15.401 Finance Theory, Solutions to Problem Sets 1–3

MIT Sloan Finance Problems and Solutions Collection Finance Theory I Part 1 Fall 2008 1 Present Value Solutions 1. (a) $10, 000 × (1.045)5 = $12, 461.82. (b) $10, 000 × (1 + (c) $10, 000 × (1 + 0.045 (4×5) ) = $12, 507.51. 4 0.045 (12×5) ) = $12, 517.96. 12 2. Using the following formula rEAR = (1 + rAP R k ) k −1 (a) Annual: ≈ 4.75% (b) Monthly: ≈ 4.85% (c) Weekly: ≈ 4.86% (d) Contious using the following formula rEAR = erAP R − 1 ≈ 4.86% 3. (a) $50, 000 × (1.04)2 = $54, 080 (b) $50, 000 × (1.01)8 = $54, 142.84 (c) $50, 000 × (1 + 0.04 (12×2) ) 12 = $54, 157.15 4. (a) $10, 000 × (1.045)5 = $12, 461.82. (b) $10, 000 × (1 + (c) $10, 000 × (1 + 0.045 (4×5) ) = $12, 507.51. 4 0.045 (12×5) ) = $12, 517.96. 12 5. e-Money. The effective annual interest rate for an A.P.R. of 5.25% is (a) 5.2500% with annual compounding (b) 5.3782% with monthly compounding (c) 5.3899% with daily compounding (d) 5.3903% with continuous compounding. We used the following two formulas for this question: rEAR = (1 + rEAR = erAP R − 1. rAP R k ) k − 1 and 6. False. The choice of the horizon date should not affect the present value. 7. (a) 1.055 × ( $10,000 )(1 − 0.055 (b) $135,835 (1+0.02)10 8. (a) PV = 1 ) 1.05510 × (1.055)10 = $135, 835 (end of year 10) = $111, 432 (NPV=79,521) 1.5 1.12 + 1.3 1.122 + 1.05 1.123 + 0.9 1.124 + 0.75 1.125 = $4.12 million. (b) Inflation affects the cash flow and the discount rate equally, and the two effects cancel out in the PV calculation. Therefore, PV = $4.12 million. 9. Assuming the rent payments are paid at the beginning of the year, the PV of the floating rent for the next 5 years is $100, 000 + $100, 000 ∗ 1.051 $100, 000 ∗ 1.054 + · · · + = $441, 285.71 1.121 1.124 This must be equal to $441, 285.71 = F + F F F F + + + 2 3 1.12 1.12 1.12 1.124 Solve for F , to get F = $109, 300.67 10. (a) Cash option PV = $40,000 Financing option PV 2% = 937.5 × 1 1.021/12 −1 Cash option costs less  × 1− 1 (1.02(1/12) ) 48  = $43, 228 (b) Cash option PV = $40,000 Financing option 15% = 937.5 × Cash option costs more 1 1.151/12 −1  × 1− 1 (1.15(1/12) ) 48  = $34, 271 1 ) = 9.7364 11. Year 1: 10 × ( 1.05×1.02 1.10 1.05×1.02 2 Year 2: 10 × ( 1.10 ) = 9.4797 3 ) = 9.2298 Year 3: 10 × ( 1.05×1.02 1.10 PV: 9.7364 + 9.4797 + 9.2298 = $28.4458 million. 12. The value of company ABC is the PV of the growing annuity of cash flow: 10 PV = 1.09−1.05 = $250 million. 13. Assume all the oil is sold at the end of each year; for example the PV of the production for the first year is $7000×$15 = $97, 222. The PV of the second year production is 1.08 $6580×$15 = $84, 619. The PV of the total production in the first 12 years is $608,254 1.082 and by then the annual production has declined to 3,544 barrels. 14. (a) Assuming first cash flow one year from now in real terms is $14m: rreal = (1.08/1.02) − 1 = 5.88% P V = 14/0.0588 = $238m (no growth) Using nominal values: 14 ∗ 1.02/(0.08 − 0.02) = $238m Alternatively, if assuming first cash flow one year from now in real terms is 14/1.02 = $13.73m, P V = (14/1.02)/0.0588 = $233.33m Using nominal values: 14/(0.08 − 0.02) = $233.33m Note: in order to get full credit, the assumptions made should have been clearly stated. (b) To calculate PV of cash flows generated in a continuous stream, starting immediately, we need to use the same perpetuity formula (PV = C / r), but substitute the continuously compounded rate New r = log(1.08/1.02) = 5.72% and P V = 14/0.0572 = $244.93m Common mistake: not using this compounding formula / forgetting that the inflation rate also must be adjusted. Using nominal values instead, 14/(log1.08 − log1.02) = $244.93m Note: in order to get full credit, the assumptions made should have been clearly stated. 15. The PV of the first scholarship from the foundations point of view is −(20, 000/0.06)[1 − (1/(1.06)4)] + (1/(1.06)5)(8, 000/0.06)[1 − (1/(1.06)10 )] = −25, 303 So it loses $25,303 every year beginning from t=0. The PV of this perpetuity is 25,303/0.06-25,303 = -447,020 This implies that the investment needed to fund this is $447,020 16. The market value of the lease is $1.25 million for 5 years, with the first payment due right now. So the value is 1.25 1.25 1.25 1.25 + (1+6%) 1.25 + 1+6% 2 + (1+6%)3 + (1+6%)4 = $5.5814 million. The value of your current lease is $1 million for 5 years, and the total PV is $4.4651 million. Therefore the difference is $1.1163 million, and this should be the compensation you asked from the owner. 17. Use growing annuity formula assuming that the payments are made at the begining of each year and you pay in full for the year that you die (unfortunately). The value if you have an expected life of T years is: PV T 1.05 + · · · + $750 ∗ = $750 + $750 ∗ 1.12 "  T # 1 1 1.05 = $750 + $750 − × 0.12 − 0.05 0.12 − 0.05 1.12  1.05 1.12   (1) Solve for T . The breakeven point is T >= 16 18. (a) Let the annual payment = C. The PV of all my payments, discounted at the dealer’s rate, must equal to the price, i.e., C 1 3500 + 0.02 (1 − 1.02 5 ) = 30000. 3500 + 4.71346C = 30000. C = $5, 622.20. (b) Since I can save at a higher rate, the cost of the financing plan in (a) is only 1 C (1 − 1.05 3500 + 0.05 5 ) = $27, 841.17. The cost of the second option is 30, 000 − 2, 500 = $27, 500. I should pay cash. 19. The fair monthly rate is (1 + 0.08)1/12 − 1 = 0.006434.  C Let C be the appropriate monthly payment. Then, 0.006434 1− C = $1, 198.58. 1 1.006434120  = 100, 000. 20. The PV of all cash inflows and outflows is −65 10 10 40 40 PV = −65 + 1.05 2 + 1.053 + 1.054 + 1.055 + . . . + 1.059 = $38, 478, 57. 1.05 I am taking a financial gain by helping him out. 21. The PV of the offer at discount rate of 12% is $359,921. With the discount rate of 8% the PV increases to $446,821. In both cases accepting the award of $500,000 is preferable. This calculation ignores other consideration such as difference tax treatment. 22. (a) EAR = (1 + .06 12 ) 12 − 1 = 0.06168. (b) Let C be the monthly payment. The effective interest rate per month is 0.06 = 12 0.005. h The PV
of the imortgage annuity must equal the amount borrowed: (12×30) C 1 1 − 1.005 = 600, 000. 0.005 C = $3, 597.30. h
(12×25) i 1 (c) 3597.30 1 − = $558, 326.10. 0.005 1.005 23. False. US treasury bond carries inflation risk.   10000 500 24. (a) PV = 0.06 1 − 1.061 10 + 1.06 10 = $9, 263.99. (b) The effective interest rate per 6 months is 1.061/2 − 1 = 0.02956. 1 10000 250 1 − 1.02956 + 1.06 PV = 0.02956 20 10 = $9, 318.39. 25. (a) EAR = (1 + .085 12 ) 12 − 1 = 0.088391. (b) The effective interest rate per month is .085 = 0.0070833. 12 Let C be the monthly payment. The PV of the mortgage annuity must equal the amount hborrowed: i (12×30) C 1 1 − = 500, 000. 0.0070833 1.0070833 C = $3, 844.57. h i (12×30−60) 3844.57 1 (c) 0.0070833 1 − 1.0070833 = $477, 451.33. 26. The PV of the current salary at the discount rate of 8% is $468,687. It is assumed that salaries are paid at the end of each year (only an approximation to the more accurate monthly payment) and he works up to the end of his 65th year. To calculate the break-even starting salary after MBA, one needs to include the negative NPV of the cost of MBA as well as the lost income for the 2-years spent in school and compare that against the positive NPV of the increased earning power. The negative NPV of the cost of MBA is $35,665 and the loss in salary is $71,330 for a total of $-106,995. Even if the starting salary after the MBA is as low as $32,687 it still pays off to get the MBA. Of course, getting an MBA makes the skill set more flexible (such as the general management skills or the network of people you know) which reduced the risk of the income as well. The aspect if not considered here but definitely would make the MBA even more valuable. 27. (a) PV at t = 5 of the annuity from t = 6 to 35: h growing
