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Valuation of Common Stocks and Bonds How to apply the PV concept FIN 819: lecture 3 1 Today’s plan Review what we have learned in the last lecture Valuing stocks • Some terms about stocks • Valuing stocks using dividends • Valuing stocks using earnings • Valuing stocks using free cash flows FIN 819: lecture 3 2 Today’s plan (Continue) Bond valuation and the term-structure of interest rates • Terminology about bonds • The valuation of bonds • The term structure of interest rates • Use duration to measure the volatility of the bond price FIN 819: lecture 3 3 What have we learned in the last lecture? Payback rule • Shortcomings IRR rule • Shortcomings Free-cash flow calculation FIN 819: lecture 3 4 Some specific questions in the calculation of cash flows Include all incidental effects Do not forget working capital requirements Forget sunk costs Include opportunity costs Be careful about inflation Depreciation Financing FIN 819: lecture 3 5 Free cash flows calculation Free cash flows = cash flows from operations + cash flows from the change in working capital + cash flows from capital investment and disposal FIN 819: lecture 3 6 Calculating cash flows from operations Method 1 • Cash flows from operations =revenue –cost (cash expenses) – tax payment Methods 2 • Cash flows from operations = accounting profit + depreciation Method 3 • Cash flows from operations =(revenue – cost)*(1-tax rate) + depreciation *tax rate FIN 819: lecture 4 7 A summary example 2 Now we can apply what we have learned about how to calculate cash flows to the IM&C’s Guano Project (in the textbook), whose information is given in the following slide. FIN 819: lecture 4 8 IM&C’s Guano Project Revised projections ($1000s) reflecting inflation FIN 819: lecture 4 9 IM&C’s Guano Project Cash flow analysis ($1000s) FIN 819: lecture 4 10 IM&C’s Guano Project NPV using nominal cash flows 1,630 2,381 6,205 10,685 10,136 NPV 12,600 1.20 1.20 2 1.20 1.20 1.205 3 4 6,110 3,444 3,519 or $3,519,000 1.20 1.20 6 7 FIN 819: lecture 4 11 New formula In chapter 4, it is argued that FCF=earnings –net investment Net investment = total investment - depreciation Do you agree with this formula? Why? FIN 819: lecture 4 12 Example A project costs $2,000 and is expected to last 2 years, producing cash income of $1,500 and $500 respectively. The cost of the project can be depreciated at $1,000 per year. If the tax rate is 50%, what are the free cash flows? FIN 819: lecture 4 13 One more question Mr. Pool is now 40 years old and plans to invest some fraction of his current annual income of $40,000 in an account with an annual real interest rate of 5%, starting next year until he retires at the age of 70 to accumulate $500,000 in real terms. If the real growth rate of his income is 2%, what fraction of his income must be invested? FIN 819: lecture 4 14 Some terms about stocks Book Value – The value of the stocks according to the balance sheet. Liquidation Value - Net proceeds that would be realized by selling the firm’s assets and paying off its creditors. Market Value Balance Sheet - Financial statement that uses market value of assets and liabilities. FIN 819: lecture 4 15 Some terms about stocks Secondary Market - market in which already issued securities are traded by investors. Dividend - Periodic cash distribution from the firm to the shareholders. P/E Ratio - Price per share divided by earnings per share. Dividend yield – Dividends per share over the price of per share FIN 819: lecture 4 16 Example IBM has a trading price of $70 per share. Its annual earnings per share is $5. Its annual dividend per share is $3.5. What are the P/E and the dividend yield? P/E=70/5=14 Dividend yield=3.5/70 or 5% FIN 819: lecture 4 17 Valuing Common Stocks using dividends (first approach) Dividend