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 as opposed to the bank 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

Future value (FV) of a dollar amount (PV) is

where

r is the amount of interest per period

n is the number of compounding periods

(1 )nFV PV r

Compounding period can be any length of time: day, month, quarter, every 6 months, 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

1 2 12 3650.010 1,140 1,151 1,160 1,1620.015 7,745 7,914 8,058 8,0860.020 52,129 54,150 55,915 56,2660.025 347,593 368,770 387,691 391,4930.030 2,296,436 2,499,540 2,685,904 2,723,9010.035 15,033,737 16,862,406 18,592,858 18,951,6730.040 97,531,946 113,225,573 128,603,489 131,853,6930.045 627,092,974 756,737,028 888,813,464 917,329,8010.050 3,996,311,022 5,034,203,714 6,137,903,569 6,381,858,4470.055 25,244,430,610 33,335,866,408 42,352,724,177 44,397,375,0280.060 158,083,653,509 219,734,177,751 292,008,020,337 308,855,885,5860.065 981,429,387,839 1,441,776,878,613 2,011,688,067,086 2,148,537,470,8220.070 6,041,113,336,886 9,417,219,235,410 13,847,750,089,745 14,945,775,188,124

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 today?

$24 compounded over 388 years

5

Present Value

Present value (PV) of a sum expected at a future time is

1

(1 )nPV = FV

r

where r is the discount rate per periodn is the number of compounding periods

Greater the discount rate, the smaller the present value.

6

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?

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?

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

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.

PV A1 (1 ) nr

r

10

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?

PV A1 (1 ) nr

r

10/10

11

2014 Federal Income Tax Rate Schedules

single

married, filing jointly

12

Single with Taxable Income = 60,000

What is 2014 FedTax?

13

Example 5

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

14

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.

15

Example:

Baa2 Valero Energy 6.625% ’37 88.250 7.652

where o Baa2 rating o 6.625% coupon rate (most likely paid in two

installments)o ’37 year of maturity o 88.250 price as a percent of paro 7.652% yield-to-maturity

How Bonds Are Expressed

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.

17

Bond Pricing

1 2(1 ) (1 ) (1 )n

C C C FPB

r r r

When first coupon payment is one period from now, this is formula

C = amount of each coupon paymentr = appropriate discount rate per periodn = total number of periodsF = principal, face value, par

Notation

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 is rarely traded, but similar bonds yield 7%.

19

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%.

20

Outstanding US MM & Bond Market Debt

Total marketcap of all listed US stocks ≈ 20.000 trillion2013 US IPO volume = 0.055 trillion

Ave daily US stock trading volume (all exchanges) ≈ 5 billion shares

(in trillions)

  Outstanding   2013 Issuance Ave Daily Trading Volume

Municipal 3.671   0.335 0.012

US Treasury 11.854   2.140 0.545

Mortgage Related 8.803   1.965 0.228

Corporate 7.451   1.384 0.018

Agency Securities 2.058   0.396 0.010

Asset-Backed 1.280   0.188 0.002

Money Market 2.714   n/a n/a

TOTAL 38.832 6.408 -

AVERAGE - - 0.809

21

Example 7

From the table, approximately:

a) How many times does US Treasury debt turnover per year?

b) What’s ave time to maturity of corporate bonds?

c) What’s ave time to maturity of mortgage related debt?

22

US IPO Volume

(so far)

23

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.

(1 )n

FPB

r

Do semiannual compounding when pricing a

zero coupon bond.

24

Example 9

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%?

25

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).

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, 2020 the following:

(a) coupon payments will restart on 1/1/16 but at half the coupon rate.

(b) will pay full rate starting on 1/1/18 until bond matures.

For what value should this bond be listed on a 10/15/14 balance sheet if discount rate to apply is 10%?

10/17

27

Example 11: A U.S. TreasuryIn 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

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

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

29

30

Example 14: Full Price

Suppose an 8% $1,000 bond (next semi-annual coupon payment on Feb 14, 2015) is quoted in the media at 123.6831. As of 10/21/14, how much would 2000 of them cost?

What is one day’s accrued interest on this purchase?

30

31

Example 15: Clean Price

Assume a 5% bond whose next semiannual coupon payment is on 11/1/14, and that on 10/17/14 someone 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?

1 24 24

40 40 10001097.37

(1 ) (1 ) (1 )r r r

Find r-value that fits. 25 terms (lot of work). Then double to obtain yield-to-maturity answer.

36

Example 17: (con’t)

1 24 24

2424

40 40 10001097.37

(1 ) (1 ) (1 )

1 (1 )40 1000(1 )

r r r

rr

r

Easier if we employ annuity formula. Only 2 terms.

37

Example 18: Realized YieldRealized 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?

6 6

66

30 30 1068995.00

(1 ) (1 ) (1 )

1 (1 )30 1068(1 )

r r r

rr

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 CalculationExpected 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

0

250

500

750

1000

1250

05101520

Pri

ce

Years to Maturity

20-Year 10% (annual) coupon bond at 8.0% and 12% Yields to Maturity

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

0 1000.00 1000.00

40

Convexity of Price-Yield Curve

0

300

600

900

1200

1500

1800

0.00 0.05 0.10 0.15 0.20

Pri

ce

Yield to maturity

20-Year 10% (annual) coupon bond atdifferent yields to maturity

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

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)

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).

43

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

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 is expressed in years.

time-weighted PV of each cash flowD

PB

45

Example 20Assume 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

• 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)

47

Example 21: Bond Price Volatility

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(0.0075) 9.55%

(1 .045)

Correcting for convexity, actual change in price of bond is a little less (next slide).

% %(1 ) (1 )

i DPB D PB i

i i

In the following, i is yield in percent per year

48

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 or US Treasury bond – semiannually

Names of items on both sides of balance sheet of Module 2.3, slide 18Know the 5 indicated items of a bond quote as in Module 2.5, slide 15Names of 7 categories of US debt on Module 2.5, slide 20

Monday 2:30 to 4:00pm accessibility. Good to send email.