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Opportunity Cost Value forgone for something else. Suppose a bank would pay 3.5%/year but you decide to keep $10,000 in your mattress. Opportunity cost of keeping in mattress as opposed to the bank is $350/yr. 1 Why Time Value of Money? • Positive time preference for consumption must be offset by adequate return. • There is a time value of money even if no inflation. Inflation just makes it more pronounced. 2 Future Value Future value (FV) of a dollar amount (PV) is FV PV (1 r ) n where r is the amount of interest per period n is the number of compounding periods Compounding period can be any length of time: day, month, quarter, every 6 months, year, etc. A hard part in using formula is getting r and n right. 3 Legend has it that Lenape Indians sold Manhattan Island to the Dutch for $24 in 1626. 4 Example 1 If the Lenape could have invested the money at X%, how much would they have today? 5 $24 Compounded Over 387 Years 1 2 12 365 1,151 1,149 1,140 1,129 0.010 7,966 7,938 7,796 7,631 0.015 55,152 54,809 53,083 51,107 0.020 381,827 378,129 359,721 339,115 0.025 2,643,401 2,606,621 2,426,208 2,229,550 0.030 18,299,869 17,954,279 16,287,360 14,525,350 0.035 126,683,914 123,569,094 108,828,886 93,780,717 0.040 876,967,412 849,774,951 723,799,621 600,088,971 0.045 6,070,632,326 5,839,161,010 4,791,627,568 3,806,010,497 0.050 42,021,630,185 40,091,261,875 31,575,343,654 23,928,370,246 0.055 290,870,953,305 275,043,913,581 207,120,537,045 149,135,522,178 0.060 921,529,941,633 1,352,439,778,495 1,885,418,123,553 2,013,336,208,866 0.065 0.070 5,645,900,314,847 8,791,074,923,951 12,914,182,772,900 13,935,441,946,107 Excel formula for monthly at 6% : =24*(1 + .06/12)^(387*12) 6 Present Value Present value (PV) of a sum expected at a future time is given by PV = FV 1 (1 r ) n where r is the discount rate per period n is the number of compounding periods Greater the discount rate, the smaller the present value. 7 Example 2 How much should US Treasury charge for a 10-year $5,000 Savings Bond if designed to earn 4.2% per annum? How much should US Treasury charge for the same bond if designed to earn 2.1% semi-annually? 8 Example 3 What is the maximum you would pay for a financial claim that pays $120,000 four years from now, if you could otherwise place your money in a bank that pays 4.00% compounded monthly? Wouldn’t pay more than what would grow to $120,000 at bank = 120,000 / ((1 + .04/12)^48) = 102,284.47 9 Annuities When same amount is paid at end of each period, with first payment one period from now, the series is an ordinary annuity whose PV is given by PV A 1 1 (1 r ) n PV A 1 (1 r ) r n r where A = amount of each payment r = appropriate discount rate per period n = total number of periods Annuity due is when first payment is now. 10 Example 4 PV A 1 (1 r ) n r Suppose an investor receives $10,000 on this date for the next 8 years, with first payment one year from now. Assume 9% per annum is the appropriate discount rate. What is the PV of this annuity? 10/10 11 2013 Federal Income Tax Rate Schedules single married, filing jointly 12 Single with TaxableIncome = 50,000 What is FedTax? 13 Example 5 You win $1 million lottery (annuitized, $50,000/yr for 20 years). How much will you get to bring home if you select lump sum payment? Assume 6% per annum discount rate. 14 What is a Bond? • Borrower (issuer) promises contractually to make periodic payments (called coupon payments) to bondholder over a given number of years. • At maturity, bondholder receives last coupon payment and principal (face value or par value). • Coupon payments are determined by the coupon rate. • Coupon rates are specified as a percentage of par. 15 How Bonds Are Expressed Example: Baa2 Valero Energy 6.625% ’37 88.250 7.652 where o Baa2 rating (beware: make sure not stale) o 6.625% coupon rate (most likely paid in two installments) o ’37 year of maturity o 88.250 price as a percent of par o 7.652% yield-to-maturity Most Treasury and corporate bonds make coupon payments twice per year (semiannually) 16 How to Compute the PB of a Bond. Compute the PV of each of the bond’s cash flows and sum. Discount rate is ascertained from yields on similar bonds. (discount rate and coupon rate are not to be confused). If price of bond (PB) is below face value, called a discount bond. If above face value, called a premium bond. A bond sells at a discount if its discount rate > coupon rate. If its discount rate < coupon rate, bond sells at a premium. 