33. as D. Moody's uses a slightly different system where issues can
go as high as Aaa and as low as C. A bond that carries a credit
rating of BB or lower by S&P and Fitch, or Ba or lower by
Moody's, is considered non-investment-grade, or junk.
Junk Bond Crisis
From the 1970’s to the 1980’s the junk bond market grew
exponentially at a a pace of around 34% per year. During this
period of time junk bonds achieved a superior risk adjusted
return. Essentially, junk bonds in this period were a superior
investment compared to other investments with the same amount
of risk.
However, this period of growth came to a sudden stop in 1989.
There is some disagreement as to what caused it, but most point
to the collapse of the US$6.75bn buyout of UAL as the main
trigger. A buyout group consists of pilots union and an
investment bank sought to take United Airlines private for
$6.79 billion couldn't secure the necessary loans.
Others point to the Ohio Mattress fiasco, a deal that would
become known as “burning bed” and remains widely considered
to be among the worst deals in modern finance. The Cleveland-
based company that Gibbons, Green bought for $1.1 billion in
April amid criticism that it was wildly overpaying. In August,
Gibbons, Green had to shelve its efforts to line up permanent
financing for the acquisition when investors in the high-risk,
high-yield ''junk bond'' market refused to buy the bonds
The culmination of the crash is considered to be the collapse of
Drexel Burnham Lambert, which was forced into bankruptcy in
early 1990, largely due to its heavy involvement in junk bonds.
At one point it had been the fifth-largest investment bank in the
US.
Credit Card and Auto Collapse
34. At the close of the 1990s, against the backdrop of the economic
boom, many low- and moderate-income families were struggling
financially and taking on credit card debt at rates unprecedented
in American history. There is growing evidence that a
combination of structural and economic trends coupled with
abusive credit card industry practices left working families with
few options other than to borrow heavily.
Between 1989 and 2001, credit card debt in America almost
tripled, from $238 billion to $692 billion. The savings rate
steadily declined, and the number of people filing for
bankruptcy jumped 125 percent.
During the 1990s, the average American family experienced a
53 percent increase in credit card debt, from $2,697 to $4,126
(all figures measured in 2001 dollars). Low-income families
saw the largest increase—a 184 percent rise in their debt—but
even very high-income families had 28 percent more credit card
debt in 2001 than they did in 1989.
With an increase in bankruptcy filings, it led to the drafting of a
bankruptcy reform bill, which was considered by the Congress,
and passed as Bankruptcy Abuse Prevention and Consumer
Protection Act of 2005. It made filing chapter 7 (liquidating)
bankruptcy, more difficult and introduced a means-test.
Effects of Deregulation
Since the late 1970s, America’s credit card industry has enjoyed
a period of steady deregulation. Two Supreme Court rulings, the
first in 1978 and the second in 1996, effectively hobbled state
usury laws that protected consumers from excessively high
interest rates and fees.
Aggressive Marketing
Relentless Credit Extension.
Between 1993 and 2000, the industry more than tripled the
amount of credit it offered to customers, from $777 billion to
almost $3 trillion.
Lowering of Minimum Payment Requirements
35. The amount of their balance customers can pay without
incurring a penalty—dropped from 5 percent to only 2 or 3
percent, making it easier for consumers to carry more debt.
Assuming an interest rate of 15 percent, it would now take more
than 30 years to pay off a credit card balance of $5,000 by
making the minimum payment. Or sometimes, never.
Skyrocketing Late Fees and Penalties.
Late fees have become the fastest growing source of revenue for
the industry, jumping from $1.7 billion in 1996 to $7.3 billion
in 2001. Late fees now average $29, and most cards have
reduced the late payment grace period from 14 days to 0 days.
In addition to charging late fees, the major card companies use
the first late payment as an excuse to cancel low, introductory
rates—often making a zero percent card jump to between 22 and
29 percent.
Asian Financial Crisis
The Asian Financial Crisis occurred in 1997 and affected
Indonesia, South Korea, Thailand, Hong Kong, Laos, Malaysia
and the Philippines. Indonesia, South Korea, Thailand being the
most affected.