i 1.03 1.03 30 100, 000 0.04−0.03 1 − 1.04 = $2, 591, 785.58. PV at t = 0 of the growing annuity: 2591785.58 × (1.04)−5 = $2, 130, 258.82 PV at t = 0 of the  annuity from t = 1 to 5: 100,000 1 1 − = $445, 182.23. 0.04 1.045 Total human capital: 2, 130, 258.82 + 445, 182.23 = $2, 575, 441.06. (b) FV at t = 5 of savings: 30,000 [1.045 − 1] = $162, 489.68. 0.04 28. (a) Assume that every year, Mr. Jones withdraws the year’s living expense at the beginning of the year. If he retires today and lives for another T years, he would need the following amount: h i 1.05 1.02 T S = 100, 000 1.05−1.02 1 − 1.05 . (i) T = 25 : S(25) = $1, 804, 335.76 (enough). (ii) T = 45 : S(45) = $2, 550, 363.68 (not enough). (iii) T = 60 : S(60) = $2, 885, 219.04 (not enough). (b) Mr. Jones can (i) work longer and retire later, (ii) downgrade his lifestyle and reduce living expense, or (iii) find better investment opportunities that give higher returns (though he must also consider the risk involved). 29. The nomilar value is $50, 000 × (1.07510 ) Inflation reduced this by a factor of (1.03)10 . So the real value is only $76,680. 30. (a) rAP R = ln(rEAR + 1) = ln(1.12) = 0.1133 (b) e0.05 − 1 = 0.0513 (c) ln(1.09) = 0.08617   1 275 1 − 1.026 + 31. (a) PV = 0.026 20 10000 1.02620 = $10, 231.64. (b) (A convex, downward sloping curve) 32. The Reborn VW Beetle. First use the annuity formula to determine the monthly payments Ca and Cb for dealerships A and B, respectively, ignoring the initial down payments: (a) Dealership A: PVa = $18,000, ra = 0.08/12, and t = 36 months ⇒ Ca = $564.05. (b) Dealership B: PVb = $15,500, rb = 0.10/12, and t = 36 months ⇒ Cb = $500.14. If the monthly discount rate is currently r, then the net present values of the two packages are   1 1 NPVa = 2,000 + Ca − r r(1 + r)36   1 1 − . NPVb = 4,000 + Cb r r(1 + r)36 It is clearly more advantageous to accept dealership A’s offer if and only if NPVa < NPVb . Substituting the expressions from above and simplifying, we have that NPVa < NPVb if and only if   1 1 − < 31.29. r r(1 + r)36 By trial and error, the cross-over point is at r = 0.00778. The conclusion is that if the current annual interest rate for a 36-month period (compounded monthly) is above 9.34%, you should choose dealership A. If the current annual interest rate for a 36-month period (compounded monthly) is below 9.34%, you should choose dealership B. 33. Monthly amount to fund from the mutual fund = 15,000-6,000 = $9,000 PV of the fund in three years = $1, 200, 000 × 1.0353 = $1, 330, 461 Number of years that it will last: $1330461 = 1.035 × 9000 × 12 × 1 0.035 × (1 − 1 (1+0.035)T ) (assume you take the money out at the beginning of the year) T = 16.4 years 1 50000 1 − 1.055 = $430, 925.89. 12 0.055   50000 1 PV−0.5 = 0.055 1 − 1.05512 = $430, 925.89. PV0 = 430, 925.89 × (1.055)0.5 = $442, 617.74.  34. (a) PV = (b)  35. IRA Accounts and Taxes (a) Yearly contribution: 2000 × (1 − 28%) = $1440: 30 X  2000 × (1 − 0.28) × [1 + 0.06 × (1 − 0.28)]30−t = $85, 218. t=1 (b) To get the total amount at end of 30 years: 30 X  2000 × (1 − 0.28) × [1 + 0.06 × (1 − 0.00)]30−t = $113, 843. t=1 of which the principal is 30 × 2000 × (1 − 0.28) = $43, 200, implying that the interest is 113, 843 − 43200 = $70, 643.79. Tax on this is 70643.79 × 0.28 implying that the take-home amount is $94,063. (c) The total amount is 30 X   2000 × (1 + 0.06)30−t = $158, 116.37. t=1 After taxes, this is only 158116.37 × (1 − 0.28) = 113, 843.789. Note: The answers to (b) and (c) are identical. Given the after–tax interest rate, it doesn’t matter whether your money is taxed on the way in or the way out. (d) Increase, because more deferred tax can be used to accrue interest. 2 Fixed Income Securities Solutions 1. (a) TRUE Coupon payments accelerate cash flows received from the investment reducing the duration. (b) FALSE Higher yield to maturity (YTM) does not mean that the bond is a good investment; NPV=0 in any case. Two good answers to this question were: 1. Bonds with longer duration have usually higher yield to maturity (YTM) as a result of increasing yield curve. 2. Bonds with higher probability of default have higher YTM to compensate for credit risk. Full answer must mention at least one of these two points. Many students concentrated on reinvestment risk only. (c) FALSE “On the run” refers to a newly issued bond. These bonds are usually more liquid and trade at a premium. 2. (a) False: yield of longer term bond also incorporates the expectation of future short term interest rates and depending on the this factor, the term structure may be up- or down-ward sloping. (b) False: If interest rates rise, future coupons of such bonds fall, and so their price (which is the present value of future coupons) falls more than the price of its straight bond counterpart. If interest rates fall, future coupons of such bonds rise, and so their price (which is the present value of future coupons) rises more than the price of its straight bond counterpart. So these bonds are more interest rate sensitive. 3. (a) False. The term structure of interest rates depends on expected future interest rates, if short term interest rates are expected to decrease substantially, the term structure can be inverted, flat, or downward sloping even when there is a liquidity premium for long term interest rates. (b) False. Bonds with higher duration are more sensitive to changes in interest rates. If we examine coupon bonds, all things equal a bond with a higher coupon rate will have a lower duration and thus be less sensitive to changes in interest rates. 4. False. The term structure depends on the expected path of interest rates (among other factors): if interest rates are expected to fall, the term structure will slope downward. 5. False. If there is liquidity premium, then a flat term structure means that investors expect interest rates to fall. 6. No. To minimize interest rate risks, we want MD(A)×V(A) − MD(L)×V(L) = 0. If V(A)>V(L), we want MD(A) < MD(L). That means we should invest in assets with shorter duration. 7. (a) Effective annual rate on 3-month T-bill:  100, 000 97, 645 4 − 1 = (1.02412)4 − 1 = .10 or 10%. (b) Effective annual interest rate on coupon bond paying 5% semiannually: (1.05)2 − 1 = .1025 or 10.25%. Therefore, the coupon bond has the higher effective annual rate. 8. The qoutes yield is calculated as y= 1F −P t P where t is in fraction of year, i.e. 1/6 in this example. Using F = $100, 000 the current  6 price is $99, 009. So the effective annual rate is $100,000 − 1 = 6.15% $99,009 9. (a) 1 + EAR = (100/98.058)2 so EAR=4% (b) Bond at par, so yield=coupon. So 1 + EAR = (1 + 0.042/2)2. So EAR=4.24% The coupon bond has the higher EAR.