17 Bond Pricing Notation C = amount of each coupon payment r = appropriate discount rate per period n = total number of periods F = principal, face value, par When first coupon payment is one period from now, this is formula PB C (1 r ) 1 C (1 r ) 2 C (1 r ) n F (1 r ) n Most Treasury and corporate bonds make coupon payments twice per year (semiannually) 18 Example 6 (Time Line Way) What is the PB of a $1,000 bond that has just made a coupon payment, has 2 years to maturity, pays interest semiannually, and has a coupon rate of 6%? Assume similar bonds yield 7%. 19 Example 6 (Using Annuity Formula) What is the PB of a $1,000 bond that has just made a coupon payment, has 2 years to maturity, pays interest semiannually, has a coupon rate of 6%? Assume similar bonds yield 7%. At this PB, what would $150 million in face value of these bonds cost? 10/15 20 Outstanding US MM & Bond Market Debt (in trillions) Outstanding 2012 Issuance Ave Daily Trading Volume Municipal 3.728 0.378 0.011 US Treasury Mortgage Related 11.286 8.118 2.308 2.055 0.519 0.284 Corporate 9.348 1.359 0.017 Agency Securities 2.074 0.677 0.010 Asset-Backed 1.648 0.199 0.002 Money Market 2.492 n/a n/a 38.474 - 6.979 - 0.843 TOTAL AVERAGE Total marketcap of all listed US stocks ≈ 20.000 trillion 2012 US IPO volume = 0.042 trillion Ave daily US stock trading volume (all exchanges) ≈ 7 billion shares Asset-Backed mostly auto, credit card, student loans, home equity 21 Example 7 From the table, approximately: a) How many times does US Treasury debt turnover per year? b) What’s average time to maturity of corporate bonds? c) What’s average time to maturity of agency MBS debt? 22 Example 8: Zero Coupon Bond Since there is no C, customary formula is PB F (1 r ) n where n is double the number of years. Do semiannual compounding when pricing a zero coupon bond. What is price of a $1,000 zero coupon bond that matures in 15 years if it is to yield 9.4%? 23 Example 9 As of 10/15/13, what is the value of a $5,000 7.5% bond (coupon payments made semi-annually) that matures on March 15, 2014 assuming the yield to use is 5.8%? 24 Fixed Income Securities (a) Fixed income securities – pay a return according to a fixed formula. Although payment amounts can vary, formula is known in advance. Fixed income securities are liabilities to their issuers. Assets Liabilities Capital on issuer’s books in here Securities issued by governments and corporations that are designed to pay contractually a specified income over a specified time horizon. 25 Fixed Income Securities (b) The payments, how they are calculated, and when they are to be paid is information known in advance. Fixed income securities generally carry lower returns because of their guaranteed income characteristics. Generally used by people for income purposes rather than for capital appreciation (as in stock market). 26 Example 10: A Distressed Bond A company trying to emerge from bankruptcy arranges with the holders of its 8.0% bonds (par $1,000) that mature on July 1, 2019 the following: (a) coupon payments will restart on 1/1/15 but at half the coupon rate. (b) will pay full rate starting on 1/1/17 until bond matures. As of 10/15/13, what is the value of this bond if discount rate is 10%? 10/17 27 Example 11: A U.S. Treasury In 1985 the US Treasury issued a 30-year bond with a coupon rate of 11.25% that matures on 2/15/15. A bank made an error with this bond on its 10/15/12 balance sheet. Using a discount rate of 2.6%, what value should the bank have used for this bond on that balance sheet? 