.15% Minimum Investment Grade Ba 1.21% Low grade B 6.53% Very speculative Caa Ca Substantial Risk Very poor quality 24.73% C Imminent default or in default D.Ding Bonds and Stocks Below investment grade. "Junk Bonds" Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Characteristics – Zeros Pure discount bonds (zero-coupon bonds) For instance, a Treasury bill (T-bill) It specifies The issuer (government) The face value/par value (e.g. $1, 000, or $10, 000) Maturity (e.g., 3 months, 6 months, with a date) Sold for less than their face value (at a discount) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Characteristics – Straight Bonds Coupon-paying Bonds E.g., government of Canada bonds, corporate bonds. Usually, pay a coupon of the face value (e.g. 8%) at regular periods, e.g., annually or semi-annually, etc. The maturity, the face value, and the issuer are also quoted. At maturity, the face value is repaid. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Cashflow, Time to Maturity, Interest Rate – Key to Pricing The value of a bond depends on the size of its coupon payments, the length of time remaining until the bond matures and the current level of interest rates. The value of a bond = present value of its cash flows (coupons + principal) discounted at a suitable interest rate(s) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Example A zero coupon bond with face value of $100 is trading for $80. It matures in six years from now. The current interest rate is 7%. In bond pricing notation for the class, P = $80 F = $100 r = 7% T=6 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Example A 8% coupon bond with a par value of $100 is trading for $110. It matures in three years from now and pays the coupon semi-annually. The current interest rate is 6%. In bond pricing notation for the class, P = $110 F = $100 r/m = 3% (effective periodic interest rate) m = 2 (the number of coupon payments per year) N = m × T = 2 × 3 = 6 (the total number of periods) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing I — The Zero Coupon Bond P= F (1+r /m)N Note that, as r ↑, price of the zero ↓ D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing — The Law of One Price (No Arbitrage) The Law of One Price (LOP) indicates that two assets with identical cash flows must trade at the same price. Suppose the interest rate on bank deposits is 9%. There are two ways to get $10, 000 in one year: 1 Deposit amount C in the bank, which gives you $10, 000 in one year: F $10, 000 C= = = $9, 174 1+r 1 + 9% 2 Buy a Treasury bill Since both strategies produce $10, 000 in one year, they should cost the same today. The T-Bill should trade for $9, 174. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (1) For the pricing, the basis for the discount rate is 360 days. With n days to maturity, and d % the discount from face value, the $-price P typically quoted as per $100 is P = 100 × 1 − n d% 360 For instance, a 91-day T-bill’s price = 4.36% means P = 100 1 − 91 4.36% 360 = 98.898% of face value So if the face value is F = $10, 000, you have to pay P = $9, 889.80 for the T-bill. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (2) d is also called bank discount yield d is calculated as: d% = 100 − P 360 × 100 n D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (3) Bond equivalent yield is used when computing T-bill yields. One uses 365 days as a basis. $-price P is typically quoted as per $100. Given $-price P and n days to maturity the formula for the bond equivalent yield y for a T-Bill is y= 100 − P 365 × P n D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (4) y= 365 100 − P × P n Note that this is not the effective annual rate but merely a simple annualization. Use simple interest rather than compound interest One can understand this rate to be the rate of return that investors require for the T-Bill investment. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (1) Pay a coupon of the face value (e.g. 8%), either annual or semi-annual (etc.). The coupon payment is quoted annual. A $1, 000 face value bond with 8% semi-annual coupons pays 1 2 × 8% × $1, 000 = $40 every half year. Note: The coupon rate is not the interest rate! interest rate is set by the market according to demand and supply, the coupon rate is determined by the bond-issuer. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (2) Bond value = PV of coupons + PV of par value. Let maturity date be T , the prevailing market interest rate be r, T coupon par value Bond value = . t+ (1 + r ) (1 + r )T t =1 or if there is m > 1 coupon payments per year, let N = m × T N Bond value = t =1 par value coupon . t+ (1 + r /m) (1 + r /m)N So the bond price is P = coupon × 1 r 1− 1 T (1 + r ) D.Ding + par value × Bonds and Stocks 1 (1 + r )T . Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (3) Example Calculate the price of a 7% coupon, 30 year maturity bond making semi-annual coupon payment with par value of $1, 000. Assume the annual interest rate is 8%. N par value coupon t+ (1 + r /m) (1 + r /m)N Bond value = t =1 30×2 = t =1 60 = t =1 $1, 000 × 1+ 8% 2 7% 2 t + $1, 000 1+ 8% 2 30×2 $1, 000 × 3.5% $1, 000 + t (1 + 4%) (1 + 4%)60 = $886.88 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (4) Example Calculate the price of a 7% coupon, 30 year maturity bond making semi-annual coupon payment with par value of $1, 000. Assume the annual interest rate increases to 1) 10%, 2) 20%. 