$100 1/2 $100 1/3 $100 − 1 = 3.95%, r2 = $91.6 − 1 = 4.45% r3 = $86.6 − 1 = 5.11% 10. (a) r1 = $96.2 (b) P V = $300M × 11. (a) Price = (b) Price = 96.2 100 + $210M × 100 = 88.90; 1.043 5 105 + 1.035 2 = 1.031 91.6 100 + $400M × 86.1 100 − $600M = $225.36 YTM = 0.040. $102.87; YTM solves 5 105 + (1+y) 2 (1+y)1 = $102.87: y = 0.03488. 6 6 6 106 (c) Price = 1.03 1 + 1.0352 + 1.043 + 1.0454 = 105.65; 6 6 106 6 YTM solves (1+y) 1 + (1+y)2 + (1+y)3 + (1+y)4 = 105.65: y = 0.04428. 12. The effective annual yield on the semi-annual coupon bonds is 8.16%. If the annual coupon bonds are to sell at par, they must offer the same yield, which will require an annual coupon of 8.16%. 13. (a) P1−year = P2−year = P3−year = $100 = $95.01 1+5.25% $5 $100 + (1+5.50%) 2 = $99.08 (1+5.50%) $6 $106 $6 + (1+6%)2 + (1+6%) 2 = $100 (1+6%) (b) The first year, is easy. It is simply the yield of the 1-year note so, r1 = 5.25%. Then solve for r2 and r3 recursively. r2 = 5.505% and r3 = 5.7%. (c) Use the fact that  1 1+r3 3 =  1 1+r2 2 × (1 + f2,3 ) so f2,3 = 6.1% 14. (a) Spot rate at 2 years is 2.0%. At 3 years is 2.58%. At 8 years is 4.24%. (b) Use this relation: (1 + rt−1 )t−1 (1 + ft−1,t ) = (1 + rt )t The one year forward rate from Aug 2004 to Aug 2005 is 3.74%. Similarly the one year forward rate from Aug 20010 to Aug 20011 is 6.33%. (c) Under the Expectation Hypothesis, the upward slope implies that interest rates are expected to go up. In practice, the long-term average term structure is upward sloping. However, the August 20, 2002 slope is very steep. So interest rates are expected to rise. 15. (a) (1 + r2 )2 = (1 + r1 )1 (1 + f1,2 ) → f1,2 = 3.5% (b) (1 + r3 )3 = (1 + r1 )1 (1 + f1,3 )2 → f1,3 = 3.95% (c) Let’s calculate the forward rate first. (1 +r4)4 = (1 + r3)3 (1 + f3,4 ) → f1,3 = 6.02% So the TA is offering too high of a rate. What you need to do is as follows: i. Enter into a contract to lend him money at 6.3% between years 3 and 4. ii. Short a 4 year strip iii. Use the proceeds form the shorting to buy 3 year strip. The amount should be based on how much you enter into a contract with the TA for. If you follow these steps you will have more than enough to cover the short position in year 4 (since the rate you lend at is higher than the fair forward rate). Hence, this is an arbitrage. The risks you are exposed are contract and counter-party risk, for example the TA may go bankrupt before year 3 if he continues to enter into contracts of this form and may not be able to honor his/her end. In that case, you will end up with hanging position and expose to various interest rate risks. 100 = 95.24. 16. (a) A: 1.05 5 105 B: 1.055 + 1.055 2 = 99.08. C: 100 (coupon rate = YTM). (b) r0,1 = 0.05. h i0.5 105 r0,2 = 99.08−5/1.05 − 1 = 0.05513. h i1/3 106 r0,3 = 100−6/1.05−6/1.05513 − 1 = 0.06041. 2 100 (c) Long 100 106 = 0.9434 C 6 Short 0.9434 105 = 0.05391 B 1 Short 100 [6 × 0.05513 − 5 × 0.06041] = 0.05391 A (d) No arbitrage means that the bond in (c) should cost 0.9434 × 100 − 0.05391 × 99.08 − 0.05391 95.24 = 83.86.  100 ∗1/3 − 1 = 0.06041. Note that 83.86 17. (a) r1 = 101.93 = 111.31 = 110 − 1 = 3.00% 106.8 105 5 + ⇒ r2 = 4.00% 1 + r1 (1 + r2 )2 10 110 10 + + ⇒ r3 = 6.00% 1 + r1 (1 + r2 )2 (1 + r3 )3 (b) f2 = (1 + r2 )2 − 1 = 5.0% 1 + r1 18. (a) 100 1 + r1 100 92.01 = (1 + r2 )2 100 87.00 = (1 + r3 )3 95.92 = 100 − 1 = 4.25% 95.92  100 1/2 = − 1 = 4.25% 92.01  100 1/3 = − 1 = 4.75% 87.00 r1 = r2 r3 (b) f2,3 (1 + r3 )3 = −1 (1 + r2 )2 = 5.76% (c) The yield to maturity is simply 4.25% since the one year and two year spot rates are roughly equivalent, more specifically when we calculate the present value of a coupon bond with a coupon rate of 4.25% this bond has a current price at par. PV 104.25 4.25 + 1 + r1 (1 + r2 )2 104.25 4.25 + = 100 = 1.0425 (1.0425)2 = 19. (a) P V = 10 × (0.9756 + 0.9518 + 0.9286) = $28.56MM (b) Duration of liability D = 10 × (0.9756 + 2 × 0.9518 + 3 × 0.9286)/28.56 = 1.98 years (c) To hedge interest rate risk ∆P/P = −D∆y/(1+y), so if we want ∆P (liability) = ∆P (asset), i.e. perfect hedge, then D(asset)P (asset) = D(liability)P (liability) Leading to 20 × D(asset) = 1.98 × 28.56 → D(asset) = 2.83year To achieve such a duration, invest x% in 2 year strips and (1-x)% in 3 year strips such as 2x + 3(1 − x) = 2.83 → x = 17% (d) Impact on net liability Since by investing in portfolio calculated in c, you are perfectly hedge, the change in interest rate has no impact on your net liability. There will be some effect due to convexity but there is not first order effect. 20. (a) P = 20(0.961538+0.924556) = $37.72. (b) Duration is given by: 20 × 0.961538 + 20 × 0.924556 × 2 = 1.490. 37.72188 The yield is 4% (we have a flat term structure) so the modified duration is: 1.490196141 = 1.433. 1.04 (c) Asset value falls by 0.1433%, i.e. by $54,050.96. (d) Let F be the dollar value of the futures contracts that we wish to short. 5F = 37.72188 × 1.433. Hence F = $10.810 million so you will short 108 futures contracts. (You can only long or short whole numbers of futures contracts.) (e) Let the yield change by y percentage points. Portfolio value will change by −5 × 10.8 × ∆y + 37.72188 × 1.433 × ∆y ≈ 0. 1 1 = 0.9662, P V2 = (1+3%) 21. (a) P V1 = 1+3.5% 2 = 0.9426 From P V3 on, sum PV = 1/(1 + 4%)3 + 1/(1 + 4%)4 + . . . 1 1 1 1 = (1+4%) 2 ( 1.04 + 1.042 + 1.043 . . .) 1 1 = (1+4%) 2 × 0.04 = 23.1139 Therefore the total = 25.02268 (b) If you increase all rates by 0.5%, then the new values become 0.9615, 0.9335 and 20.3500. The total is 22.2446. Maturity 1 2 22. 3 4 Price $943.40 $898.47 $847.62 $792.16 YTM Forward Rates 6.00% 5.50% (1.0552/1.06 − 1) = 5% 5.67% (1.05673/1.0552 − 1) = 6% 6.00% (1.064 /1.05673 − 1) = 7% 23. (a) No, the rate of 5.50% is too low. To see why, use the identity that (1 + rt )t × ft,t+1 = (1 + rt+1 )t+1 So the forward rate for year 1 to 2 should be: f1,2 = 1.0552 − 1 = 5.75% 1.0525 (b) Borrow PV of $10 million due in one year now for one year. Invest the amount for two years. The amount you have to borrow is $10M/(1 + 5.25%) = $9, 501, 187. In one year, you willl have enough to pay back your loan. To borrow for one year, you can short sell the 1-year strip and to invest for 2-years, you can buy the 2-year strip. 