The causes are disputed but most agree that current account
deficits, foreign currency denominated debt and a fixed
exchange rate contributed to the crisis. In the mid-1990s, the
maintenance of fixed exchange rates encouraged external
borrowing and led to excessive exposure to foreign exchange
risk in both the financial and corporate sectors.
Most recognize the devaluation of the Chinese Renminbi, the
devaluation of the Japanese Yen as a result of the Plaza accord
and a strengthen US dollar and a rise in US interest rates as
triggers for the collapse.
Asian Financial Crisis
36. This made the United States a more attractive investment
destination relative to Southeast Asia, which had been attracting
hot money flows through high short-term interest rates, and
raised the value of the U.S. dollar. For the Southeast Asian
nations which had currencies pegged to the U.S. dollar, the
higher U.S. dollar caused their own exports to become more
expensive and less competitive in the global markets.
The resulting panic among lenders led to a large withdrawal of
credit from the crisis countries, causing a credit crunch and
further bankruptcies. In addition, as foreign investors attempted
to withdraw their money, the exchange market was flooded with
the currencies of the crisis countries,
putting depreciative pressure on their exchange rates. To
prevent currency values collapsing, these countries'
governments raised domestic interest rates to exceedingly high
levels (to help diminish flight of capital by making lending
more attractive to investors), and to intervene in the exchange
market - buying up any excess domestic currency at the fixed
exchange rate with foreign reserves. Neither of these policy
responses could be sustained for long.
Thailand
On 14 May and 15 May 1997, the Thai baht was hit by massive
speculative attacks. However, Thailand lacked the foreign
reserves to support the USD–Baht currency peg, and the Thai
government was eventually forced to float the Baht, on 2 July
1997, allowing the value of the Baht to be set by the currency
market.
As a result of high interest rates, Thailand's booming economy
came to a halt amid massive layoffs in finance, real estate, and
construction that resulted in huge numbers of workers returning
to their villages in the countryside and 600,000 foreign workers
being sent back to their home countries. The baht devalued
swiftly and lost more than half of its value. The baht reached its
lowest point of 56 baht to the U.S. dollar in January 1998. The
37. Thai stock market dropped 75%.
Indonesia
In July 1997, when Thailand floated the baht, Indonesia's
monetary authorities widened the rupiah currency trading band
from 8% to 12%. The rupiah suddenly came under severe attack
in August. On 14 August 1997, the managed floating exchange
regime was replaced by a free-floating exchange rate
arrangement. The rupiah dropped further.
Although the rupiah crisis began in July and August 1997, it
intensified in November when the effects of that summer
devaluation showed up on corporate balance sheets. Companies
that had borrowed in dollars had to face the higher costs
imposed upon them by the rupiah's decline, and many reacted by
buying dollars through selling rupiah, undermining the value of
the latter further. Before the crisis, the exchange rate between
the rupiah and the dollar was roughly 2,600 rupiah to 1 U.S.
dollar. The rate plunged to over 11,000 rupiah to 1 U.S. dollar
on 9 January 1998, with spot rates over 14,000 during 23–26
January and trading again over 14,000 for about six weeks
during June–July 1998. On 31 December 1998, the rate was
almost exactly 8,000 to 1 U.S. dollar. Indonesia lost 13.5% of
its GDP that year.
South Korea
The banking sector was burdened with non-performing loans as
its large corporations were funding aggressive expansions.
During that time, there was a haste to build
great conglomerates to compete on the world stage. Many
businesses ultimately failed to ensure returns and profitability.
The chaebol, South Korean conglomerates, simply absorbed
more and more capital investment. Eventually, excess debt led
to major failures and takeovers.