Discount Model - Computation of today’s stock price which states that share value equals the present value of all expected future dividends plus the selling price of the stock. Div1 Div 2 Div H PH P0 ... (1 r ) (1 r ) 1 2 (1 r ) H H - Time horizon for your investment. FIN 819: lecture 4 18 Example George has bought one IBM share in the beginning of this year and decides to hold this share until next year. The expected dividend this year is $10 per share and the stock is expected to sell at $110 per share in the end of the year. If the cost of the capital is 10%, what is the current stock price? FIN 819: lecture 4 19 Solution P0=(110+10)/(1+0.1)=$109.1 FIN 819: lecture 4 20 Valuing common stocks using dividends Example Current forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return? FIN 819: lecture 4 21 Solution 3.00 3.24 3.50 94.48 P0 (1 .12) (1 .12) 1 2 (1 .12) 3 P0 $75.00 FIN 819: lecture 4 22 Valuing common stocks using dividends If we forecast no growth, and plan to hold out stock indefinitely, we will then value the stock as the PV of a PERPETUITY. Div1 EPS1 PV ( perpetuity) P0 or r r Assumes all earnings are paid to shareholders. FIN 819: lecture 4 23 Example Suppose that a stock is going to pay a dividend of $3 every year forever. If the discount rate is 10%, what is the stock price for the following cases: • (a) you invest and hold it forever? • (b) you invest and hold it for two years? • (c) you invest and hold it for 20 years? FIN 819: lecture 4 24 Solution (a) P0=3/0.1=$30 (b)P0=PV (annuity) + PV( the stock price at year 2) = 3/1.1 + 3/1.12+(3/0.1)/1.12 = 3/0.1=$30 (c) P0=PV (annuity of 20 years) + PV (the stock price at the year of 20) =$30 FIN 819: lecture 4 25 Valuing Common Stocks Gordon Growth Model: A version of the dividend growth model in which dividends grow at a constant rate (Gordon Growth Model). Stocks can be valued as a perpetuity with a growth rate, if you want to hold this stock forever, that is Div1 P0 rg FIN 819: lecture 4 26 Example Suppose that a stock is going to pay a dividend of $3 next year. Dividends grow at a growth rate of 3%. If the discount rate is 10%, what is the stock price for the following cases: • (a) you buy and hold it forever? • (b) you buy and hold it for two years? • (c) you buy and hold it for 20 years? FIN 819: lecture 4 27 Solution (a) P0=3/(0.1-0.03)=$42.86 (b)P0=PV (annuity) + PV( the stock price at year 2) = 3/1.1 + 3*1.03/1.12+(3*1.032/(0.1- 0.03))/1.12 = 3/(0.1-0.03)=$42.86 (c) P0=PV (annuity of 20 years) + PV (the stock price at the year of 20) =$42.86 FIN 819: lecture 4 28 Capitalization rate Expected Return - The percentage yield that an investor forecasts from a specific investment over a set period of time. Sometimes called the market capitalization rate. Div1 P1 P0 Expected Return r P0 FIN 819: lecture 4 29 Example If Fledgling Electronics is selling for $100 per share today and is expected to sell for $110 one year from now, what is the expected return if the dividend one year from now is forecasted to be $5.00? FIN 819: lecture 4 30 Solution According to the formula, 5 110 100 Expected Return .15 100 FIN 819: lecture 4 31 Capitalization rate The formula for the capitalization rate can be broken into two parts. Capital. Rate = Dividend Yield + Capital Appreciation Div1 P1 P0 Expected Return r P0 P0 FIN 819: lecture 4 32 Using dividends models to derive the capitalization rate Capitalization Rate can be estimated using the perpetuity formula, given minor algebraic manipulation. Div1 P0 rg Div1 r g P0 FIN 819: lecture 4 33 Valuing Common Stocks Example- If a stock is selling for $100 in the stock market, the cost of capital is 12% and the next year dividend is $3, what might the market be assuming about the growth in dividends? $3.00 $100 .12 g g .09 FIN 819: lecture 4 34 Some terms about dividend