Finance Lecture3

MBAM 614 Finance Bonds and Term Structure • MBAM614 Class 3 - 1 Summary of Last Class $1 Tomorrow is Worth Less Than $1 Today Basic PV Equation: PV = FVt * 1 (1 + r)t If We Know Any 3 Variables, Can Solve For the Last Summing the PV of Individual CF’s is Equivalent to Discounting Back One Period at a Time • MBAM614 Class 3 - 2 1 Summary of Last Class 1 1. 1 (1 + r)t r = $C * PVIFA(r,t) C r C r-g APV = $C * 2. PPV = GPPV = 3. 4. NPV = PV(Cash Inflows) – PV(Cash Outflows) EAR = (1 + QR/m)m - 1 Class 3 - 3 5. • MBAM614 Agenda 1. Bonds and Bond Valuation 2. Term Structure of Interest Rates 3. Forward Rates 4. Expectations Hypothesis 5. Liquidity Preference Hypothesis • MBAM614 Class 3 - 4 2 Bonds A long term debt security issued by a government or corporation is generally called a Bond. In return for the loan, the issuer promises to pay: – Regular Coupon (interest) payments every period until the bond matures – The Face Value or Par Value of the bond when it matures • MBAM614 Class 3 - 5 Bond Terminology The date on which a bond’s face value is repaid is called the bond’s Maturity Date The time remaining before a bond matures is its Term to Maturity The bond’s Coupon Rate is its annual coupon divided by it face value The discount rate that equates the present value of a bond’s cash flows with its market price is called the bond’s Yield to Maturity (YTM) or simply Yield • MBAM614 Class 3 - 6 3 Example: Consolidated Moose Pastures Consolidated Moose Pastures issued a 15 year bond on January 26, 1994 on which they pay $80 interest per bond annually. When these bonds mature, CMP will have to pay $1000 for each bond. These bonds have: - Maturity Date of January 26, 2009 - Term to Maturity of 5 years (Jan 26, 2009 - Jan 26, 2004) - Par Value of $1000 - A Coupon of $80 and a Coupon Rate of 8% ($80/$1000) If the current interest rate for bonds of this type is 10% (that is, the YTM), what is the value of this bond? • MBAM614 Class 3 - 7 Valuing a Bond Valuing a bond is straight-forward. Simply find the PV of the bond’s cash flows. 2004 t=0 Coupon Face Value 2005 1 2006 2 2007 3 2008 4 2009 5 $80 $ 80 $80 $ 80 $80 $ 80 $80 $ 80 $80 $1000 $1080 Sum of PV of individual Coupons and Face Value PV of Coupon Annuity plus PV of Face Value • MBAM614 Class 3 - 8 4 Example: CMP Revisited CMP’s bonds have a term of 5 years, an annual coupon of $80, and a par value of $1000. The appropriate discount rate is 10%. PV of Coupon Annuity: APV = $80 * PVIFA(10%,5) = $80 * {1 - [1/(1.10)5]}/0.10 = $80 * 3.7908 = $303.26 PV of Face Value: PV = $1000 * PVIF(10%,5) = $1000 * 1/(1.10)5 = $1000 * 0.6209 = $620.90 Bond Value: $303.26 + $620.90 = $924.16 • MBAM614 Class 3 - 9 Example: Wilhite Company Wilhite Co. plans to issue bonds with a face value of $1000, a term to maturity of 20 years, and an annual coupon of $110. Similar bonds have a YTM of 11%. What will Wilhite’s bonds sell for? PV of Coupons = $110 * {1 - [1/(1.11)20]}/0.11 = $110 * 7.9633 = $875.97 PV of face value = $1000/(1.11)20 = $124.03 Price = $875.97 + $124.03 = $1000 Notice that since the Coupon Rate and the YTM are the same, the price = face value For bonds: price = face value ⇔ coupon rate = YTM • MBAM614 Class 3 - 10 5 Discount Bonds If a bond’s coupon rate is less than current YTM, what will its price be? Since “the market” wants interest equal to the YTM and the bond is paying less than that rate, these bonds will not be as valuable as other investments that do pay YTM. In this case, the bond’s price will be less than its face value and the bond is said to be a Discount Bond or a bond Selling at a Discount • MBAM614 Class 3 - 11 Example: Wilhite’s Bonds What would Wilhite’s bonds sell for if the required YTM was 13%? Recall