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