In the wake of the Asian market downturn, Moody's lowered the
38. credit rating of South Korea on 28 November 1997, and
downgraded again on 11 December. That contributed to a
further decline in South Korean shares since stock markets were
already bearish in November. The Seoul stock exchange fell by
4% on 7 November 1997. On 8 November, it plunged by 7%, its
biggest one-day drop to that date. And on 24 November, stocks
fell a further 7.2% on fears that the IMF would demand tough
reforms. In 1998, Hyundai Motors took over Kia
Motors. Samsung Motors' $5 billion venture was dissolved due
to the crisis, and eventually Daewoo Motors was sold to the
American company General Motors (GM).
The IMF provided US$57 billion as a bailout package. In return,
Korea was required to take restructuring measures. The ceiling
on foreign investment in Korean companies was raised from 26
percent to 100 percent. In addition, the Korean government
started financial sector reform program. Under the program, 787
insolvent financial institutions were closed or merged by June
2003.
The South Korean won, meanwhile, weakened to more than
1,700 per U.S. dollar from around 800, but later managed to
recover. South Korea’s national debt-to-GDP ratio more than
doubled (approximately 13% to 30%) as a result of the crisis.
What is the current account?
The components of the current account: goods, services, income
and current transfers.
1. Goods - These are movable and physical in nature, and for
a transaction to be recorded under "goods," a change of
ownership from/to a resident (of the local country) to/from
a non-resident (in a foreign country) has to take place. Movable
goods include general merchandise, goods used for processing
other goods, and non-monetary gold. An export is marked as a
credit (money coming in), and an import is noted as a debit
(money going out).
39. 2. Services - These transactions result from an intangible
action such as transportation, business services, tourism,
royalties or licensing. If money is being paid for a service, it is
recorded like an import (a debit), and if money is received, it is
recorded like an export (credit).
3. Income - Income is money going in (credit) or out (debit)
of a country from salaries, portfolio investments (in the form
of dividends, for example), direct investments or any other type
of investment. Together, goods, services, and income provide an
economy with fuel to function. This means that items under
these categories are actual resources that are transferred to and
from a country for economic production.
4. Current Transfers - Current transfers are unilateral
transfers with nothing received in return. These include
workers' remittances, donations, aids and grants, official
assistance and pensions. Due to their nature, current transfers
are not considered real resources that affect economic
production.
Mortgage Crisis of 2008
The immediate cause or trigger of the crisis was the bursting of
the United States housing bubble which peaked in
approximately 2005–2006. An increase in loan incentives such
as easy initial terms and a long-term trend of rising housing
prices had encouraged borrowers to assume risky mortgages in
the anticipation that they would be able to quickly refinance at
easier terms.
However, once interest rates began to rise and housing prices
started to drop moderately in 2006–2007 in many parts of the
U.S., borrowers were unable to
refinance. Defaults and foreclosure activity increased
dramatically as easy initial terms expired, home prices fell,
and adjustable-rate mortgage (ARM) interest rates reset higher.
Several other factors set the stage for the rise and fall of
housing prices, and related securities widely held by financial
43. 24
24
k is the promised gross return
of = direct fees (origination fees)
BR + ø = loan interest rate
b= Compensating balance
RR = Reserve Rate
Note that the text displays 1+ E(r) = p(1+k) + (1-p)0 but this
simplifies to the form displayed above.