growth rates If a firm elects to pay a lower dividend, and reinvest the funds, the stock price may increase because future dividends may be higher. Payout Ratio - Fraction of earnings paid out as dividends Plowback Ratio - Fraction of earnings retained by the firm. FIN 819: lecture 4 35 Deriving the dividend growth rate g Growth can be derived from applying the return on equity to the percentage of earnings plowed back into operations. ROE Return on Equity EPS ROE Book Equity Per Share g = return on equity X plowback ratio FIN 819: lecture 4 36 Example Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision? FIN 819: lecture 4 37 Solution Without growth 5 P0 $41.67 0.12 With growth g 0.4 * 0.2 0.08 5 * 0.6 P0 $75 0.12 0.08 FIN 819: lecture 4 38 Example (continued) If the company did not plowback some earnings, the stock price would remain at $41.67. With the plowback, the price rose to $75.00. The difference between these two numbers (75.00-41.67=33.33) is called the Present Value of Growth Opportunities (PVGO). FIN 819: lecture 4 39 Valuing common stocks using earnings We often use earnings to value stocks as EPS1 P0 PVGO r What is the relationship between this formula and the dividend growth formula? FIN 819: lecture 4 40 Example Firm A has a market capitalization rate of 15%. The earnings are expected to be $8.33 per share next year. The plowback ratio is 0.4 and ROE is 25%. Every investment in year i is to yield a simple perpetuity starting in year (i+1) with each cash flow equal to total investment times ROE. All the investments have the same capitalization rate. • (a) Using the formula P=ESP1/r + PVGO to calculate the stock price • (b) If ROE is increased, what will happen to the stock price? Why? • (c) Use the dividend model to calculate the stock price? • (d) What have you found? • (e) Think about why you have this kind of result? FIN 819: lecture 4 41 Simple Solution (a) g=10%, EPS1/r=8.33/0.15=$55.56 PVGO=NPV1/(r-g)=2.22/(0.15- 0.1)=$44.44, P=$100 (b) The price will be increased (c) P=Div1/(r-g)=5/(0.15-0.1)=$100 FIN 819: lecture 4 42 Valuing common stocks using FCF (free cash flows) The value of a business or stock is usually computed as the discounted value of FCF out to a valuation horizon (H). The horizon value is sometimes called the terminal value . FCF1 FCF2 FCFH PVH PV ... (1 r ) (1 r ) 1 2 (1 r ) H (1 r ) H FIN 819: lecture 4 43 FCF and PV FCF1 FCF2 FCFH PVH PV ... (1 r ) (1 r ) 1 2 (1 r ) H (1 r ) H PV (free cash flows) PV (horizon value) FIN 819: lecture 4 44 FCF and PV Free Cash Flows (FCF) should be the theoretical basis for all PV calculations. FCF is a more accurate measurement of PV than either Div or EPS. The market price does not always reflect the PV of FCF. When valuing a business for purchase, always use FCF. FIN 819: lecture 4 45 FCF and PV Example Given the cash flows for Concatenator Manufacturing Division, calculate the PV of near term cash flows, PV (horizon value), and the total value of the firm. r=10% and g= 6% Year 1 2 3 4 5 6 7 8 9 10 Asset Value 10.00 12.00 14.40 17.28 20.74 23.43 26.47 28.05 29.73 31.51 Earnings 1.20 1.44 1.73 2.07 2.49 2.81 3.18 3.36 3.57 3.78 Investment 2.00 2.40 2.88 3.46 2.69 3.04 1.59 1.68 1.78 1.89 Free Cash Flow - .80 - .96 - 1.15 - 1.39 - .20 - .23 1.59 1.68 1.79 1.89 .EPS growth (%) 20 20 20 20 20 13 13 6 6 6 FIN 819: lecture 4 46 FCF and PV Solution 1 1.59 PV(horizon value) 6 22 .4 1.1 .10 .06 .80 .96 1.15 1.39 .20 .23 PV(FCF) - 1.1 1.1 1.1 1.1 1.1 1.16 2 3 4 5 3.6 FIN 819: lecture 4 47 FCF and PV PV(busines s) PV(FCF) PV(horizon value) -3.6 22.4 $18.8 FIN 819: lecture 4 48 How to estimate the horizon value? It is very difficult to forecast or estimate the horizon value. There are several ideas that may be used to estimate the horizon value. • Competition • Constant growth rate FIN 819: lecture 4 