Opportunity Cost/Time Value of Money Opportunity cost -- value forgone by doing 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 $_____ /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.

Future Value and Present Value Let 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.

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

$24 compounded over 392 years Example 1 If the Lenape could have invested the $24 at X%, how much would they have today? $24 compounded over 392 years Excel formula for monthly at 6% : =24*(1 + .06/12)^(12*392)

Example 2 How much should US Treasury charge for a 10-year $5,000 Savings Bond if it is to pay at the rate of 4.2%/yr, interest compounded annually? semi-annually? 10/8

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 = discount rate per period n = total number of periods (n must be integer when using this formula) An annuity due is different, it is when first payment is now.

Example 3 Starting 6 months from now, suppose an investor receives $10,000 every 6 months for the next 10 years. Assume 8% per annum is the appropriate discount rate. What is the PV of this annuity?

Example 4 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.

What is a Bond? Bond is a loan you can buy. 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.

2018 Federal Income Tax Brackets

Typical Bond Quote Example: Baa2 Valero Energy 6.625% ’37 88.250 7.652 where Baa2 rating 6.625% coupon rate (most likely paid in two installments) ’37 year of maturity 88.250 price as a percent of par 7.652% yield-to-maturity Treasury and corporate bonds typically make coupon payments twice per year (semiannually). MBSs and ABSs, typically twelve times a year (monthly). 10/10

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.

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

Example 5 (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%.

Example 5 (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, and has a coupon rate of 6%? Assume is rarely traded, but similar bonds yield 7%. What would $150 million in face value of these bonds cost?

Example 6: Zero Coupon Bond 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. What is price of a $1,000 zero coupon bond that matures in 15 years if it is to yield 9.4%?

US IPO Volume

Outstanding US MM & Bond Market Debt (2017 in trillions)   Outstanding Issuance Ave Daily Trading Volume Municipal 3.871 0.263 0.010 US Treasury 14.468 1.608 0.501 Mortgage Related 9.304 1.246 0.205 Corporate 9.001 1.145 0.027 Agency Securities 1.934 0.475 0.005 Asset-Backed 1.472 0.347 0.001 Money Market 6.225 n/a TOTAL 46.275 5.084 - AVERAGE 0.748 Total marketcap of all listed US stocks ≈ 30.000 trillion 2017 US IPO volume = 0.039 trillion Ave daily US stock trading volume (all exchanges) ≈ 6 billion shares 10/13

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

Example 8 As of today, what is the value of a $5,000 7.5% bond (coupon payments made semi-annually) that matures 5 months from now assuming yield to use is 5.8%?

Example 9: Accrued Interest Clean price, Dirty price. Full price also known as “dirty price”. Clean Price = Full Price – Accrued Interest Accrued Interest = days since last coupon payment 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? 10/15

What is one day’s accrued interest on this purchase? Example 10: Full Price Suppose an 8% $1,000 bond (next semi-annual coupon payment on Feb 14, 2019) is quoted in the media at 123.6831. As of 10/15/18, how much would 2000 of them cost? Assume 184 days in coupon period. What is one day’s accrued interest on this purchase? 22

Cut-off for the Quiz 23

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. 10/19

Fixed Income Securities Fixed income securities generally carry lower returns because of their contractual income characteristics. Generally used by people for income purposes rather than for capital appreciation (as in stock market).

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 12 1150.72 876.11 10 1134.20 887.00 8 1114.93 900.65 6 1092.46 917.77 4 1066.24 939.25 2 1035.67 966.20 1000.00

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 0.13 789.26 0.14 735.07 0.15 687.03 0.16 644.27

Bond Theorems A bond’s price is inversely related to its yield. The longer the time to maturity, the greater the bond’s volatility (the more sensitive the PV of the bond is to yield rates). The lower the coupon rate, the greater the bond’s volatility.

Price-Yield Relationship

Risks Faced by Holder of a Bond Credit or default risk. Interest rate risk. Two components: Reinvestment risk (chance lender will not be able to reinvest coupon payments at yield-to-maturity in effect at time instrument was purchased) Price risk (chance interest rates will change thereby affecting price of the bond) a) and b) offset one another. Duration is the number of years from now at which a) and b) exactly counterbalance one another. 30

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

Example 13 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)?

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.

Example 14: Bond Price Volatility In the following, i is yield in percent per year 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?

Example 15: Portfolio Duration: Assume $4,000 in D = 5, $10,000 in D = 7, and $6,000 in D = 9 bonds. What is Portfolio D?

Example 11: Clean Price Assume a 5% bond whose next semiannual coupon payment is on 11/8/18. As of 10/22/18, assume that someone just paid $995.47 for the bond. What clean price corresponds to this sale assuming 184 days in current coupon period?

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

Cut-off for the Exam Be sure to know the seven categories of US debt from slide 18. 38