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Time Value of Money Bond Valuation Risk and Return Stock Valuation

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## Time Value of Money Bond Valuation Risk and Return Stock Valuation

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**WEB CHAPTER 28Basic Financial Tools: A Review**• Time Value of Money • Bond Valuation • Risk and Return • Stock Valuation**Time lines show timing of cash flows.**0 1 2 3 i% CF0 CF1 CF2 CF3 Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.**Time line for a $100 lump sum due at the end of Year 2.**0 1 2 Year i% 100**Time line for an ordinary annuity of $100 for 3 years.**0 1 2 3 i% 100 100 100**What’s the FV of an initial $100 after1, 2, and 3 years if**i = 10%? 0 1 2 3 10% 100 FV = ? FV = ? FV = ? Finding FVs (moving to the right on a time line) is called compounding.**After 1 year:**FV1 = PV + INT1 = PV + PV (i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV2 = PV(1 + i)2 = $100(1.10)2 = $121.00.**After 3 years:**FV3 = PV(1 + i)3 = $100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n.**What’s the FV in 3 years of $100 received in Year 2 at**10%? 0 1 2 3 10% 100 110**What’s the FV of a 3-year ordinary annuity of $100 at 10%?**0 1 2 3 10% 100 100 100 110 121 FV = 331**N**I/YR PV PMT FV Financial Calculator Solution INPUTS 3 10 0 -100 331.00 OUTPUT Have payments but no lump sum PV, so enter 0 for present value.**What’s the PV of $100 due in 2 years if i = 10%?**Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 10% 100 PV = ?**Solve FVn = PV(1 + i )n for PV:**2 1 PV = $100 = $100 PVIF i, n 1.10 = $100 0.8264 = $82.64.**What’s the PV of this ordinary annuity?**0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.69 = PV**N**I/YR PV PMT FV INPUTS 3 10 100 0 OUTPUT -248.69 Have payments but no lump sum FV, so enter 0 for future value.**How much do you need to save each month for 30 years in**order to retire on $145,000 a year for 20 years, i = 10%? months before retirement years after retirement 1 2 360 1 2 19 20 0 ... ... PMT PMT PMT -145k -145k -145k -145k**How much must you have in your account on the day you retire**if i = 10%? years after retirement 0 1 2 19 20 ... ... -145k -145k -145k -145k How much do you need on this date?**N**I/YR PV FV You need the present value of a20- year 145k annuity--or $1,234,467. INPUTS 20 10 -145000 0 PMT OUTPUT 1,234,467**How much do you need to save each month for 30 years in**order to have the $1,234,467 in your account? You need $1,234,467 on this date. months before retirement 1 2 360 0 ... ... PMT PMT PMT**N**I/YR PV FV You need a payment such that the future value of a 360-period annuity earning 10%/12 per period is $1,234,467. INPUTS 360 10/12 0 1234467 PMT OUTPUT 546.11 It will take an investment of $546.11 per month to fund your retirement.**Key Features of a Bond**1. Par value: Face amount; paid at maturity. Assume $1,000. 2. Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed. (More…)**3. Maturity: Years until bond**must be repaid. Declines. 4. Issue date: Date when bond was issued.**PV annuity**PV maturity value PV annuity $ 614.46 385.54 $1,000.00 = = = The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10: INPUTS 10 10 100 1000 N I/YR PV PMT FV -1,000 OUTPUT**What would happen if expected inflation rose by 3%, causing**r =13%? INPUTS 10 13 100 1000 N I/YR PV PMT FV -837.21 OUTPUT When rd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.**What would happen if inflation fell, and rd declined to 7%?**INPUTS 10 7 100 1000 N I/YR PV PMT FV -1,210.71 OUTPUT If coupon rate > rd, price rises above par, and bond sells at a premium.**The bond was issued 20 years ago and now has 10 years to**maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%,or at 7%?**Bond Value ($)**rd = 7%. 1,372 1,211 rd = 10%. M 1,000 837 rd = 13%. 775 30 25 20 15 10 5 0 Years remaining to Maturity**At maturity, the value of any bond must equal its par value.**• The value of a premium bond would decrease to $1,000. • The value of a discount bond would increase to $1,000. • A par bond stays at $1,000 if rd remains constant.