Return on Loan Equation
Return = inflow/outflow
1+k = 1+(of + (BR + ø))/(1-[b(1-RR)])
Of = origination fee, this is the fee paid by the customer to
initiate and process the loan application
BR = Base rate
Ø = risk premium of the customer
b = compensating balance requirement
RR = reserve requirement
k = the gross return on the loan
Example 10-1
Suppose a Bank does the following
Sets a loan rate of 10% (BR = 6% and Ø =4%)
Charges a .125% origination fee
Imposes an 8% compensating balance requirement to be held in
non-interest accounts
Sets aside reserves of 10% per Federal Reserve
1 + k = 1 + (.00125+ (.06+.04))/(1-(.08)(.9)
1 + k = 1 + (.10125)/(.928)
1 + k = 1.1091
k = .1091 or 10.91%
44. Expected Return on a Loan
Expected return: 1 + E(r) = p(1+k) + (1- p) 0
where p equals probability of complete repayment
1- p is the probability of non-payment or default
This can be considered and binomial option as there are only
two possible outcomes
It’s important to note that Ø and p are not completely
independent
Loan originators consider the probability of default when
setting the risk premium. This is to compensate for default risk
To an extent it can be self-reinforcing, high interest rates equal
higher payments, all else equal
Higher fixed payments increases the likelihood of default which
leads to higher interest rates
Note that realized and expected return may not be equal
Example
Calculate the promised return (k) on a loan if the base rate is
13%, the risk premium is 2%, the compensating balance
requirement is 5%, origination fees are .5% and the reserve
requirement is 10%
1+k = 1+(of + (BR + ø))/(1-[b(1-RR)])
1+k = 1 +((.005+(.13+.02))/(1-[.05(.9))
1+k = 1+.155/ .955
1 + k = 1.1623
What is the expected return on the loand is the probability of
default is 5%
Retail versus Wholesale Credit Decisions
At retail
Usually a simple accept/reject decision rather than adjustments
to the rate
Credit rationing
48. Total assets $700 Total liabilities and equity
$700
Also assume sales = $500,000 cost of goods sold =
$360,000 taxes = $56,000 interest payments = $40,000 net
income = $44,000 the dividend payout ratio is 50 percent, and
the market value of equity is equal to the book value.
Part a
What is the Altman discriminant function value for MNO, Inc.?
Recall that:
Net working capital = Current assets - Current
liabilities.
Current assets = Cash + Accounts receivable +
Inventories.
Current liabilities = Accounts payable + Accruals +
Notes payable.
EBIT = Revenues ‑ Cost of goods sold ‑ Depreciation.
Net income = EBIT ‑ interest ‑ taxes.
Retained earnings = Net income (1 ‑ Dividend payout
ratio
Part A solution
Altman’s discriminant function is given by: Z = 1.2X1 + 1.4X2
+ 3.3X3 + 0.6X4 + 1.0X5
X1 = (20+90+90‑30‑30-90)/ 700 = .0714 X1 = Working
capital/total assets (TA)
X2 = 44(1-.5) / 700 = .0314 X2 = Retained earnings/TA
X3 = (500-360) / 700 = .20 X3 = EBIT/TA
X4 = 400 / 150 = 2.67 X4 = Market value of equity/long
49. term debt
X5 = 500 / 700 = .7143 X5 = Sales/TA
Z = 1.2(0.07) + 1.4(0.03) + 3.3(0.20) + 0.6(2.67) +
1.0(0.71) = 3.104
= .0857 + .044 + .66 + 1.6 +
.7143 = 3.104
Part b
Based only on the Altman’s Z-score, should you approve MNO,
Inc.'s application to your bank for a $500,000 capital expansion
loan?
Since the Z-score of 3.104 is greater than 2.99, ABC Inc.’s
application for a capital expansion loan should be approved.
Part c
If sales for MNO were $300,000, the market value of equity
were only half of book value, and all other values are
unchanged, would your credit decision change?
ABC’s EBIT would be $300,000 - $360,000 = -$60,000.
X1 = (20 + 90 + 90 ‑ 30 ‑ 30 ‑ 90) / 700 = .0714
X2 = 22 / 700 = 0.0314
X3 = ‑60 / 700 = ‑0.0857
X4 = 200 / 150 = 1.3333
X5 = 300 / 700 = 0.4286
Z = 1.2(0.0714) + 1.4(0.0314) + 3.3(-0.0857) + 0.6(1.3333) +
1.0(0.4286) = 1.0754
Since ABC's Z‑score falls to 1.0754 < 1.81, credit should
75. Bonds are usually issued with a high enough coupon rate to
induce investors to purchase the bond. However there are zero-
coupon bonds, where the buyer only receives the face value at
the maturity date but receive no coupons payments.
When these bonds are issued they are priced considerably lower
than the par value. They may be issued by federal, state or local
governments or by corporations. Then there are the tax
exemptions. If issued by a government entity, the interest
generated by a zero-coupon bond is often exempt from federal
income tax, and often from stateand local income taxes too.