24. (a) P = 100 (1+y/2)18 So P=$67.17 (b) P = 100 (1+y/2)18 + c/2 (1 y/2 − 1 ) (1+y/2)18 So P=$106.79 25. Not available but identical (different numbers) to question 000202 26. (a), (d) 27. Not available but almost identical to question 02MM03 28. (a) Assuming a par value of $100. The price is the quoted price plus accrued interests: $109 + 20/32 + 5 ∗ 41/365 = $110.2 (Any reasonable approximation to the accrued interest was considered correct). You will receive $2.5 in Feb and August every year from 2003 to 2010. In Feb 2011, you will receive $102.5 Common mistake: Most people didnt take into account the difference between clean and dirty price (b) You were expected to give EAR. Using the quoted yield (semi-annually compounded rate): r2010 = (1 + 0.0365/2)2 − 1 = 3.68% r2011 = (1 + 0.0377/2)2 − 1 = 3.81% Common mistake: give the APR (-2/4), or compute it using prices without taking correct time to maturity (-2/4). (c) f2010,2011 = 1.03818.4 1.03687.4 −1 = B2010 B2011 = 76.6875 73.25 = 4.6% (d) The Feb 2010 Note with 6.5 coupon rate has the shortest duration, because its coupons (given out early) are bigger than the Feb 2011 bond and the principal is also given earlier. 29. (a) Price = PV of cash flows = 100(1 + r)/(1 + r) + 100(1 + r)2 /(1 + r)2 + · · · =400.00 (b) Duration = (1 + 2 + 3 + 4)/4 = 2.5 years (c) Put 50% in each. Then the duration of your liability = 2.5 = duration of your assets = 0.5(1)+0.5(4) 30. (a) PV = 20 20 + = 12.28 + 6.94 = 19.22million 1 (1 + 5%) 0 (1 + 5%)3 0 (b) D= 12.28 ∗ 10 + 6.94 ∗ 30 = 17.22 19.22 MD = D = 16.40 1+y y = 5% because the yield curve is flat (c) When rates drop by 0.25%, the PV of liabilities will go up by 0.25% ∗ MD ∗ P V = 0.7881 million (d) First, you calculate the desired MD of your assets. We want: MDasset ∗ P Vasset = MDliabilites ∗ P Vliabilities Therefore, MDasset = 16.4∗19.22 = 17.51 18 Now we can determine the allocation of our portfolio. Suppose we invest a fraction of x of our portfolio into 1-year bond and the rest into treasury bond, then the MD of our portfolio will be: 1 MDportf olio = x ∗ MD1yrbond + (1 − x) ∗ MDtbond = x ∗ 1+5% + (1 − x) ∗ 20 Equating MDportf olio = 17.51, we get x = 13.05% 31. (a) Using the perpetuity formula P VLiability = 10M 10M = = 200M r 0.05 (b) P VLiability = 10M 10M = = 204.0816M r 0.049 The value of the liabilities would increase by 4.0816M. (c) Pnew = Pold − Pold × MD × ∆y Pold − Pnew Pold × ∆y 200 − 204.0816 = 20.4082 = 200 × −0.001 → MD = (d) You should match the modified duration to neutralize first order interest rate risk. MD=20.4082 (e) We must match both the modified duration and the present value of the zero coupon bond. Let P AR be the par value of the zero coupon bond and t be the maturity. Since the duration of a zero coupon bond is its maturity t = 21.4286. We can then solve for the P AR value P AR (1.05)t → P AR = 200(1.05)20.4082 = 541.331M 200M = 32. (a) r1 = 4.5%, r3 = 5.5% B2 = 100 (coupon = YTM), so 100 = 5/(1 + r1 ) + 105/(1 + r2 )2 , so r2 = 5.0126% Therefore the price of the 5% 3-year bond is 5/(1 + 4.5%) + 5/(1 + 5.0126%)2 + 105/(1 + 5.5%)3 = 98.7382 (b) f3 = (1 + r3 )3 /(1 + r2 )2 − 1 = 6.4817% (c) MD(bond 1) = 1/(1+4.5%) = 0.9569, MD(bond 3) = 3/(1+5.5%) = 2.8436, so MD(portfolio) = 30%*0.9569+70%*2.8436 = 2.2776 (d) That means the portfolio will decrease in value by 0.15%*2.2776 = 0.3416% 33. (a) We have the following system of equations: 100 = (100 + 25)/(1 + f0,1 ) 422.61 = 50/(1 + f0,1 ) + (500 + 50)/((1 + f0,1 )(1 + f1,2 )) 232.28 = 20/(1+f0,1 )+20/((1+f0,1 )(1+f1,2 ))+(300+20)/((1+f0,1 )(1+f1,2 )(1+ f2,3 )) 192.31 = 1000/((1 + f0,1 )(1 + f1,2 )(1 + f2,3 )(1 + f3,10 )7 The solutions are f0,1 = 0.25, f1,2 = 0.15, f2,3 = 0.10, f3,10 = 0.18.54. (b) NP V = −21 + 9/1.25 + 10/((1.25)(1.15)) + 11/((1.25)(1.15)(1.10)) = 12 34. (b) the entire value of the treasury strip is in the principal repayment in the distant future; it has the highest duration and is most sensitive to a change in the interest rate. (c) despite having the same maturity as (b), (c) has 5.5% coupon payments thatdampen its sensitivity to interest rate changes. (d) higher coupon payment → lower duration (a) T-bills are only for 1 to 6 months; they have the smallest duration. 35. Since we want portfolio D = 9, we can invest x in the 6-year bond (D = 6) and (1 − x) in 16-year bond (D = 16). Then Dp = x ∗ 6 + (1 − x) ∗ 16, x = 70% (*) Therefore invest 70% (350,000 or 3500 shares) in 6-year bond and 30% (150,000 or 1500 shares) in 16-year bond. *: This equation holds for D here because the bonds have the same YTM. 36. Let the annual interest payment for the two-year forward loan be C. You can arrange the loan and lock in the forward interest rate loan as follows • Buy $15M in par value of 1-year zero, costing (15M)(943.40/1000) = $14,151,000. • Sell $C in par value of 2-year zero, receiving $C × (898.47/1000) • Sell $C + 15M in par value of 3-year zero, receiving $(C + 15M)(847.62/1000) • Choose C such that proceeds from the selling of 2-year and 3-year zeros just cover the cost of buying the 1-year zero. Thus, (15M)(943.40/1000) = C(898.47/1000) + (C + 15M)(847.62/1000) which gives C = $822, 810. The CF’s of the transactions are: Transaction Buy 1-year zero Sell 2-year zero Sell 3-year zero Net CF today CF in year 1 CF in year 2 CF in year 3 -14,151,000 15,000,000 0 0 739,270 -822,810 0 13,411,730 0 0 -15,822,810 0 15,000,000 -822,810 -15,822,820 The forward interest rate for the 2-year forward loan is f = 822, 810/15, 000, 000 = 5.4854% which is locked in now. 37. (a) The current price of the bond is P0 = $60 $1060 $60 + + = $986.76 2 1.065 1.065 1.0653 You can sell it in one year for P1 = $60 $1060 + = $990.90 1.065 1.0652 But there is also the $60 of coupon. So the total return is RoR = ($990.90 + $60)/$986.76 − 1 = 6.5% (b) The bond now has the same coupon rate as the yield to maturity so it is trading at par. So the new price P1 = $1, 000 and RoR = ($1, 000 + $60)/$986.76 − 1 = 7.4% 38. (a) D = P P i 1.04i i=120 1 i 1.04 i=120 = 9.21 years (b) Need to match the duration and also the value of investment today should be equal to the total liabilities. So have the following two equations: V5 ∗ 5 + V20 ∗ 20 = D ∗ (V5 + V20 ) V5 + V20 = $13.59M Annuity formula Solving gives V5 = $9.78M and V20 = $3.81M P5 = V5 ∗ (1.04)5 = $11.9M P20 = V20 ∗ (1.04)20 = $8.36M (c) No. Interest rates increase, so bigger fraction of present value given early. So duration decreases. So more in 5 year strips and less in 20 year strips. 