### Module2.5

```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.
1
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.
Hard part is in getting r and n right.
2
Legend has it that Lenape
Indians sold Manhattan
Island to the Dutch for \$24 in
1626.
3
Example 1
If the Lenape could have invested the \$24 at X%, how
much would they have today?
\$24 compounded over 388 years
1
2
12
365
0.010
1,140
1,151
1,160
1,162
0.015
7,745
7,914
8,058
8,086
0.020
52,129
54,150
55,915
56,266
0.025
347,593
368,770
387,691
391,493
0.030
2,296,436
2,499,540
2,685,904
2,723,901
0.035
15,033,737
16,862,406
18,592,858
18,951,673
0.040
97,531,946
113,225,573
128,603,489
131,853,693
0.045
627,092,974
756,737,028
888,813,464
917,329,801
0.050
3,996,311,022
5,034,203,714
6,137,903,569
6,381,858,447
0.055
25,244,430,610
33,335,866,408
42,352,724,177
44,397,375,028
0.060
158,083,653,509
219,734,177,751
292,008,020,337
308,855,885,586
0.065
981,429,387,839 1,441,776,878,613 2,011,688,067,086 2,148,537,470,822
0.070 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
Present Value
Present value (PV) of a sum expected at a future time is
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.
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
1  (1  r )  n
PV  A
r
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
1  (1  r )  n
PV  A
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
10
2014 Federal Income Tax Rate Schedules
single
married, filing
jointly
11
Single with Taxable Income = 60,000
What is 2014 FedTax?
12
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.
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
How Bonds Are Expressed
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 par
o 7.652% yield-to-maturity
Most Treasury and corporate bonds make coupon
payments twice per year (semiannually)
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
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
C


1
2
(1  r ) (1  r )

CF
(1  r ) n
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 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?
10/15
19
Outstanding US MM & Bond Market Debt
(in trillions)
Outstanding
2013 Issuance Ave Daily Trading Volume
Municipal
3.671
0.335
0.012
US Treasury
Mortgage Related
11.854
8.803
2.140
1.965
0.545
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
38.832
-
6.408
-
0.809
TOTAL
AVERAGE
Total marketcap of all listed US stocks ≈ 20.000 trillion
2013 US IPO volume = 0.055 trillion
Ave daily US stock trading volume (all exchanges) ≈ 5 billion shares
20
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?
21
US IPO Volume
(so far)
22
Example 8: Zero Coupon Bond
Since there is no C, customary formula is
F
PB 
(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 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%?
24
Fixed Income Securities
Fixed income securities – pay a return according to a
fixed formula. Although payment amounts can vary,
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, 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
26
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?
27
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?
28
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
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
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?
31
Example 16: 3 In-class Exercises
10/22
32
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
33
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
34
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?
40
40
1000
1097.37 
 

1
24
(1  r )
(1  r )
(1  r ) 24
Find r-value that fits. 25 terms (lot of work). Then
35
Example 17: (con’t)
40
1097.37 

1
(1  r )
40
1000


24
(1  r )
(1  r ) 24
Easier if we employ annuity formula. Only 2 terms.
1  (1  r ) 24 
24
 40 

1000(1

r
)

r


36
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
1068

(1  r ) 6 (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
37
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?
38
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
39
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
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
Price-Yield Relationship
42
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
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.
43
Duration
Duration is given by a time-weighted average of a bond’s
cash flows over price of bond. Formula for duration is
time-weighted PV of each cash flow

D
PB
D is expressed in years.
44
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)?
45
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.
46
Example 21: Bond Price Volatility
In the following, i is yield in percent per year
 i 
%PB   D 

(1

i
)


%PB 
D
i
(1  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

(0.0075)  9.55%
(1  .045)
Correcting for convexity, actual change in price of bond
is a little less (next slide).
47
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 18
Know the 5 indicated items of a bond quote as in Module 2.5, slide 15
Names of 7 categories of US debt on Module 2.5, slide 20
Monday 2:30 to 4:00pm accessibility. Good to send email.
48
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