60 P1 = t =1 = $716.06 60 P2 = t =1 = $1, 000 $1, 000 × 3.5% + t 60 (1 + 5%) (1 + 5%) $1, 000 × 3.5% $1, 000 + t 60 (1 + 10%) (1 + 10%) $352.13 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing — Prices and Interest Rates (1) Prices and Interest Rates have an inverse relationship When interest rates get very high the price of the bond will be very low When interest rates approach zero, the price of the bond approaches the sum of the cash flows D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing — Prices and Interest Rates (2) A bond sells at par only if its coupon rate equals the interest rate. A bond sells at a premium only if its coupon rate above the interest rate. A bond sells at a discount only if its coupon rate below the interest rate. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond Pricing between Coupon Dates When a bond is bought between two coupon payments, the buyer has to pay more than just the price (flat/clean price). Implicitly, whoever held the bond for some time between the two coupons is entitled to receive a reward, that is a share of the upcoming coupon payment. This is referred to as accrued interest and computed as follows: AI = days since last coupon date × coupon amount days between two coupon dates Invoice/Dirty/Full price = flat price + accrued interest D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bonds are quoted on a flat price basis in units of 100. Fractions of 1 a dollar are quoted in units of 32nds. So for example, 100 − 07 4 means 100 + 7.25/32 = 100.226563. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion D.Ding Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond yields I Current yield ycurrent = Cash income of a bond (i.e., Annual coupon receipts) Bond price Example a bond with a par value of $1, 000 sell for $886.88, mature in 30 years, and has a 7% annual coupon rate paid semiannually. ycurrent = 7% × $1, 000 = 7.89% $886.88 Ignores any prospective capital gains or loss Ignores reinvest value of coupon payment D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bond yields — Yield to Maturity (YTM) The YTM is a measure of average rate of return that will be earned on a bond if it is bought now and held until maturity. It is the one discount rate that equates the PV of future payments to the current bond price. YTM is the interest rate that investors are using to value the bond. It is an implicit measure that one can obtain only from the current bond-price. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Example A 20-year bond with a face value of $1, 000 pays an annual coupon of 8% and trades at 90.871%. What is its YTM? we need to solve: 20 $908.71 = t =1 $1, 000 8% × $1, 000 + → y = 9% t (1 + y ) (1 + y )20 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure YTM More generally Price and YTM are negatively related If YTM increases, price drops Intuition: Higher yield = more discounting = less willing to pay for far-distant payments. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure YTM More generally The bonds features: F face value C annual coupon m per annum coupon payments — n period remaining y yield to maturity/interest rate C m are physical payments Price of the bond will be n P= t =1 = C y C m 1+ 1− yt m + 1 y 1+ m F y 1+ m n + n F y 1+ m Every variable is known, except for y , solve for y . D.Ding Bonds and Stocks n Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Precise Pricing between Coupon Dates I The formula applies only when there are exactly n payment periods. Between two periods the formula must be adjusted as follows: Assume there are n full periods and the length between payment periods is 365/m. There are k days to an (n + 1)st payment. For k = 0 (the n + 1 coupon payment is about to happen), the present value of this stream is n t =1 C m 1+ yt m + F y 1+ m n + C m thus for k > 0, the present value of this stream is 1 y 1+ m k /(365/m) n t =1 D.Ding C m 1+ yt m + Bonds and Stocks F y 1+ m n + C m Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Precise Pricing between Coupon Dates II However, an investor, will not see all of the upcoming payment as he has to pay accrued interest! Taking this into account, the precise formula is 1 y 1+ m p= − k /(365/m) n t =1 C m 1+ yt m 365/m − k C 365/m m D.Ding Bonds and Stocks + F y 1+ m n + C m Class Roadmap Bonds Stocks Conclusion D.Ding Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Spot Rates, The Term Structure, The Yield Curve I A Spot Rate is the interest rate on a T-year loan to be made today. y1 = 5% means the current rate for a one-year loan today is 5% y2 = 7.01% means the current rate for a two-year loan today is 7.01% Spot rates ≡ YTM on risk-free zero coupon bonds D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Spot Rates, The Term Structure, The Yield Curve II The Term Structure of Interest Rates is the series of spot rates y1 , y2 , y3 , ... Essentially linking interest rate to investment term. The Yield Curve is a plot of the term structure: Interest rate/spot rates against investment term/maturity Zero-Coupon Yield Curve: zero-coupon bond yields (STRIPS) Coupon Yield Curve: coupon bond yields (Treasuries) Corporate Yield Curve: corporate bond yields of the same class (i.e., similar risks or the same credit rating) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The US yield curve in May D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The Current US yield curve D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The Spot Curve I The spot curve is based on the YTM of zero-coupon bonds (or the price of T-bills). If F is the principal repayment, and there is maturity n then P= F (1 + yn )n The spot rate curve consists of the sequence of rates y1 , y2 , ..., yT . Naturally, with the spot rates, one can compute the present value of a large variety of cash flows. E.g., cash-flows are CF1 , CF2 , ..., CFt , then PV = CF1 CF2 CFT + ... + + 2 (1 + y1 ) (1 + y2 ) (1 + yT )T D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The Spot Curve II YTM computations typically assume that the same interest rate