39. (a) P = D= $8 $108 $8 + 1.05 2 + · · · + 1.0510 = $123.16 1.05 $8 1 $8 2 $108 10 + 1.05 2 $123.16 + · · · + 1.0510 $123.16 1.05 $123.16 = 7.54 (b) Using pricing equation, the price will change to Pnew = $122.28. Recall the approximation that ∆P/P ≈ −MDy + CX × (∆y)2 MD = D/(1 + y) = 7.54/1.05 = 7.18 Usign the first two terms only we have ∆P = −P × MD × ∆y = −$123.16 × 7.18 × 10 × 10−5 = 0.88 → Pnew = $122.28 (c) Using pricing equation: Pnew = $107.02. Using duration approximation: Pnew = $105.47. (d) Addign the convexity term the new prices are Pnew = $122.28, and Pnew = $107.13 So not much has changed in the first case (part b) but the approximation is not much better for part c. In short, convexity is more important issue for large interest rate movements and hence one has to be hedged in terms of convexity in order to be immune to large interest rate movements. 40. (a) 4% per year (b) No. The spot rates at t = 5 are not known at t = 0. 3 %) = 13609.38 41. (a) 10000 × (136 32 (b) $437.50 in Feb. 2007, Aug. 2007, Feb. 2008, ..., Aug. 2020 $10,437.50 in Aug. 2020. (c) $43,750 in Feb. 2007 $-9,956,250 in Aug. 2007 $206,250 in Feb. 2008 $10,206,250 in Aug. 2008. 42. (a) The 1, 2, 3, 4, 6 and 10-year spot interest rates are 5.08%, 4.93%, 4.91%, 4.80%, 4.87% and 5.05%, respectively. (b) f1,2 = 1.04932 /1.0508 − 1 = 0.0478. f3,4 = 1.04804 /1.04913 − 1 = 0.0447. (c) P V = 437.50/(1.04804) = 362.69. 43. P = Annuity(218.75, (1.0488)0.5, 12) + 10000/(1.0488)6 = $9, 769.56. 10 = $9.5166 million worth of one-year strip. 44. (a) Short 1.0508 Long $9.5166 million worth of two-year strip. Cash flow at t = 1: 0. Cash flow at t = 2: 9.5166 × (1.0493)2 = $10.478 million. 10 (b) Short 1.0508 = $9.5166 million worth of one-year strip. Long $9.5166 million worth of six-year strip. Cash flow at t = 1: 0. Cash flow at t = 5: 9.5166 × (1.0487)6 = $12.659 million. 45. (a) 4% (b) 200 × (1.04)3 = 224.97. 46. Not available but we have many other pension questions 47. Bond underwriting If the underwriter purchases the bonds from the corporate client, then it assumes the full risk of being unable to resell the bonds at the stipulated offering price. In other words, the underwriter bears the risk of interest rate movement between the time of purchase and the time of resale. For long maturity bonds, it is generally true that its duration is also long. Thus, bonds with long maturities are more exposed to interest rate movement risk. Therefore, the underwriter demands a larger spread (higher underwriting fees) between the purchase price and stipulated offering price. 48. (a) Use the formula that D= T X t=1 t × wt where wt = CFt /(1 + y)t BondP rice Durations are 1.97, 4.61, 8.35, 14.13 for A through D, respectively. (b) You can either do this question by aggregating the cashflows as each time and using an approach identical to what we did in part a. Here is an alternative. To calculate the duration of a portfolio, it is easier to calculate its modified duration first. Recall that modified duration if MD = D/(1 + y) and that ∆P/P = −MD × ∆y Is is not have to show that the modified duration of a portfolio is simply the weighted sum of the modified durations of the individual bonds where the weights are proportional to the value of the bond to the total value. In this case, the modified durations A through D are 1.85, 4.35, 7.88, and 13.33, respectively. For the portfolio we have: 20 20 10 10 × 1.85 + × 4.35 + × 7.88 + × 13.33 = 6.6 60 60 60 60 Hence the duration is MDportf olio = Dportf olio = 6.6 × 1.06 = 7 (c) ∆P/P = −MD × ∆y = 7 × 20bps = 140bps = 0.14% 49. (a) PV bond = PV semiannual payments + PV principal payment 10,000 1 1 × (1 − (1+0.05/2) = 300 × 0.05/2 20 ) + 1.02520 = $10, 779.45 Note: if 5% is an effective annual rate, the semiannual rate is 1.05 = (1 + r/2)2 so r/2 = 2.47% 10,000 1 1 Then, PV bond = 300 × 0.0247 × (1 − (1+0.0247) 20 ) + 1.024720 = $10, 828.57 (b) To calculate the IRR sovle this following equation 300 300 300 0 = −11240 + (1+IRR/2) 1 + (1+IRR/2)2 + · · · + (1+IRR/2)20 So: IRR/2 = 2.23% 50. (a) D = 200∗0.5+300∗1+500∗5−900∗1 = 20 100 It means that the equity has the same sensibility to interest rates as a 20 year strip (b) dP/P = −D/(1 + y) ∗ dy = −20/1.03 ∗ 0.01 = −0.1941. Given the $100 value, dP = −$19.41M So the new price is $80.58M. Because of convexity, the change will be smaller. 51. Not available but there are several other pension related question dealing with duration matching. 105 − 1 = 0.040016. 52. (a) A: y = 100.96 6.5 6.5 106.5 B: y solves (1+y) 1 + (1+y)2 + (1+y)3 = 106.29; y = 0.042238. 2 2 102 C: y solves (1+y) 1 + (1+y)2 + (1+y)3 = 93.84; y = 0.042294. 105 (b) r0,1 = 100.96 = 1.0400. r0,2 : consider the portfolio of +1 B, − 106.5 = −1.0441 C and 2×1.0441−6.5×1 = 102 105 −0.042017 A. The portfolio pays 0 at time t = 1, 3 and 6.5×1−2×1.0441 = 4.4118 at time t = 2. The portfolio costs 100.96 × (−0.042017) + 106.29 × 1 − 93.84 ×  1/2 1.0441 = 4.0680. Then, r0,2 = 4.4118 = 1.041398. 4.0680 i1/3 h − 1 = 0.042323. r0,3 = 93.84−6.5/(1+r106.5 2 0,1 )−6.5/(1+r0,2 ) f0,1 = 0.040016. f1,2 = 1.0413982/1.040016 − 1 = 0.042781. f2,3 = 1.0423232/1.0413982 − 1 = 0.044175. 53. (a) Zero-coupon bond: y = 4.5%. 5 5 Coupon bond: P = 1.035 + 1.04 2 + 101.46; y = 0.044675. 105 1.0453 = 101.46; y solves 5 1+y + 5 (1+y)2 + 105 (1+y)3 = 100 100 (b) Zero-coupon bond: P0 = 1.045 3 = 87.63; P1 = 1.042 = 92.46; realized return = 92.46 − 1 = 0.055072. 87.63 5 105 101.91+5 + 1.04 −1 = Coupon bond: P1 = 1.035 2 = 101.91; realized return = 101.46 0.053659. (1) (2) (3) Yrs Pmt PV of Pmt 1 $1 M $0.9091 M 54. 2 $2 M $1.6529 M 3 $1 M $0.7513 M Total $3.3133 M (4) Wt of Pmt 0.2744 0.4989 0.2267 1.0000 (5) (1) × (4) 0.2744 0.9978 0.6801 1.9523 Duration is 1.9523 years. 55. Not available but there is another very similar question ID:020205 56. (Assuming semiannual coupon payment: YTM per six months is (1.06)0 .5 = 1.02956.) (a) P = Annuity(2.5, 0.02956, 20) + 100/1.0610 = $93.18. (b) D = 2.5 1 1.02956 +2 MD = 2.5 1.029562 2.5 + 2.5 2 1.02956 1.02956 15.8067 1.02956 2.5 +...+20 1.02956 2.5 +...