Accrued Interest and Quoted Prices
The bond prices quoted in financial papers are not the actual
prices that an investors pays for the bond. This is because the
quoted price doesn’t include the interest that accrues between
coupon payments.
Accrued Interest = Annual Coupon Payment/2 *Days since last
coupon/Days between payment
For example, a bond with a coupon rate of 8%. The annual
coupon payment is $80 and the semi-annual coupon is $40. 30
days have passed since the last coupon payment. What is the the
accrued interest?
$40 x (30/182) = $6.59
This value would be added to the quoted price when the bond is
sold.
Call Provision and Callable Bonds
Some corporate bonds are issued with a call provision. A call
provision allows the issuers to repurchase a bond at a specified
call price before the maturity date. Why might a company issue
76. a callable bond?
When corporate bonds are issued in a high interest rate
environment, they will most likely issue bonds with a high
coupon rate. Over the course of the bonds life, interest rates
might fall. A corporation might take advantage of the call
provision to retire the bond early and issue a new bond at a
lower coupon rate
Callable Bonds typically come with a period of call protection,
an initial period of time when the bond cannot be called.
The option to call a bond is useful to the issuer, but what is
beneficial to an issuer is detrimental to bond holders. To
compensate for this, callable bonds are often issued with higher
coupons and promised yield to maturity than non callable bonds.
Convertible bonds
Convertible bonds give bondholders an option to exchange each
bond for a specified number of shares. The conversion ratio is
the number of shares for which each bond may be exchanged.
For example, a convertible bond is issued at a par value of
$1,000 and is convertible in 40 shares of stock. If the current
stock price is $20 it is not profitable to convert. As $20*40 =
$800 is less than the par value of the bond. If the stock price
increases to $30, the conversion is profitable ($30*40 = $1,200)
The market conversion value is the current value of shares for
which the bonds may be exchanged.
Convertible bond holders benefit from price appreciation of the
company’s stock. Because of this benefit, convertible bonds
tend to have lower coupon rates and lower yields to maturity.
Floating Rate Bonds
Floating rate bonds make interest payments based on some
measure of current market rates. For example, the might might
be adjusted annually to the current T-bill rate plus 2%. If the
One year T-Bill rate is 4%, then the coupon rate over the next
77. year will be 6%.
The major risk with floating-rate bonds is that while the spread
between the market interest rate and the floating coupon rate is
fixed, the adjustment is not connected to changes in the issuing
firm’s financial position. For example, if a company runs into
financial distress, investors might demand a greater yield
premium than is offered by the security. In this case the price of
the bond will fall.
Bond Pricing
Bond Value = Present Value of Coupons + Present Value of par
value
For example consider an 8% coupon, 30-year maturity bond
with a par value of $1,000, paying 60 semiannual coupon
payments of $40 each. Suppose that the market interest rate is
8% annually.
The present value of all the coupon payments is $904.94 and the
present value of the par value is $95.06. When the market
interest rate is equal to the coupon rate, the par value and
equals the bond price.
If the market interest rate were to rise to 10%. The bond price
would fall by $189.29 to $810.71
Yield to Maturity
As you may have guessed, the coupon rate and the market
interest rate are rarely equal. As such bonds usually do not sell
for par value. Therefore we need to measure or rate of return
that accounts for both current income (coupon payments and the
return of principle) and the bond price increase or decrease over
the lifetime of the bond.
The yield to maturity is the interest rate that makes the present
value of a bond’s payments equal to its current price. This
interest rate is interpreted as the average rate of return of a
78. bond if held to maturity.
Example: An 8% coupon, 30-year bond is selling at $1,276.76.
What is the yield to maturity?
In this example, the semiannual yield to maturity is .03.
However, yield to maturity is generally quoted as an annual
figure. Therefore we must annualize the yield. 1.03^2 =1.0609.
In this case the yield to maturity in 6.09%
YTM vs Current Yield
A bond’s yield to maturity (YTM) is the internal rate of return
on the investment in the bond. YTM differs from the current
yield of a bond. The current yield of a bond is defined as the
annual coupon payment divided by the current price of a bond.