is applied for discounting. With spot curves, there can be differences. However, the no-arbitrage principle must apply! Thus a bond investment based on spot-rates cannot give you a different PV than the current bond-price: = CF CF CF F + ... + + + ∗ )T (1 + y ∗ ) (1 + y ∗ )2 (1 + y (1 + y ∗ )T CF CF F CF + + ... + + 2 T (1 + y1 ) (1 + y2 ) (1 + yT ) (1 + yT )T where yt are the spot rates and y ∗ is the yield to maturity (YTM) for this bond with maturity T D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The Spot Curve III What if not, e.g. if bond price is smaller? Buy bond. Strip the coupons Sell zero-coupon bonds with face value CFt and maturity t , (t = 1, 2, ...T − 1) and one with face value CF + F and maturity T . D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure The Spot Curve IV Spot rate Yield to maturity on zero-coupon bonds It is the rate that prevails today for a time period corresponding to the zero’s maturity Short rate The interest rate available for a given time interval (e.g., 1 year) at different points in time. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Short rates vs. Spot rates D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Forward Rates The forward rates signifies a commitment rate for a future investment/borrowing. It derives from the spot rate: Suppose you know y1 and y2 and you care to know the forward rate f2 , i.e. the rate at which you can pre-contract to borrow a period from now. Then (1 + y2 )2 = (1 + y1 ) (1 + f2 ) The more distant ft is computed in the same manner (1 + yt )t = (1 + yt −1 )t −1 (1 + ft ) f2 , ..., fT is then the forward curve Claim: forward rates are unbiased estimates of expected future interest rates. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion D.Ding Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Spot rates, short rates and forward rates D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Spot rates, short rates and forward rates D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Interest rate uncertainty and forward rates In a certain world, different investment strategies with common terminal dates must provide equal rates of return, e.g., two consecutive 1 year investments in zeros would need to offer the same total return as an equal sized investment in a 2 year zero. (1 + r1 ) (1 + r2 ) = (1 + y2 )2 However, future short-rate is unknown (1 + r1 ) (1 + E (r2 )) = (1 + y2 )2 D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Example Suppose r1 = 5% , E (r2 ) = 6%. Now consider a short-term investor who wishes to invest only for one year. In a certain world, she should be indifferent between i ) buying a 1 year bond and ii ) buying a 2-year bond then selling it after 1 year. However, if we take risk into account, Option i ) rate of return = 5% Option ii ) At the end of year 1 if r2 > E (r2 ) , bonds price decreases, rate of return < 5%; if r2 < E (r2 ) , bonds price increases, rate of return > 5%; So, Option ii ) is more risky Clearly, this investor will not hold the 2-year bond unless the offered price of 2-year bond is lower! D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Bond Pricing Bond Yields Yield to Maturity and Term Structure Theories of the term structure The expectation theory: fn = E (rn ) The liquidity preference theory: fn = E (rn ) + risk premium A upward sloping yield curve because Investors expect rising interest rate (expectation hypothesis) Large risk premium for holding longer-term bond (liquidity preference theory) D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Common Stocks Issuer: Corporations. Ownership with residual claim and limited liability D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Valuation of Common Stocks The value of any asset is the present value of its expected future cash flows. Stock ownership produces cash flows from: Dividends Capital Gains The price you are willing to pay depends on size and timing of future dividends, and risks of the stock. Discount rate, r , of the stock is the rate of return investors can expect to earn on stocks with similar risk. Valuation of Different Types of Stocks Zero Growth Constant Growth Differential Growth D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Valuation of Different Types of Stocks I Zero growth P0 = Div r Constant growth P0 = D.Ding Div r −g Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Valuation of Different Types of Stocks II Differential Growth Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. To value a Differential Growth Stock, we need to: Estimate future dividends in the foreseeable future. Estimate the future stock price when the stock becomes a Constant Growth Stock Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate. D.Ding Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Valuation of Different Types of Stocks III Suppose dividends will grow at rate g1 for N years and grow at rate g2 thereafter To value a Differential Growth Stock, we can use P= Div1 (1 + g1 )N 1− r − g1 (1 + r )N D.Ding + DivN +1 / (r − g2 ) Bonds and Stocks (1 + r )N Class Roadmap Bonds Stocks Conclusion Characteristics Stock Valuation Example A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. If stocks of similar risk earn 12% effective annual return, what is the stock worth? P= (1 + g1 )N Div1 1− r − g1 (1 + r )N + 2 × 1.08 1.083 p= 1− + 0.12 − 0.08 1.123 = 28.89 D.Ding DivN +1 (r −g2 ) (1 + r )N 2×1.083 ×1.04 0.12−0.04 1.123 Bonds and Stocks Class Roadmap Bonds Stocks Conclusion Recap Pricing is just the PV of all cash flows that you expect to receive. Bonds Bond Pricing: Zeros and Coupon bonds YTM The Yield Curve and Theories Stocks Valuation - Discounted Dividend Model D.Ding Bonds and Stocks ...
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