that the Wilhite bonds had a coupon rate of 11% (< YTM), so we would expect these bonds to be selling at a discount. PV of Coupons = $110 * {1 - [1/(1.13)20]}/0.13 = $110 * 7.0248 = $772.72 PV of face value = $1000/(1.13)20 = $86.78 Price = $772.72 + $86.78 = $859.50 The difference between these bonds and an investment paying the required YTM (13%) is $1000 * (13%-11%) or $20 per year: APV = $20 * PVIFA(13%,20) = $20 * 7.0248 = $140.50 • MBAM614 Class 3 - 12 6 Premium Bonds If a bond’s coupon rate is greater than the current YTM, what will its price be? Since “the market” wants interest equal to the YTM and the bond is paying more than that, these bonds will be more valuable than other investments that do pay YTM. In this case, the bond’s price will be greater than its face value and the bond is said to be a Premium Bond or a bond Selling at a Premium • MBAM614 Class 3 - 13 Example: Wilhite’s Premium Bonds What would Wilhite’s bonds sell for if the required YTM was 9%? Since Wilhite bonds had a coupon rate of 11% (> YTM), we expect these bonds to be selling at a premium. PV of Coupons = $110 * {1 - [1/(1.09)20]}/0.09 = $110 * 9.1286 = $1004.14 PV of face value = $1000/(1.09)20 = $178.43 Price = $1004.14 + $178.43 = $1182.57 Again the difference between these bonds and an investment paying the required YTM (9%) is $1000 * (11%-9%) or $20 per year but in the bondholder’s favor. What do you think the PV of this additional $20 per year is? Why? • MBAM614 Class 3 - 14 7 Rules of Thumb for Bond Yields and Prices Bonds Selling at Par Coupon Rate = YTM ⇔ Price = Face Value Discount Bonds Coupon Rate < YTM ⇔ Price < Face Value (Raising YTM will cause the price to fall) Premium Bonds Coupon Rate > YTM ⇔ Price > Face Value (Lowering the YTM will cause the price to rise) • MBAM614 Class 3 - 15 Bond Valuation In general, the value of a bond is given by Bond Value = PV of Coupons + PV of Face Value 1Bond Value = C * 1 (1 + YTM)t 1 + F* YTM (1 + YTM)t If YTM ↑ then Bond Value ↓ and vice versa • MBAM614 Class 3 - 16 8 More Than One Coupon per Year If a bond has more than one coupon payment per year, simply treat the quoted YTM as an APR. For example, find the value of a 10 year, $1000 bond with an 8% coupon rate and semi-annual coupon if the quoted yield is 10%. There are 20 semi-annual coupon payments of $40, and the semiannual yield is 5%. Since YTM>C, should be at a discount: PV of Coupons = $40 * {1 - [1/(1.05)20]}/0.05 = $40 * 12.4622 = $498.49 PV of Face Value = $1000/(1.05)20 = $376.89 Bond Value = $498.49 + $376.89 = $875.38 • MBAM614 Class 3 - 17 Interest Rate Risk Bond prices change as interest rates (YTMs) change bonds are subject to Interest Rate Risk Given two bonds and a fixed change in rates, the bond with the largest price change is said to have more interest rate risk All else equal: – the longer the time to maturity, the greater the interest rate risk – the lower the coupon rate, the greater the interest rate risk • MBAM614 Class 3 - 18 9 Example: Government Bonds Consider two U.S. Treasury bonds both with an 8.5% coupon rate paid semi-annually. One has 2 years to maturity and the other has 28 years. If the required yield on both bonds is 8.5%, both will be selling at par (why?). What will their prices be if yields rise to 9%? The price of both bonds will drop - how far? 2 yr bond: $42.50 * {1 - [1/(1.045)4]}/0.045 + $1000/(1.045)4 = $152.47 + $838.56 = $991.03 $42.50 * {1 - [1/(1.045)56]}/0.045 + $1000/(1.045)56 = $864.15 + $85.01 = $949.16 28 yr bond: • MBAM614 Class 3 - 19 Finding a Bond’s YTM A matter of trial-and-error although rules-of-thumb help – make a guess based on coupon rate and price relative to par – calculate bond price based on your guess for YTM – if your