For example, take the 8% coupon bond selling for $1,276.76.
The current yield would be 80/ $1,276.76 or .0627
(6.27%). Recall that the YTM on this bond was 6.09%. This
bond is considered to be selling at a premium. In this case the
coupon rate is higher than the current yield which is higher than
the YTM.
The reason for this is that the Coupon rate is divided par value,
the current yield is divided by the current price. The current
yield is higher than the YTM because the YTM accounts for the
capital loss as the bond will eventual repay only $1,000 at
maturity.
As a general rule, for premium bonds the Coupon rate > Current
Yield > YTM. For discount bonds the relationship is reversed.
Yield to Call
Yield to maturity assumes that the bond will be held to
maturity. However, a bond maybe be retired prior to that time.
This is especially true if the bond has a call provision. For
example, a $1,000, 30 year bond with a coupon payment of 8%
has a callable provision at 110% of par value and has call
protection for 10 years. If the bond currently sells for $1,150
79. what is the yield to call?
In this example, the Yield to Call (YTC) is equal to 6.64% (3.32
x 2) and the YTM is 6.82%
Yield to CallYield to MaturityCoupon Payment4040Number of
Periods2060Final Payment$1,100$1,000Price$1,150 $1,150
Zero-Coupon Bonds
US Treasury Bill are common for of short term zero-coupon
bonds. Since these have no coupon the return on these bonds is
completely from price appreciation. However, Long-Term Zero
coupon bonds are usually created by separating a coupon
payment stream from the principal repayment. An investor can
request that the US Treasury split or strip the coupon payment
from the principal repayment. In this case each component is
assigned a CUSIP number, the CUSIP allows the each security
to trade on the FEDWIRE system.
The process is governed by the US Treasury Program called
STRIPS (Separate Trading of Registered Interest and Principal
of Securities)
The primary purpose of strips is to appeal to different investor
types. Much of the reason is related to cashflow matching.
Term Structure of Interest Rate
The term structure of interest rates refers to the the process of
discounting cash flows of different maturities.
Most often investors will plot the YTM against Maturity. This is
called the yield curve. There are three common “types” of yield
80. curve; rising, flat and inverted. A rising yield curve is the most
common in normal economic times, and it suggests that interest
rates will rise in the future. The yield curve is derived from
plotting zero-coupon
bonds.MaturityYTMPrice15%$952.3826%$89037%$816.3048%
$735.03
Yield Curve and Future Interest Rates
In order to derive the Yield curve we’ll be making some
assumptions. First, we assume that there is no possibility of
arbitrage. Under this scenario, yields from must be identical.
To see what this means, consider the following. There are two
strategies, a choice to buy and hold a two year zero-coupon
bond and to buy a one year zero coupon bond and reinvest in
another one year zero coupon.
For the two year bond, assume a 6% market rate for a holding
period of years. In this case the bond price would be $890. For
the second strategy, assume the interest rate for a one year
holding period is 5%. What will the one year rate be in one year
We can set this up as the following:
$890 x 1.062 = $890 x 1.05 x (1+r)
Or 1.062/1.05 = 1+r = 1.0701 or 7.01%
Finding Future Short Rates
Now let’s compare a three year strategy. One will be to
purchase a 3 year zero coupon bond with a YTM of 7%. This
bond would be priced at $816.30
The alternative strategy is to buy a 2 year zero and reinvest in a
1 year zero.
We can structure this as follows
1.073 = 1.062 x 1+r
1.073/1.062 = 1+r
This equals 1.09025 or 9.025%
81. Additionally, 1.07 = (1.05 x 1.0601 x 1.09025)1/3
Interest Rate Types
One thing you might have noticed is that there are a lot of
interest rates when doing these calculations. In order to
differentiate between them, investors coined two terms to
describe them.
The spot rate refers to the yield to maturity for a zero coupon
bond that prevails today.
The forward rate (also called the forward yield) is the
theoretical, expected yield on a bond several months or years
from now. It is common to denote a forward rate as xRy this can
be read as the x forward rate y years from today.