**Economy**Prob. T-Bill HT Coll USR MP Recession 0.10 8.0% -22.0% 28.0% 10.0% -13.0% Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0 Average 0.40 8.0 20.0 0.0 7.0 15.0 Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0 Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00 Assume the FollowingInvestment Alternatives**What is unique about the T-bill return?**• The T-bill will return 8% regardless of the state of the economy. • Is the T-bill riskless? Explain.**Do the returns of HT and Collections move with or counter to**the economy? • HT moves with the economy, so it is positively correlated with the economy. This is the typical situation. • Collections moves counter to the economy. Such negative correlation is unusual.**Calculate the expected rate of return on each alternative.**^ r = expected rate of return. ^ rHT = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%.**^**r HT 17.40% Market 15.00 USR 13.80 T-bill 8.00 Collections 1.74 • HT has the highest rate of return. • Does that make it best?**What is the standard deviationof returns for each**alternative? = Standard deviation. .**.**HT: = ((-22 - 17.4)2 0.10 + (-2 - 17.4)2 0.20 + (20 - 17.4)2 0.40 + (35 - 17.4)2 0.20 + (50 - 17.4)2 0.10)1/2 = 20.0%. T-bills = 0.0%. Coll = 13.4%. USR = 18.8%. M = 15.3%. HT = 20.0%.**The coefficient of variation (CV) is calculated as follows:**^ /r. CVHT = 20.0%/17.4% = 1.15 1.2. CVT-bills = 0.0%/8.0% = 0. CVColl = 13.4%/1.74% = 7.7. CVUSR = 18.8%/13.8% = 1.36 1.4. CVM = 15.3%/15.0% = 1.0.**Prob.**T-bill USR HT 0 8 13.8 17.4 Rate of Return (%)**Standard deviation measures the stand-alone risk of an**investment. • The larger the standard deviation, the higher the probability that returns will be far below the expected return. • Coefficient of variation is an alternative measure of stand-alone risk.**Expected Return versus Risk**Expected Risk, CV Security return HT 17.4% 20.0% 1.2 Market 15.0 15.3 1.0 USR 13.8 18.8 1.4 T-bills 8.0 0.0 0.0 Collections 1.74 13.4 7.7 • Which alternative is best?**Portfolio Risk and Return**Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections. ^ Calculate rp and p.**Portfolio Return, rp**^ ^ rp is a weighted average: n ^ ^ rp = wiri i = 1 ^ rp = 0.5(17.4%) + 0.5(1.74%) = 9.6%. ^ ^ ^ rp is between rHT and rColl.**Alternative Method**Estimated Return Economy Prob. HT Coll. Port. Recession 0.10 -22.0% 28.0% 3.0% Below avg. 0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0 10.0 Above avg. 0.20 35.0 -10.0 12.5 Boom 0.10 50.0 -20.0 15.0 ^ rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%. (More...)**p = ((3.0 - 9.6)2 0.10 + (6.4 - 9.6)2 0.20 + (10.0 -**9.6)2 0.40 + (12.5 - 9.6)2 0.20 + (15.0 - 9.6)2 0.10)1/2 = 3.3%. • p is much lower than: • either stock (20% and 13.4%). • average of HT and Coll (16.7%). • The portfolio provides average return but much lower risk. The key here is negative correlation.**Portfolio standard deviation in general**p= Portfolio standard deviation. Where w1 and w2 are portfolio weights and r1,2 is the correlation coefficient between stock 1 and 2.**Two-Stock Portfolios**• Two stocks can be combined to form a riskless portfolio if r = -1.0. • Risk is not reduced at all if the two stocks have r = +1.0. • In general, stocks have r 0.65, so risk is lowered but not eliminated. • Investors typically hold many stocks. • What happens when r = 0?**Portfolio beta**bp = Portfolio beta bp = w1b1 + w2b2 Where w1 and w2 are portfolio weights, and b1 and b2 are stock betas. For our portfolio of 50% HT and 50% Collections, bp = 0.5(1.30) + 0.5(-0.87) = 0.215 0.22.**What would happen to the riskiness of an average portfolio**as more randomly picked stocks were added? • p would decrease because the added stocks would not be perfectly correlated, but rp would remain relatively constant. ^**Prob.**Large 2 1 0 15 Return 135% ; Large20%.**p (%)**Company-Specific (Diversifiable) Risk 35 Stand-Alone Risk, p 20 0 Market Risk 10 20 30 40 2,000+ # Stocks in Portfolio**Stand-alone Market Diversifiable**= + . risk risk risk Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification. Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification.**Conclusions**• As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio. • p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M . • By forming well-diversified portfolios, investors can eliminate about half the riskiness of owning a single stock.