price is too high, increase estimate of YTM (this will decrease your calculated price) – if your price is too low, decrease YTM estimate (thus increasing calculated price) – repeat until your price is “reasonably close” • MBAM614 Class 3 - 20 10 Corporate Bond Quotes Bond ATT ATT BellsoT BellsoT BethSt 9 yrs ChaseM DukeEn 25 yrs DukeEn IBM IBM Cur Yld 7.1 8.1 6.6 8.0 8.5 7.0 7.9 8.0 6.4 7.2 Vol 13 133 40 17 5 3 46 20 5 75 Close 100 100 883/8 981/8 99 933/8 853/4 931/8 9927/32 1003/8 Net Chg -1/4 -7/8 -5/8 +3/8 +1/4 +1/2 -1/2 -1/8 2 yrs 71/802 81/822 57/809 77/832 83/801 61/209 63/425 71/225 63/800 71/402 (Bond quotes on Jan 17/00) NOT YTM Class 3 - 21 • MBAM614 Term Structure of Interest Rates We have assumed that interest rates are fixed over time. However, interest rates vary across times to maturity Nominal interest rates (what we actually get paid) have two components: – real interest rate (increase in purchasing power) – inflation rt ≈ ( rrt + it ) Variation across time due to differences in expected inflation • MBAM614 Class 3 - 22 11 Term Structure of Interest Rates The t year interest rate that we can borrow or invest at today to be repaid in year t is called the t year Spot Rate, rt Eg. if we can borrow for one year at 8.5% or three years at 9.25% then the one year spot rate is r1 = 8.5% and the three year spot rate is r3 = 9.25% Collectively, the set of spot rates for different terms to maturity at any one point in time is called the Term Structure of Interest Rates • MBAM614 Class 3 - 23 Spot Rates vs Yield to Maturity If the current spot rates for one and two years were r1 = 10% and r2 = 11%, what would a 2 year, 8% bond be worth today? Assume $1000 par value and an annual coupon. Alternatively, if you could “sell” the first coupon payment and the final coupon + face value separately, how much would you get? (this is known as “stripping” a bond) t=0 1 2 Total Value = Bond Value = $949.28 $80 $72.73 $876.55 • MBAM614 $1080 r1 = 10% r2 = 11% Class 3 - 24 12 Spot Rates vs Yield to Maturity The YTM for this bond is that rate that satisfies $80 (1 + YTM) $1080 (1 + YTM)2 $949.28 = + YTM = 10.96% Not the same as either spot rate - it is a “time weighted” average due to multiple cash flows in multiple periods YTM on a 0 coupon bond is a spot rate • MBAM614 Class 3 - 25 Forward Rates The Forward Rate from year 1 to year 2, denoted f1,2, is the interest rate you can arrange today for an investment from year 1 to year 2 (one year) Eg. Suppose you know you will be getting $5,000 for your, er, 29th birthday on Jan 23, 2004. You would like to “lock in” the interest rate you get paid on this money until you graduate in 2005. Since you won’t have the money for a year, today’s spot rate does not apply. Your bank may offer you a one year rate one year forward. Banks (and financial markets) usually offer a complete menu of forward rates • MBAM614 Class 3 - 26 13 Example: Forward Rates You have set aside $10,000 as a down-payment for a house which you intend to buy 4 years from now (after graduation and a trip round the world!). Your bank has offered you 10% compounded annually for 4 years but before leaving, your branch manager let slip that they were also offering a 2 year rate 2 years forward of 11%. This got you thinking so you called your broker and found that you could purchase 2 year T-bonds that paid 9% annually. What should you do? What is the EAR of the second option? t=0 × 1.10 r4 = 10% • MBAM614 1 × 1.10 2 × 1.10 3 × 1.10 4 = (1.10)4 = 1.4641 Class 3 - 27 Example: Forward Rates t=0 × 1.09 r2 = 9% 1 × 1.09 r2 = 9% 2 × 1.11 f2,4 = 11% 3 × 1.11 f2,4 = 11% 4 = (1.09)2 * (1.11)2 = 1.4639 r4 is better • MBAM614 Class 3 - 28 14 Implied Forward Rates Suppose the current spot rates