28 Example 12: Accrued Interest Clean price, Dirty price. Full price also known as “dirty price”. Clean Price = Full Price – Accrued Interest days since last coupon payment Accrued Interest = coupon payment x -------------------------------------days in coupon period What is accrued interest on a 5% $1000 bond if 181 days in coupon period and last coupon payment was 136 days ago? 29 Example 13: Saw-Tooth Pattern When buy a bond, what you pay is full price. But clean price is what is reported in the media. Clean Price + Accrued Interest = Full Price Full price has a saw-tooth pattern. Clean price smoothes this out. Plot saw-tooth pattern of the full price of a 3-year 6% bond (semi-annual payments) yielding 6%. Bond price and yield inversely related 30 Example 14: Full Price Suppose an 8% $1,000 bond (next semi-annual coupon payment on Feb 10, 2014) is quoted in the media at 123.6831. As of 10/17/13, how much would 2000 of them cost? What is one day’s accrued interest on this purchase? 31 Example 15: Clean Price Assume a 5% bond whose next semiannual coupon payment is on 11/1/13. As of 10/17/13, assume that someone just paid $995.47 for the bond. What clean price corresponds to this sale assuming 184 days in current coupon period? 32 Example 16: 3 In-class Exercises 10/22 33 When Full Price = Clean Price When no interest has accrued Clean Price = Full Price Can happen: At absolute beginning of a coupon period Zero coupon situation Coupon payments have been suspended 34 Three Bond Yields 1. Yield-to-maturity. Assumes • Issuer makes all payments as promised • Coupon payments are reinvested at the rate that the bond yielded when purchased • Investor holds bond to maturity 2. Realized yield. An ex post calculation of the bond’s yield while holding it. For instance, holder sells a bond before maturity. 3. Expected yield. An ex ante calculation of a bond’s expected yield based upon anticipated cash flows. All trial-and-error “number crunching” calculations 35 Example 17: Yield-to-Maturity Yield-to-maturity. The annual rate that causes all cash flows to discount back to the bond’s market price. Solve by trial-and-error. What is the yield-to-maturity on a 12-year, 8% coupon bond (semi-annual payments) whose price is $1,097.37? 1097.37 40 (1 r ) 1 40 (1 r ) 24 1000 (1 r ) 24 Find r-value that fits. 25 terms (lot of work). Then double to obtain yield-to-maturity answer. 36 Example 17: (con’t) 1097.37 40 (1 r ) 1 40 (1 r ) 24 1000 (1 r ) 24 Easier if we employ annuity formula. Only 2 terms. 1 (1 r ) 2 4 24 40 1000(1 r ) r 37 Example 18: Realized Yield Realized yield. Rate that causes all cash flows to discount back to the purchase price. What did bond project yield (annual rate) now that it is over? Paid $995 for a new 6% coupon bond (semi-annual payments). Sold after 3 years for $1,068 (minutes after coupon payment). What was realized yield? 995.00 30 (1 r ) 30 (1 r ) 6 1068 (1 r ) 6 1 (1 r ) 6 6 30 1068(1 r ) r Principle: If trial r makes RHS too low, decrease r If trial r makes RHS too high, increase r 38 Example 19: Expected Yield Calculation Expected yield -- the discount rate that causes the sum of the PV’s of all expected cash flows to equal purchase price. Solve by trial-and-error. Let’s do 6-month clock version of prob on p. 148 in book: Purchased a new 8% 10-yr, semiannual coupon payment bond at par. Plan to sell in 2 yrs when bond expected to yield 6% (at which point PB = 1,126) . What is project’s expected yield? 39 Price-Time to Maturity Relationship When bond’s yield differs from coupon rate, price of bond moves toward par as time to maturity decreases. 20 1196.36 850.61 18 1187.44 855.01 16 1177.03 860.52 14 1164.88 867.44 1250 12 1150.72 876.11 1000 10 1134.20 887.00 750 8 1114.93 900.65 500 6 1092.46 917.77 250 4 1066.24 939.25 0 2 1035.67 966.20 0 1000.00 1000.00 20 15 10 Years to Maturity 5 0 40 Price 20-Year 10% (annual) coupon bond at 8.0% and 12% Yields to Maturity Convexity of Price-Yield Curve Bond prices goes up if its yield goes down, and vice versa. “Bowed” shape of curve is known as convexity. 0.05 1623.11 0.06 1458.80 0.07 1317.82 0.08 1196.36 0.09 1091.29 0.1 1000.00 0.11 920.37 0.12 850.61 600 0.13 789.26 300 0.14 735.07 0.15 687.03 0.16 644.27 20-Year 10% (annual) coupon bond at different yields to maturity 1800 1500 Price 1200 900 0 0.00 0.05 0.10 0.15 0.20 Yield to maturity 41 Bond Theorems 1. A bond’s price is inversely related to its yield. 2. The longer the time to maturity, the greater the bond’s volatility (the more sensitive the PV of the bond is to yield rates). 