1 Opportunity Cost/Time Value of Money Opportunity cost -- value forgone for something else. Example: Suppose a bank would pay 3.5%/year but you decide to keep $10,000 in your mattress. Opportunity cost of keeping in mattress is $350/yr. Time value of money -- because of “time preference ffffor consumption” There is a time value of money even if no inflation. Inflation just makes it more pronounced.

2 Future Value, Present Value Let FV be future value, PV be present value where r is the amount of interest per period n is the number of compounding periods Compounding period can be: day, month, quarter, semi- annually, year, etc. Hard part is in getting r and n right.

3 Legend has it that Lenape Indians sold Manhattan Island to the Dutch for $24 in 1626.

Example 1 Excel formula for monthly at 6% : =24*(1 +.06/12)^(12*388) 4 If the Lenape could have invested the $24 at X%, how much would they have as of a year ago? $24 compounded over 388 years

5 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?

6 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?

7 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

8 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 where A = amount of each payment r = appropriate discount rate per period n = total number of periods (n must be integer) Annuity due is when first payment is now.

9 Example 4 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

Federal Income Tax Rate Schedules individual married, filing jointly

11 Single with Taxable Income = 60,000 What is 2015 FedTax?

12 Example 5 You win $1 million lottery (annuitized, $50,000/yr for 20 payments, first now). How much will you bring home if you select lump sum? Assume 6% per annum discount rate.

13 What is a Bond? Borrower (issuer) promises 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.

14 Example: Baa2 Valero Energy 6.625% ’ where o Baa2 rating o 6.625% coupon rate (most likely paid in two installments) o ’37 year of maturity o price as a percent of par o 7.652% yield-to-maturity How Bonds Are Expressed Most Treasury and corporate bonds make coupon payments twice per year (semiannually), most MBSs and ABSs, monthly

15 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.

16 Bond Pricing When first coupon payment is one period from now, this is formula C = amount of each coupon payment r = appropriate discount rate per period n = total number of periods F = principal, face value, par Notation

17 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 is rarely traded, but similar bonds yield 7%.

18 Example 6 (Using Annuity Formula) What would $150 million in face value of these bonds cost? 10/15 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 is rarely traded, but similar bonds yield 7%.

19 US IPO Volume (so far)

20 Outstanding US MM & Bond Market Debt Total marketcap of all listed US stocks ≈ trillion 2014 US IPO volume = trillion (a big year) Ave daily US stock trading volume (all exchanges) ≈ 5 billion shares (in trillions) Outstanding 2014 IssuanceAve Daily Trading Volume Municipal US Treasury Mortgage Related Corporate Agency Securities Asset-Backed Money Market n/a TOTAL AVERAGE

21 Example 7 From the table, approximately: a)How many times does US Treasury debt turnover per year? b)How many times does US Corporate debt turnover per year? c)What’s ave time to maturity of mortgage related debt?

22 Example 8: 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%? Since there is no C, customary formula is where n is double the number of years. Do semiannual compounding when pricing a zero coupon bond.

23 Example 9 As of today, what is the value of a $5, % bond (coupon payments made semi-annually) that matures 5 months from now assuming yield to use is 5.8%?

24 Fixed Income Securities Fixed income securities – pay a return according to a fixed formula. Although payment amounts can vary, formula is 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).

25 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, 2021 the following: (a)coupon payments will restart on 1/1/17 but at half the coupon rate. (b) will pay full rate starting on 1/1/19 until bond matures. For what value should this bond be listed on a 10/15/15 balance sheet if discount rate to apply is 10%? 10/17

26 Example 11: Accrued Interest Clean price, Dirty price. Full price also known as “dirty price”. Clean Price = Full Price – Accrued Interest Accrued Interest = What is accrued interest on a 5% $1000 bond if 181 days in coupon period and last coupon payment was 136 days ago? days since last coupon payment coupon payment x days in coupon period

27 Example 12: 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 27

28 Example 13: Full Price Suppose an 8% $1,000 bond (next semi-annual coupon payment on Feb 14, 2016) is quoted in the media at As of 10/21/15, how much would 2000 of them cost? Assume 184 days in coupon period. What is one day’s accrued interest on this purchase? 28

29 Example 14: Clean Price Assume a 5% bond whose next semiannual coupon payment is on 11/1/15, and that on 10/17/15 someone paid $ for the bond. What clean price corresponds to this sale assuming 184 days in current coupon period?

30 Things for Sure To Know for Quiz … Everything on slides of Modules 2.3, 2.4, 2.5 until this slide Study end-of-chapter problems and solutions for Chapters 3, 4, and 5 How to price a zero coupon corporate, Treasury, or agency bond – semiannually Monday 2:30 to 4:00pm accessibility. Good to send .

31 Price-Time to Maturity Relationship When bond’s yield differs from coupon rate, price of bond moves toward par as time to maturity decreases

32 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

33 Price-Yield Relationship

34 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) b)Price risk (chance interest rates will change thereby affecting price of the bond) 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). 34 a) and b) offset one another. Duration is the number of years from now at which a) and b) exactly counterbalance one another. 10/29

35 Duration Duration is given by a time-weighted average of a bond’s cash flows over price of bond. Formula for duration is D is expressed in years.

36 Example 15 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)?

37 Duration Properties 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. D is sum of discounted time-weighted cashflows divided by PB (with time measured in years)

38 Example 16: Bond Price Volatility Consider a 20-year, 5% bond (annual payments) yielding 4.5% whose D = If interest rates change causing yield to rise 75 basis points, what happens to price of bond? Correcting for convexity, actual change in price of bond is a little less (next slide). In the following, i is yield in percent per year

39 Example 17: Portfolio Duration: Assume $4,000 in D = 5, $10,000 in D = 7, and $6,000 in D = 9 bonds. What is Portfolio D?

40 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.

41 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

42 Things for Sure To Know for Exam … Everything on slides of Modules 2.3, 2.4, 2.5 until this slide How to price a zero coupon corporate, Treasury, or agency bond – semiannually Names of items on both sides of balance sheet of Module 2.3, slide 10 Know the 5 indicated items of a bond quote as in Module 2.5, slide 14 Names of 7 categories of US debt on Module 2.5, slide 20 1:30-3:30 accessibility Monday. Good to send if coming.