for one and two years were r1 = 10% and r2 = 11%. If you don’t care how you invest you money, what would the forward rate be from year 1 to year 2? You have two choices for your investment: – one year at r1 and the second year at f1,2 – two years at r2 Since you don’t care how you invest your money, both must pay the same total interest or have the same FV: $1(1 + r1)(1 + f1,2) = $1(1 + r2)2 or (1 + f1,2) = (1.11)2/1.10 = 1.12 f1,2 = 12% • MBAM614 Class 3 - 29 Explaining the Term Structure Two theories are most common – Expectations Hypothesis says that the forward rates reflect everyone’s (collective) “best guess” about what spot rates will be tomorrow rising term structure means interest rates are expected to rise – Liquidity Preference Hypothesis states that, due to the greater interest rate risk of investing for long periods, people prefer to invest for short periods. Must pay a premium to attract long term investments even if interest rates are NOT expected to rise, the term structure will be rising • MBAM614 Class 3 - 30 15 Key Points 1. 2. 3. 4. 5. Bond Value = PV of Coupons + PV of Face Value YTM is the discount rate that satisfies Bond Value = C * PVIFA(YTM,t) + F * PVIF(YTM,t) Bond Value drops as YTM rises Bond Value increases as YTM decreases All else equal: - longer term means greater interest rate risk - lower coupon means greater interest rate risk Spot rates and forward rates are linked (1 + r1)(1 + f1,2) = (1 + r2)2 6. • MBAM614 Class 3 - 31 MBAM 614 Finance Stocks and NPV • MBAM614 Class 3 - 32 16 Agenda 1. Stock and Stock Valuation 2. Dividends and Constant Dividend Growth 3. Non-constant Dividend Growth 4. Estimating Growth • MBAM614 Class 3 - 33 Stock Valuation As with bonds, simply PV the cash flows Problems arise because: – Cash flows are not known with certainty – Unlike bonds, stock doesn’t mature so its term to maturity is infinite – Bond interest rates are reasonably easy to find (newspapers, etc) but the “required return” (or market rate) on stock is not • MBAM614 Class 3 - 34 17 Stock Cash Flows In theory, include dividends and selling price Eg. If we expect a dividend of $1.50 in one year and we plan to sell the stock in one year for $30, what is the stock worth today if our required return is 18%? Simply find PV of D1 = $1.50 and P1 = $30 t=0 1 P0 = $26.69 • MBAM614 × 1/1.18 D1 = $ 1.50 P1 = $30.00 $ 31.50 Class 3 - 35 Future Stock Prices P0 = D1/(1 + r) + P1/(1 + r) How do we know what the stock price, P1, will be next year? Same approach P1 = D2/(1 + r) + P2/(1 + r) P0 = D1/(1 + r) + D2/(1 + r)2 + P2/(1 + r)2 P0 = D1/(1 + r) + D2/(1 + r)2 + … + Dt /(1 + r)t + ... • MBAM614 Class 3 - 36 18 Future Dividends In practice, means we need to estimate and discount an infinite number of dividends. Clearly need to simplify. A simple assumption is constant dividends (0 growth) D1 = D2 = D3 = … = D (a constant) t=0 P0 = ? 1 D 2 D 3 D Just a perpetuity so: • MBAM614 P0 = D/r Class 3 - 37 Example: Preferred Shares CitiGroup Inc., the parent of CitiBank, has a preferred share outstanding (C.PR.G) that pays total dividends of $3.11 per year. If the dividend is not expected to change and the required rate of return on these shares is 7%, what will they be selling for? P0 = D/r = ($3.11/4)/(0.07/4) = $44.43 If these shares are actually selling for $51.80 right now, what is the required rate of return on them? Since P0 = D/r , r = D/ P0 so r = D/ P0 = ($3.11/4)/$51.80 = 0.01501 or 6.00%/yr Remember: Dividend is Quarterly so Rate is Quarterly • MBAM614 Class 3 - 38 19 Constant Growth Dividends grow by g% per year forever D1 = D0 * (1 + g) D2 = D1 * (1 + g) = D0 * (1 + g)2 and generally, Dt = D0 * (1 + g)t Eg. Sears, Roebuck & Co. (S) shares paid $0.92 in dividends