+ 1.02956 = 15.8067 periods (of 6 months). = 15.3529. (c) ∆P = 93.18(1 − 15.3529 × (1.070.5 − 1.060.5 )) = $86.25. (d) P = Annuity(2.5, (1.07)0.5, 20) + 100/1.0710 = $86.56. 57. (a) The net cash flow in May-June, 2010 will be −(1.04)0.5 + 0.055/2 = −$0.99230 billion. (b) The transaction’s NPV is 1-1=0. 58. (a) There is a one-to-one mapping between a bond’s price and YTM (y). y is the solution to P = Annuity(4.4375, y, 20) + 100/(1 + y)20 . (b) Annuity(4.4375, 0.02315, 20) + 100/(1.02315)20 = 133.67. 59. (a) iv; (b) iv; (c) iv. 60. (a) (i) Increased. r ≈ R − i. (b) (i) A. Bond A has a smaller coupon payment, so its value depends more heavily on the principal repayment, which occurs far into the future. Therefore, it is more sensitive to change in the yield. (c) (i) y > 6%. Since the bond is sold at a discount, y > c. (d) (i) Profit. f1,2 = 1.062 /1.05 − 1 = 0.07010 > 0.06. The price has gone up. 100 3 = 94.79; D = 3; MD = 1.055 = 2.844. 1.0553 h 1 5 +2 5 +3 105 2 5 5 105 1 1.0553 + + 1.0553 = 98.65; MD = 1.055 1.055 1.055 1.055 1.0552 94.79 61. (a) Zero-coupon bond: P = Coupon bond: P = 2.7094. i = (b) Zero-coupon bond: ∆P = −2.844 × 0.001 = −0.0028436. Actual % change = 100/1.0563 −94.79 = −0.0028382. 94.79 Coupon bond: ∆P = −2.7094 × 0.001 = −0.0027094. New price = $98.38; actual = −0.0027043. % change = 98.38−98.65 98.65 (c) Zero-coupon bond: ∆P = −2.844 × 0.02 = −0.056872. Actual % change = 100/1.0753 −94.79 = −0.054782. 94.79 Coupon bond: ∆P = −2.7094 × 0.02 = −0.054187. New price = $93.50; actual % change = 93.50−98.65 = −0.052228. 98.65 (d) Portfolio: MD = 2.844 × 0.4 + 2.7094 × 0.6 = 2.7631. 62. (a) The present value of the annuities is $1 M /0.1 = $10 M. The duration is 1.10/0.10 = 11 years. Let x = weight of 5 year zeros, and 1 − x = weight of 20 year zeros. Then 11 = 5x + 20 (1 − x) and so x = 0.60 (in 5 year zeros), and 1 − x = 0.40 (in 20 year zeros). 5 year zeros: $10M × 0.60 = $6M market value; 20 year zeros: $10 M ×0.40 = $4M market value. (b) Face value of 5 year zeros: $6M × (1.10)5 = $9.66M; Face value of 20 year zeros: $4M × (1.10)20 = $26.91M. 63. (a) Depends. If the bond is held to maturity and all coupon payments are reinvested at the YTM, then higher YTM means higher return, ceteris paribus. (b) No. Forward rates are expectations of future spot rate; they are unrelated to the current spot rate. (c) No. Since the market expects a rate cut, it is already incorporated into today’s prices. 64. (a) PV of obligation: PV = 2 X $10, 000 t=1 Duration of obligation: (1) Yr 1 2 Total (2) (3) Pmt PV of Pmt $10, 000 $9, 259.26 $10, 000 $8, 573.39 $17, 832.65 (1.08)t (4) Wt of Pmt 0.51923 0.48077 1.0000 Duration of obligation is 1.4808 years. = $17, 832.65. (5) (1) × (4) 0.51923 0.96154 1.48077 (b) Zero coupon bond with a duration of 1.4808 years would immunize the obligation. The present value of this bond must be $17, 832.65, thus the face value (feature redemption value) must be: $17, 832.65 × (1.08)1.4808 = $19, 985.26. (c) If interest rates increase to 9%, the value of the bond would be: $19, 985.26 = $17, 590.92. (1.09)1.4808 The tuition obligation would be: PV = 2 X $10, 000 t=1 (1.09)2 = $17, 591.11. The net position changes by only $0.19, If interest rates decline to 7%: The value of the zero coupon bond would increase to: $19, 985.26 = $18, 079.99. (1.07)1.4808 The tuition obligation would increase to PV = 2 X $10, 000 t=1 (1.07)2 = $18, 080.18. The net position changes by $0.19. As interest rates change, so does the duration of the stream of tuition payments, thus the slight net differences. 65. (a) Real return is 1.05/1.02 − 1 = 0.029412. (b) Real return is 1.05/1.03 − 1 = 0.019417. (c) (i) The cash flows at time t = 1, 2, 3, 4 and 5 are 5.10, 5.20, 5.31, 5.41 and 115.93, respectively. (ii) The cash flows at time t = 1, 2, 3, 4 and 5 are 5.15, 5.30, 5.46, 5.63 and 121.72, respectively. 3 Common Stock Solutions 1. (a) UNCERTAIN/FALSE We can increase the present value of a share of common stock with a new investment only if ROE > r , where r is a discount rate (capitalization rate) If a new investment results in ROE < r, the price of the stock will decline even though earnings could be higher. Many students missed this point. 2. (a) False: the market value of a share of stock equals the discounted value of the stream of future dividends per share. It “works” with earnings if and only if PVGO = 0, and the plowback ratio = 0. 3. (a) False. The discount rate on one asset is determined by its risk characteristics. (b) False. A growth company has investment opportunities with expected return higher than the required rate of return. However, earnings may not be growing. For example, the firm could be investing heavily, leading to lower current earnings and dividend. 4. False. The goal of a manager should always be to maximize the market value of the firm. The firm’s risk will be reflected in its market value and shareholders can always unload any undesirable risk in the market. (Caveat: It could be true if shareholders can’t sell or have blocked access to financial markets like in private firms). 5. False. DDM use future dividend for discounting, so it works even with firms that have not paid dividends in the past. 6. (a) Growth stocks are stock of companies that have access to growth opportunities, where investment opportunities earn expected returns higher than the required rate of return on capital, or when PVGO > 0. A stock with growing dividends may not be a growth stock. A growth stock may be a stock with DPS growing slower than the required rate of return. Comment: to get a full credit, its not enough just to mention that growth stock dont usually have rapidly growing dividends. Many students said that growth stocks have DPS growing slower than the required rate of return. This is not necessarily true. 7. (a) False: To be a growth stock, its PVGO must be positive. It can have a positive PVGO, and yet not have made a profit since its IPO. OR Depends: If its PVGO is also non-positive, it is not a growth stock. 8. (a) False. A growth company has ROE greater than r, its cost of capital. (b) False. Higher volatility are not necessarily associated with higher returns as part of the volatility may be due to idiosyncratic risks. 