49 Another example Imagine Corporation has just paid a dividend of $0.40 per share. The dividends are expected to grow at 30% per year for the next two years and at 5% per year thereafter. If the required rate of return in the stock is 15% (APR), calculate the current stock price. FIN 819: lecture 4 50 Solution Answer: First: visualize the cash flow pattern; • C1, C2 and P2 Then, you know what to do? P0 = [(0.4 *1.3)/1.15] + [(0.4 * 1.3^2)/(1.15^2)] + [(0.4 * 1.3^2*1.05)/((1.15^2 * (.15 -.05))] = $6.33 FIN 819: lecture 4 51 Another cash flow problem! Firm Excellent has an old packaging machine that can be used for another 3 years. The remaining book value for this old machine is $15,000, which can be depreciated in the next three years by using the straight-line depreciation approach. If the old packaging machine is used, the annual total cost of operation, labor and maintenance is $10,000. In the market, the new packaging machine is available now at the price of $50,000, which will increase by 5% annually. Based on Excellent’s experience, the new packaging machine will be used forever. The capital investment cost of the new package machine will be depreciated in the next 10 years by using the straight-line depreciation approach. The annual total cost of operation, labor and maintenance will be $8,000 when the new package machine is used. Questions: a. What is the valuation horizon used in this problem? b. Should Excellent invest in the new packaging machine now or wait three years later? FIN 819: Lecture 2 Bonds Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during some time horizon. Coupon - The interest payments made to the bondholder. Face Value (Par Value, Principal or Maturity Value) - Payment at the maturity of the bond. Coupon Rate - Annual interest payment, as a percentage of face value. FIN 819: lecture 4 53 Bonds A bond also has (legal) rights attached to it: • if the borrower doesn’t make the required payments, bondholders can force bankruptcy proceedings • in the event of bankruptcy, bond holders get paid before equity holders FIN 819: lecture 4 54 An example of a bond A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. • The coupon payment is $100 annually • The discount rate is different from the coupon rate. • In the third year, the bondholder is supposed to get $100 coupon payment plus the face value of $1000. • Can you visualize the cash flows pattern? FIN 819: lecture 4 55 Bonds WARNING The coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce. Since the coupon rate is listed as a %, this misconception is quite common. FIN 819: lecture 4 56 Bond Valuation The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return. cpn cpn 1,000 cpn PV ... (1 r )1 (1 r ) 2 (1 r ) N FIN 819: lecture 4 57 Zero coupon bonds Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds) You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity ) How much is a 10-yr zero coupon bond worth today if the face value is $1,000 and the effective annual Face rate is 8% ? value PV Time=0 Time=t FIN 819: lecture 4 58 Zero coupon bonds (continue) P0=1000/1.0810=$463.2 So for the zero-coupon bond, the price is just the present value of the face value paid at the maturity of the bond Do you know why it is also called a discount bond? FIN 819: lecture 4 59 Coupon bond The price of a coupon bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return. cpn cpn (cpn par) PV .... (1 r )1 (1 r ) 2 (1 r )t 1 1 cpn par PV (annuity) PV ( par) r r (1 r )t (1 r )t FIN 819: lecture 4 60 Bond Pricing Example What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%. FIN 819: lecture 4 61 Bond Pricing Example What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%. 60 60 1,060 PV 1 2 (1.056 ) (1.056 ) (1.056 )3 PV $1,010 .77 FIN 819: lecture 4 62 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 6 %? 