Theories of the Term Structure
The expectations hypothesis is the theory that the forward rate
equals the market consensus expectations of what future short-
term rates will be. This means that there is no liquidity
preference.
This can be stated as E(r) = f2 or that the expected rate in
period two will be the rate in period two.
Using this hypothesis is what allows use to generate a yield
curve, as no other information is required aside from current
spot rates.
Theories of Term Structure
Liquidity Preference: Essentially states that investors or buyers
of fixed income securities have a preference of either short term
or long term securities. Part of the reason for these preferences
can be seen as attempts to match cash inflows to cash outflows.
In any event, short term investors would require a premium to
invest in longer term securities, or the f2 > E(r2) or the rate in
82. period 2 must be greater than the expected rate in period 2.
Advocates of liquidity preference believe that short term
investors dominate the market which pushes the short term
interest rate down.
Interest Rate Risk
As we’ve seen bond prices move inversely to changes in the
market interest rate. Therefore, bond investors are particularly
concerned with the sensitivity of bond prices. We also note that
bond prices are convex and therefore decreases in YTM have
bigger impacts on price than increases in YTM for the same
magnitude. A summary of bond price observations
Bond prices and yields are inversely related.
An increase in a bond’s yield to maturity results in a smaller
price change than a decrease
Prices of long-term bonds tend to be more sensitive to changes
in interest rate.
Sensitivity to price changes increases at a decreasing rate. A 30
year bond is not 6x more sensitive than a 5 year
Interest rate risk is inversely related to a bond’s coupon rate
The sensitivity of a bond’s price to a change in its yield is
inversely related to the YTM at which the bond is currently
selling.
(Macaulay’s) Duration
Frederick Macaulay termed the effective maturity concept the
duration of a bond. Macaulay’s Duration equals the weighted
average of the times to each coupon or principal payment. The
weight associated with each payment time should be related to
the “importance” of that payment to the value of the bond.
Timing of cash flows is designated in years
Wt = PV of CFt/Bond Price
D = Σ T x Wt
83. Duration ExampleTime untilPV of CFColumn
CPaymentDiscount rate =timesPeriod(Years)Cashflow5% per
periodWeightColumn FA. 8% Coupon Bond10.540 $
38.10 0.0394960.01972140 $ 36.28
0.0376150.037631.540 $ 34.55
0.0358240.0537421040 $ 855.61 0.8870651.7741Sum:
$ 964.54 1.8852B Zero-Coupon10.50 $
- 00.0000210 $ - 00.000031.50 $
- 00.0000421000 $ 822.70 12.0000 $ 822.70
2.0000
Why is duration important
There are three primary reasons why duration is important.
First it is a simple summary statistic of the effective average
maturity of the portfolio
It is an essential tool in immunizing portfolios from interest rate
risk
Duration measures interest rate sensitivity of a portfolio
Modified Duration
Reason for – sign The price-yield relationship is negatively
correlated; when prices go down, the implied yield goes up. The
minus sign allows the modified duration to be positive for a
normal bond.
23
Modified Duration Example
Consider the 2-year maturity, 8% coupon bond, selling at a
price of $964.54 for a YTM of 10%. The semi-annual duration
of this bond is 1.8852 years. The annual duration is 3.7704.
84. Therefore modified duration is 3.7704/1.05 = 3.591.
No suppose that the semiannual interest rate increases to 5.01%
-3.591x .01% = -.03591%
This would be interpreted as a .01% increase in interest rates
would cause the price of a bond to fall by .03591%
Rules for Duration
The duration of a zero-coupon bond is its time to maturity
A coupon bond can have a duration less than one.
Holding maturity constant, a bond’s duration is lower when the
coupon rate is higher
A bond’s duration increases with its time to maturity.
The duration for a coupon bond is higher when the bond’s yield
to maturity is lower
Duration of a perpetuity (1+r)/r
Convexity
Convexity Formula
Duration with Convexity Example
Immunization
Immunization techniques refer to strategies used by investors to
shield their overall financial status from interest rate risk.