9. False. The price of a stock equals the present value of all expected future dividends per share, discounted at the appropriate rate. 10. No. P = E/r + PVGO. So a higher P/E ratio may just mean higher PVGO. 11. (a) g = ROE × b = 12% × 0.5 = 6%. It is the same for both earnings and dividends. (b) P = D1 r−g == BV P S×ROE×p r−g = $0.6 12%−6% = $10 (c) We have assumed that the first payment of dividend happens at the end of the current year. D1 D2 (d) P = (1+r) 1 + (1+r)2 + · · · . Note that Dt = BV P St × ROEt × p, where p is the payout ratio. If all the quantities are restated in real terms, we will have a higher discount rate in the denominator of the DCF. But the BVPS will also be higher by the same factor so they will cancel out and the price won’t change. We have assumed that the inflation will affect the assets and the discount factor in the same way here. (e) No change. The price will remain as $10/share as the ROE of the firm is the same as its cost of capital, i.e. it does not have access to any growth opportunities so the PVGO=0. 12. Price of Qs stock = $2.2/9.5% = $23.16 Price of Rs stock = $1.4/(9.5%-3%) = $21.54 Price of Ss stock = 1/(1 + 9.5%) + 1.5/(1 + 9.5%)2 + 2/(1 + 9.5%)3 + 2.5/9.5%/(1 + 9.5%)3 = $23.73 2 3 4 1.1 1.1 1.1 1.1 1 + 1.08 13. P V0 of the first 5 years = 1.08 2 + 1.083 + 1.084 + 1.085 = 4.8043. 1 1.15 ×1.02 P V0 of all dividends after the first 5 years = 1.08 5 0.08−0.02 = 16.9395. P0 = 4.8043 + 16.9395 = 21.7438. 14. (a) p0 = D1 r−g = $5·0.3 0.1−0.08 = $75. $5 (b) Yes, it has growth opportunities, and b > 0. P V GO = p0 − Er1 = $75 − 0.1 = $25. 15. (a) g solves 5×0.4×(1+g) 0.10−g (b) PVGO = 80 − 5 0.1 = 80; g = 0.073171. = 30. 16. The growth rate of dividend after year 4 is g2 = 0.10 · (1 − 0.6) = 0.04. The following tables give answers to part (a) (b) and (c). We see that today’s price in part (a) is the same as that in part (c), but higher than that in part (b). The reason is that the ROE after year 4 is the same as the cost of capital (both at 0.10). Therefore, after year 4, it does not matter what the payout ratio is, and the price will remains the same. However, since from year 1 to year 3, the ROE is higher than cost of capital, a increase in payout ratio will reduce the growth rate of earnings (and dividends), therefore, the price in part (b) is lower. Year 0 1 Price and P/E 2 3 4 BVPS0 100.000 110.500 122.103 134.923 EPS 15.000 16.575 18.315 13.492 DivPS 4.500 4.972 5.495 8.095 BVPS1 100.000 110.500 122.103 134.923 140.320 Price 113.698 120.568 127.653 134.923 140.320 P/E 7.274 6.970 10.000 10.000 Payout Ration for the first three years is 0.3000, and is 0.600 after year 4. Year 0 1 Price and P/E 2 3 4 BVPS0 100.000 106.000 112.360 119.102 EPS 15.000 15.900 16.854 11.910 DivPS 9.000 9.540 10.112 7.146 BVPS1 100.000 106.000 112.360 119.102 123.866 Price 113.147 115.461 117.467 119.102 123.866 P/E 7.262 6.970 10.000 10.000 Payout Ration for the first three years is 0.6000, and is 0.600 after year 4. Year 0 1 Price and P/E 2 3 4 BVPS0 100.000 110.500 122.103 134.923 EPS 15.000 16.575 18.315 13.492 DivPS 4.500 4.972 5.495 12.143 BVPS1 100.000 110.500 122.103 134.923 136.272 Price 113.698 120.568 127.653 134.923 136.272 P/E 7.274 6.970 10.000 10.000 Payout Ration for the first three years is 0.3000, and is 0.900 after year 4. 17. P0 = D1 /(r − g) where, P0 = current price D1 = expected dividend next period r = cost of equity or market capitalization rate g = expected growth rate in dividends In this example, we have 20/(0.12 − 0.10) = 1000 A 23% drop in P0 will cause it to drop from 1000 to 770. This can be caused by a 23% drop in the estimate of D1 (from 20 to 15.4), or it can be caused by a 5% increase in the estimate of the cost of equity (from 12% to 12.6%), or it can be caused by a 6% decrease in the estimate of the expected growth rate in dividends (from 10% to 9.4%). Though the first possibility is not a slight change, the last two possibilities are. (5 points) 18. (a) P = D1 r−g = $5 10%−6% = $125 (b) P = EP S1 r + P V GO = $8 10% + P V GO → P V GO = $45 19. First, calculate the nominal growth rate: 1 + g = (1.04)(1.02) so g = .0608 PV = 4.5 0.095−0.0608 = $131.58 20. The table below summerizes the calculations: Year 1 2 3 BB BVPS Investment EB BVPS EPS DPS PV(DPS) PV(TV) Total share price $35.00 $ 4.20 $39.20 $ 5.60 $ 1.40 $ 1.28 $41.79 $47.08 ROE g Payout Ratio Plowback Ratio Cost of Capital 16% 12% 25% 75% 9.50% PVGO 4 5 $39.20 $43.90 $49.17 $55.07 $ 4.70 $ 5.27 $ 5.90 $ 2.20 $43.90 $49.17 $55.07 $57.28 $ 6.27 $ 7.02 $ 7.87 $ 5.51 $ 1.57 $ 1.76 $ 1.97 $ 3.30 $ 1.31 $ 1.34 $ 1.37 16% 12% 25% 75% 16% 12% 25% 75% 16% 12% 25% 75% 10% 4% 60% 40% $(11.87) If we calculate PVGO as P rice − EP S/r, we obtain a negative PVGO. How come that PVGO is negative while ROE > cost of capital at any time? Because the ROE on current assets decreases over time, which means that, by taking current ROE, we overestimate the value of assets in place. So, calculate an average ROE from now to the infinite = 10%*35 = 3.5 Value of assets in place = 3.5/0.095 = $36.84 “real” PVGO = $47.08 - $36.84 = $10.24 21. The table below summerizes the calculations: Year 1 Year BB BVPS Investment EB BVPS EPS DPS Net cash needed PV(DPS) PV(TV) Total share price 2 1 $35.00 $ 4.20 $39.20 $ 5.60 $ 3.36 $ 1.96 $ 3.07 $41.79 $54.49 ROE G payout ratio plowback ratio cost of capital 4 5 2 3 4 5 $39.20 $43.90 $49.17 $55.07 $ 4.70 $ 5.27 $ 5.90 $ 2.20 $43.90 $49.17 $55.07 $57.28 $ 6.27 $ 7.02 $ 7.87 $ 5.51 $ 3.76 $ 4.21 $ 4.72 $ 3.30 $ 2.20 $ 2.46 $ 2.75 $$ 3.14 $ 3.21 $ 3.28 16% 12% 60% 75% 9.5% PVGO = 3 16% 12% 60% 75% 16% 12% 60% 75% 16% 12% 60% 75% 10% 4% 60% 40% $(4.46) At end of year 1, the company needs $1.96 per share of additional cash, or $9,800,000. At the new price of $54.49 per share, this represents 179,850 shares to be issued at t=1. The only effect on the ex-dividend price per share today comes from the change in the payout policy, which increases the price per share from $47.08 to $54.49. Dividend policy does not create value. What creates value is that the company gets money at 9.5% and invests it at 10%. Once the company announces this measure, its market cap goes up to 5,000,000*54.49. Once it announces it will issue shares at the fair price to cover the additional cash needs, the new price per share becomes 5,000,000*(54.49+1.96)/(5,000,000+179,850)= $54.49! The reason why this issue does not impact the price today is that is is done at the fair market value of the share, and every dollar invested shows up either as growth or as dividend. 