60 60 1,060 PV 1 2 3 (1.06 ) (1.06 ) (1.06 ) PV $1,000 FIN 819: lecture 4 63 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 15 %? 60 60 1,060 PV 1 2 3 (1.15 ) (1.15 ) (1.15 ) PV $794 .51 FIN 819: lecture 4 64 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? FIN 819: lecture 4 65 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? 30 30 30 1,030 PV 1 2 ... (1.028 ) (1.028 ) 5 (1.028 ) (1.028 )6 PV $1,010 .91 FIN 819: lecture 4 66 Bond Pricing Example (continued) Q: How did the calculation change, given semi- annual coupons versus annual coupon payments? FIN 819: lecture 4 67 Bond Yields Current Yield - Annual coupon payments divided by bond price. Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond. cpn cpn (cpn par ) P .... (1 y )1 (1 y ) 2 (1 y )t FIN 819: lecture 4 68 An example of a bond A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the present value of the bond at the discount rate of 8%. • What is the current yield? • What is the yield to maturity. FIN 819: lecture 4 69 My solution First, calculate the bond price P=100/1.08+100/1.082+1100/1.083 =$1,051.54 Current yield=100/1051.54=9.5% YTM=8% FIN 819: lecture 4 70 Bond Yields Calculating Yield to Maturity (YTM=r) If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r. cpn cpn (cpn par) P .... 1 2 (1 r ) (1 r ) (1 r )t FIN 819: lecture 4 71 Bond Yields Example What is the YTM of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,010.77 60 60 1,060 PV (1 r ) (1 r ) 1 2 (1 r ) 3 PV $1,010.77 FIN 819: lecture 4 72 Bond Yields In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam You may use the trial by errors approach get it. FIN 819: lecture 4 73 Bond Yields (3) Can you guess which one is the solution? (a) 6.6% (b) 7.1% (c) 6.0% (d) 5.6% My solution is (d). FIN 819: lecture 4 74 The rate of return on a bond Coupon income + price change Rate of return = investment or bond price profit Rate of return = cost of investment Example: An 8 percent coupon bond has a price of $110 dollars with maturity of 5 years and a face value of $100. Next year, the expected bond price will be $105. If you hold this bond this year, what is the rate of return? FIN 819: lecture 4 75 My solution The expected rate of return for holing the bond this year is (8-5)/110=2.73% • Price change =105-110=-$5 • Coupon payment=100*8%=$8 • Profit=8-5=$3 • The investment cost or the initial price=$110 FIN 819: lecture 4 76 Some new terms So far, we consider one discount rate for all the cash flows In fact, the discount rate for one period cash flows can be different from the discount rate for two-period cash flows. Spot interest rate: the actual interest rate available today (t=0) Future interest rate: the spot rate in the future (t>0) FIN 819: lecture 4 77 Example Spot rates (r) Investment Horizon r 1 6% 2 6.5% 3 7% 4 7.2% FIN 819: lecture 4 78 The Yield Curve Term Structure of Interest Rates: is the relationship between the spot rates and their maturity dates Yield Curve - Graph of the term structure. FIN 819: lecture 4 79 The term structure of interest rates (Yield curve) FIN 819: lecture 4 80 Value the bond (revisit) If we are given the term structure of interest rates, we know the discount rates for cash flows in different time periods. Then cpn cpn (cpn par ) PV .... (1 r1 ) (1 r2 ) 1 2 (1 rt ) t Here r1, r2, …, rt are spot rates for period 1, 2, …, t, respectively. FIN 819: lecture 4 81 Question Which kind of the yield curve can make you use a single discount rate for the bond valuation? For what kind of bonds, YTM is the same as spot rates? FIN 819: lecture 4 82 Example Please use the following information to value a 10%, four years coupon bond, if the spot rates are: Year Spot rate 1 5% 2 5.4% 3 5.7% 4 5.9% FIN 819: lecture 4 83 Solution The interest payment is $100 every year. 100 100 100 (100 1,000) PV (1 .05)1 (1 .054) 2 (1 .057)3 (1 .059) 4 $1,144.5 FIN 819: lecture 4 84 A problem A 6 percent six-year bond yields 12% and a 10 percent six year bond yields 8 percent. Please calculate six-year spot rate. FIN 819: lecture 4 85 Forward rate Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time according to the term structure of the interest rates. The forward rate is implied by the term structure of interest rates and doesn’t exist in a financial market FIN 819: lecture 4 86 Forward rate Another way to look at the forward rate is that bonds can be priced in the following way: cpn cpn (cpn par ) PV .... (1 r1)1 (1 r2 ) 2 (1 rt )t cpn cpn (cpn par ) PV .... (1 r1)1 (1 r1)(1 f 2 ) (1 rt 1)t 1(1 ft ) FIN 819: lecture 4 87 Forward rate calculation One period forward rate can be calculated by using the spot rates as follows: n (1 rn ) 1 fn n 1 (1 rn 1) Where fn is the forward rate from period (n-1) to period n, rn is the n-period spot rate and rn-1 is the spot rate for the (n-1)-period spot rate. FIN 819: lecture 4 88 Example Using the information for spot rates given in the previous example, what is the forward rate(f2) from year 1 to year 2? FIN 819: lecture 4 89 Solution Here n=2, (1 r2 ) 2 1.0542 1 f2 1.058 (1 r1)1 1.05 f 2 5.8% FIN 819: lecture 4 90 What moves the interest rate? Nominal interest rate • What is it? Real interest rate • What is it? Inflation • What is it? Can the nominal interest rate be less than zero? Can the real interest rate be less zero? FIN 819: lecture 4 91 What moves the interest rate? Irving Fisher’s theory • Nominal r = (1+Real r)(1+ expected inflation)-1 • Real r is theoretically somewhat stable • Change in inflation drives the change in the interest rate This theory doesn’t work bad in the past 50 years. Why do we care about the movement of the interest rate? FIN 819: lecture 4 92 Bond price volatility When the interest rate changes, what will happen to the bond price? cpn cpn ( cpn par ) P .... (1 y )1 (1 y ) 2 (1 y )t Then what decides the sensitivity of the bond price to the interest rate change? FIN 819: lecture 4 93 Duration To capture the sensitivity of the bond price to the interest rate change, financial economists have defined a measure called Duration (or Macaulay Duration) Duration is a weighted average of the maturity for the cash flows of the bond FIN 819: lecture 4 94 Example A two-year 5% coupon bond with YTM of 10%. Maturity Cash flows 1 50 2 50+1000 FIN 819: lecture 4 95 Duration t Mathematically Duration iwi i 1 PV (Ci ) t wi and wi 1 P i 1 Volatility (percent) =Duration/(1+YTM) Change in bond price = volatility*change in interest rates. If YTM is small, Change in bond price = Duration*change in interest rates. FIN 819: lecture 4 96 Example Calculate the Duration, volatility and the change of the bond price for 1% change in the interest rate for a bond that is 6 7/8%, 5-year coupon bond with 4.9% YTM. FIN 819: lecture 4 97 Solution i Ci PV(Ci) wi __ _i* wi _______ 1 68.75 65.54 .060 0.060 2 68.75 62.48 .058 0.115 3 68.75 59.56 .055 0.165 4 68.75 56.78 .052 0.209 5 68.75 841.39 .775 3.875 1085.74 1.00 4.424 FIN 819: lecture 4 98 Solution Volatility=4.424/(1.049)=4.22 Price change=4.22*1%=4.22% FIN 819: lecture 4 99 Example (practice question 8) Please use the following information to answer the questions in the next slide. Year Spot rate 1 5% 2 5.4% 3 5.7% 4 5.9% 5 6.0% FIN 819: lecture 4 100 Questions (a) What are the discount factors for each date? (b) What is the forward rates for each period? (c) Calculate the PV of 5 percent and 10 percent five-year note? (d) Which note is going to have a higher YTM? FIN 819: lecture 4 101 Solution (a)df1=1/1.05=0.95 df2=1/1.0542=0.90 (b) f2=1.0542/1.05-1=5.8% f3=1.0573/1.0542-1=6.3% (c) PV(5%)=$959.34 PV(10%)=$1171.43 (d) 5% note has a higher YTM. FIN 819: lecture 4 102