this year. Assuming annual dividends and that everyone expects dividends to grow by 10% per year forever, what will Sears’ dividend be in 5 years? D5 = D0 * (1 + g)5 = $0.92 * (1.10)5 = $1.482 An amount that grows at a constant rate forever is called a Growing Perpetuity • MBAM614 Class 3 - 39 Share Price With Constant Growth P0 = D0(1+g)1/(1+r)1 + D0(1+g)2/(1+r)2 + D0(1+g)3/(1+r)3 + … When g < r, this simplifies to: P0 = D0* (1 + g)/(r - g) = D1/(r - g) The price at any time t is: Pt = Dt* (1 + g)/(r - g) = Dt+1/(r - g) This is called the Constant Growth Model of stock price VERY IMPORTANT • MBAM614 Class 3 - 40 20 Example: Sears If people require a 16.5% return on Sears, Roebuck & Co. shares, what will you be able to buy them for? This year’s dividend (paid annually) was $0.92 and growth is expected to continue at 15% per year forever. First, check: is g < r ?15% < 16.5% so we can use the constant growth model. Since expect D0 = $0.92 (this year’s dividend) D1 = $0.92 * (1 + 0.15) = $1.058 P0 = D1/(r - g) = $1.058/(16.5% - 15%) = $1.058/0.015 = $70.53 Since these shares are actually selling for $32, people either expect lower growth, require higher returns, or both. • MBAM614 Class 3 - 41 Changing Growth What happens to P0 as g → r ? (recall that CGM doesn’t apply if g > r) Stock Price Sensitivity to Growth, g 50 45 40 35 30 25 20 15 10 5 0 10% 11% 12% 13% 14% 15% 16% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% Stock Price D0 = $0.20 r = 16.5% Grow th • MBAM614 Class 3 - 42 21 Changing Required Return What happens to P0 as r increases? Stock Price Sensitivity to Required Return, r 25 20 Stock Price 15 10 5 0 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% D0 = $0.20 g = 10.0% Re quire d Re tu rn • MBAM614 Class 3 - 43 Example: NCG Company The NCG Company has just paid a $5 dividend. Management (and everyone else) believes their dividends will grow at 10% for the next two years, 8% the year after, then at 6% indefinitely. If the required return is 12%, what should NCG’s stock sell for? Time 0 1 2 3 4 at t = 3, and Growth 10% 10% 8% 6% Dividend $5.000 $5.500 $6.050 $6.534 $6.926 Super-normal Growth Normal Growth Terminal Value P3 = D4/(r - g) = $6.926/(0.12 - 0.06) = $115.43 P0 = D1/(1 + r) + D2/(1 + r)2 + D3/(1 + r)3 + P3/(1 + r)3 = $5.50/1.12 + $6.05/(1.12)2 + $6.534/(1.12)3 + $115.43/(1.12)3 = $96.55 • MBAM614 Class 3 - 44 22 Required Rates of Return Rearranging the CGM equation high-lights the components of the required rate of return P0 = D1/(r - g) r = D1/P0 + g r = expected capital + dividend yield gains yield • MBAM614 Class 3 - 45 Example: Dillon’s Dilemma Dillon prefers his returns as capital gains rather than dividends because he can wait to pay taxes on capital gains (dividends are taxed in the year received – this may change if Mr. Bush has his way!). If Shasta Soft Drinks stock sells for $50, has just paid a $2 dividend, and has expected dividend growth of 10%, is it a reasonable choice for Dillon? What is the required return on SSD stock? First, D1 = D0 * (1 + g) = $2.00 * (1.10) = $2.20 Capital Gains Yield = g = 10% Dividend Yield = D1/P0 = $2.20/$50 = 4.4% Required Return = r = D1/P0 + g = 14.4% • MBAM614 Class 3 - 46 23 Estimating Growth All else equal from year to year, corporate earnings only grow when a firm invests in new assets E1 = E0 + (RE0 * RRE) E1 RE0 = (1 + g) = 1 + * RRE E0 E0 Earnings Retention Ratio Estimate with ROE Increase in Earnings g ≈ Retention Ratio * ROE • MBAM614 Class 3 - 47 Key Points 1. P0 = D1/(1 + r) + D2/(1 + r)2 + … + Dt /(1 + r)t + … 2. if g = 0, 3. if g < r, P0 = D/r P0 = D1/(r - g) Pt = Dt+1/(r - g) 4. r = D1/P0 + g = (Dividend Yield) + (Capital Gains) 5. g ≈ Retention Ratio * ROE • MBAM614 Class 3 - 48 24