22. (a) P = 5 1.12 + 5×1.1 1.122 + (b) PVGO = 77.75 − (c) P/E = 77.75 5/0.3 5×1.12 1.123 5/0.3 0.12 + 1 5×1.12 ×1.05 1.123 0.12−0.05 = 77.75. = −61.14. = 4.6650. 2 2 1 5×1.1 ×1.05 + 5×1.1 + 1.12 = 82.08. (d) P1 = 5×1.1 2 0.12−0.05 1.12 1.122 E2 = 5.5/0.3 = 18.33. P/E1 = 82.08 = 4.4770. 18.33 (e) 10%: (a) 109.09 (b) -57.58 (c) 6.5455 (d) 6.2727 14%: (a) 60.34 (b) -58.70 (c) 3.62060 (d) 3.4795 23. (a) We compute the market value for the two division separately. For division A, the cash flow is risk-free, so we need to use the treasury rate to discount. The PV of the division A is P VA = $2m = $66.667m. 0.05 − 0.02 (2) The PV of division B is the difference between the currently market value of the whole firm and the PV of division A: P VB = P V − P VA = $87m − $66.667 = $20.3333m. (3) The cost of capital for division B is rB = EB $2m + gB = + 0.04 = 0.138. P VB $20.333m (4) (b) The NPV of the new project is NP V = −$5m + 0.8 0.8 = −$5m + = $3.133m. rB − g B 0.138 − 0.04 (5) So the new market value of the firm is $87m + $3.133m = $90.133m. 24. (a) Cost of capital is 4 + 1.2 × 8 = 13.6%. (b) P = 5/0.136 = $36.76; hence P/E = 7.35. (c) 0.1 × 0.5 σi = = 0.25. σm 0.2 Hence the required rate of return on the new project is 4 + 0.25 × 8 = 6%. β=ρ (d) Present value of the growth opportunity is −3 + 0.8/0.06 = $10.33 as of year 1. Thus price will be 10.33 P = 36.76 + = $46.51 1.06 and P/E = 9.30. 25. Not available. We have at least two other questions very similar to this. 26. The analyst uses a constant growth one-stage DCF model: P0 = DIV1/(r − g) But the growth rate is actually going to decline over the next years. The analyst should use a two-stage DCF analysis to value the company; assuming constant growth rate leads to overestimating the cost of capital. For example, if we assume that growth rate stays at 16% next year, but falls at 10% starting year 2, then the right discount rate is r such as $62 = 0.5/(1 + r) + 0.55/(r − 0.1)/(1 + r) which leads to r = 10.81% 27. (a) The company should issue 200,000 shares. (b) The company’s value with the new project is 50 + = $52.78. The share price is then 63.3333 1+0.2 1.2 0.12−0.03 = $63.3333 million. 1.2 − 10 = $3.3333 million. In anticipation (c) The NPV of the new project is 0.12−0.03 to this project, the stock price moves to $53.33 even before the new equity issue. = 187, 500 shares, and the stock price will The company needs to issue 10,000,000 53.33 not change after the issue. 28. It is not a fair offer. If you have market prices you should use them. NPV is just a tool to estimate market prices when they do not exist or market anomalies occur. Value per share from your aunt = share price-dividend foregone = 22−1.4/1.08 = $20.7 Major mistakes: a. Not use of the market price, b. Not consider the dividend, c. Not discount the dividend 29. (a) d1 = EP S(1 − b) = 5(1 − 0.4) = 3; g = ROE × b = 12(0.4)% = 4.8%. So P = d1 /(r − g) = 3/(0.09 − 0.048) = 71.43. So P/E = 71.43/5 = 14.29 (b) P/E will increase, because ROE ¿ r here, that is, investors are getting more than what they want (c) Plowback. Do not give dividends 30. (a) Dividend next year = Earnings * (1 - Plowback) = 0.5 Dividend growth rate = 12%*plowback = 6% (b) Pre-announcement stock price = price under no growth = Post-announcement stock price: Using DDM, P = 1 10% = 10 D1 .5 = = 12.5 r−g 10% − 12% ∗ .5 (c) Pre-announcement PE = 10, post-announcement PE = 12.5 Since ROE is higher than the cost of capital, the company is a growth company. 31. (a) Using the Gordon Growth Model, we can solve for the value of g. D1 r−g 0.5 20 = 0.1 − g 0.5 g = 0.1 − = 7.50% 20 P = (b) g = ROE × b where b is the plowback ratio b = ROE = 0.75 1.25 = 0.6, thus g 0.075 = = 12.5% b 0.6 (c) MW Co. is a growth company since ROE on its new investments (12.5%) is higher than the cost of capital (10%), and b > 0. 32. From the original valuation, we can find r = EPS/P = 12.5% Now if the earnings grow forever at 2.5%, then the new value is EPS1/(r-g) = 2.5*(1+2.5%)/(12.5%-2.5%) = 25.625 million So the value of the company will increase by 5.625 million *Here we assumed the company always payout all the earnings as dividend. 33. (a) g = ROE*b = 12%. Therefore, price = D1/(r-g) = 4/(12.5%-12%) = 800. PVGO = 800 -$20/12.5% = 640 (b) As ROE < r, the company should stop expanding. Then its value will be $160 ($20 per year forever). 34. No. EPS is only growing at 12.7% for the next three years, not forever. The expected rate of return can only increase by less than that amount. In addition, there may be a cost to the rapid growth (e.g. part of the current earnings may be retained), so the rate of return is lowered further. 35. We will calculate the price of the stock a year from today. At that point, the firm can either not take the project or if it does, the project will reduce the current EPS by $3 but as a result the EPS moving forward will be higher by $3 × 20% = $0.6. In the first case, the price is the PV of the perpetuity of $5 discounted at 10%. So P = $55 (note that the first earning is at time zero, but this is more of less a timing assumption and other assumption as long as clearly markes would be valid as well). In the second case, the price is $2 plus the PV of a perpetuity of $5.6 starting in 1 year discounted at 10%. So price will be P = $58 36. We assume that dividends are paid continuously and discounting is also done continuously. We will divide each year into N parts each with length of 1/N. Assume dividend is paid at the end of each little time section. Fix a time horizon T and let’s calculate the PV: "  i !  −i # TN X 0.06 0.12 0.6 1+ 1+ P V (0, T ) = N N N i=0 " ! #  N Ni  N −i TN X N 0.6 0.06 0.12 = 1+ 1+ N N N i=0 Now, taking the limit at N → ∞ and usign the trick that ert = limn→∞ (1 + r/n)nt , we get: P V (0, T ) = = = Z Z T 0.6e0.06t e−0.12t dt 0 T 0.6e−0.06t dt 0 0.6 (e−.06T − 1) −0.06 Now as T → ∞ the PV become equal to $10. So the combination of continously paying dividend, continous compounding and contimous growth cancel each other out and we have the same PV.