3. The lower the coupon rate, the greater the bond’s volatility. 42 Price-Yield Relationship 43 Risks Faced by Holder of a Bond 1. Credit or default risk. 2. Interest rate risk. Two components: a) Reinvestment risk (chance lender will not be able to reinvest coupon payments at yield-to-maturity in effect at time instrument was purchased) Recall 11.25% Treasury of Example 11. Say bond yielded 11.35% when bought. But now coupon payments can only be reinvested at about 1%. (Was discount bond when issued, now premium bond). b) Price risk (chance interest rates will change thereby affecting price of the bond) a) and b) offset one another. 10/29 Duration is the number of years from now at which a) and b) exactly counterbalance one another. 44 Duration Duration is given by a time-weighted average of a bond’s cash flows over price of bond. Formula for duration is D tim e-w eighted P V of each cash flow PB D is expressed in years. 45 Example 20 Assume 1-yr clock. With 4 years to maturity and annual coupon payments, what are the durations of (4% coupon rate, 5% required yield)? (4% coupon rate, 10% required yield)? (8% coupon rate, 10% required yield)? 46 Duration Properties D is sum of discounted time-weighted cashflows divided by PB (with time measured in years) • Higher coupon rates mean shorter duration • D of a zero coupon bond is time to maturity. • The greater the required yield, the less the duration. • Longer maturities generally mean longer durations. 47 Example 21: Bond Price Volatility In the following, i is yield in percent per year i % PB D (1 i ) % PB D (1 i ) i Consider a 20-year, 5% bond (annual payments) yielding 4.5% whose D = 13.31. If interest rates change causing yield to rise 75 basis points, what happens to price of bond? 13.31 (1 .045) (0.0075) 9.55% Correcting for convexity, actual change in price of bond is a little less (next slide). 48 Convexity Correction Yield Straight line is what we get with %ΔPB formula (underestimates when yield drops, over-estimates when rises) Greater a bond’s convexity, the more valuable it is (because price increases are more, and price declines are less.) 49 Not To Be Naïve about Duration 1. The duration D we have been discussing also known as Macaulay duration. 2. First derivative of price-yield curve is D (1 i ) and is known as modified duration. Found in %ΔPB formula. 3. Convexity is second derivative of price-yield curve. Is a complicated expression (not studied here). 50 Duration Properties (2) • Duration is a measure of interest rate risk. The greater D, the more sensitive a bond’s price is to changes in the bond’s required yield. 51 Closer in the Payments, Less the Duration Holding a bond’s yield fixed (say at 10% in the below), D increases with maturity and varies inversely with coupon rate. 52 Managing Interest Rate Risk (a) Duration is the holding period for which reinvestment risk exactly offsets price risk. Designed to give investor the YTM that was in effect at time bond purchased. A way duration is used: If have a $5 million liability 7.5 years from now, buy a bond (or a portfolio of bonds) today that has a duration of 7.5 years. 10/31 53 Example 22: Portfolio Duration: N P ortfolio D wD i i i 1 Assume $4,000 in D = 5, $10,000 in D = 7, and $6,000 in D = 9 bonds. What is Portfolio D? 54 Example 23: Rebalancing Bond Portfolio Consider the $20,000 portfolio of Example 22. How much in D = 9 bonds should be sold, and how much in D = 5 bonds should be purchased, to reduce Portfolio D to 6.80? 55 Eliminating Interest Rate Risk (b) • Zero-coupon approach (best way). Buy high quality “zeros” with maturity equal to desired holding period. Locks in YTM. No reinvestment risk because no coupons payments, no price risk when held to maturity. • Duration matching (next best way). Selecting a portfolio of bonds whose duration matches desired holding period. Theoretically perfect, but only approximately perfect in real world as per footnote 8 on p. 162. • Maturity matching (don’t use). That is, selecting bonds with terms to maturity equal to desired holding period. Don’t use. Doesn’t work for eliminating interest rate risk. 56