Many banks and thrifts have naturally occurring mismatches
between their liabilities and assets. Much of there liabilities are
deposits that are short term and have low duration. Bank assets
85. are primarily consumer and commercials loads and have higher
duration. This means that banks are sensitive to changes in
interest rate as they can directly effect new worth.
Immunization Example
Consider an insurance company offering a Guaranteed
Investment Contract for $10,000 and guarantees and interest
rate of 8%. If the GIC has a maturity of 5 years. The future
value of the liability is $14,693.28.
If the company chooses to fund the liability with a 10,000 8%
coupon bond. Then as long as interest rates remain at 8% the
liability will be exactly matched.
But what happens when interest rate rise or fall?
Immunization ExamplePayment NumberYears Remaining until
ObligationAccumulated Value of Invested PaymentA. Rates
Remain at 8%14800 x (1.08)^4=1088.39123801 x
(1.08)^3=1007.7732802 x (1.08)^2=933.1241803 x
(1.08)^1=86450804 x (1.08)^0=800Sale of
Bond010800=1000014693.28A. Rates Fall to 7%14800 x
(1.07)^4=1048.63723801 x (1.07)^3=980.034432802 x
(1.07)^2=915.9241803 x (1.07)^1=85650804 x
(1.07)^0=800Sale of Bond010800=10093.4614694.05A. Rates
Fall to 7%14800 x (1.09)^4=1129.26523801 x
(1.09)^3=1036.02332802 x (1.09)^2=950.4841803 x
(1.09)^1=87250804 x (1.09)^0=800Sale of
Bond010800=9908.25714696.03
Immunization
In this example because we have successfully matched the
duration of our asset with our liability we can be considered
immune from interest rate risk. However, immunization
investors have a risk trade off between price risk and
86. reinvestment risk.
Price risk is essentially the capital loss or gain that occurs
because of a change in interest rate. Now a key consideration
with immunization is the need to rebalance. As interest rate
change a portfolio manager must rebalance as the duration will
have changed.
Additionally, even if interest rates stay the same durations will
change solely because of the passage of time.
Constructing and Immunized Portfolio
A bank must make a payout of $19,487 in seven years. The
current market interest rate is 10%, so the PV of the payout is
$10,000. The portfolio managers wants to fund the obligation
with a 3 year zero coupon bond and a perpetuity paying 10%.
How can the manager immunize the portfolio?
Calculate the duration of the liability. In this case because it is
a single payment obligation, the duration is 7 years
Calculate the duration of the asset portfolio. The portfolio
duration will be the weighted average of each assets duration. In
this case the zero coupon bond has a duration of 3 years and the
perpetuity a duration of 11 years.
Asset Duration = w * 3 + (1-w)*11
Find the asset mix that sets the duration of assets equal to 7
years
w * 3 + (1-w)*11 = 7 in this case w = .5
Fully fund the obligation. In this case it means that 5,000
should be invested in the zero coupon bond and 5,000 should be
invested in the perpetuity.
Duration Gap Analysis
Duration Gap:
From the balance sheet, A = L+E, which means E = A-L.
Therefore, DE = DA-DL.
87. In the same manner used to determine the change in bond
prices, we can find the change in value of equity using duration.
DE = -[DA - DLk]A(DR/(1+R))
DLk is total liabilities / (Total Liabilies + Equity) or the
proportion of assets funded by liabilities
Suppose a manager has the following situation:
Duration of Assets = 5 years and Duration of Liabilities = 3
years
The current interest rate is 10% and it is expected to rise to 11%
what is the impact on the net worth of the company
Duration Gap AnalysisAssetsLiabilities and EquityAssets
100Liabilities 90Equity 10
First let’s calculate the potential impact on net worth.
DE = -[DA - DLk] x A x (DR/(1+R))
-[5-3*.9] x 100 x (.01/1.10)
or -2.09 decrease in equity
What about Asset and Liability Accounts?
DA = -5(.01/1.10) = -.04545 = -4.545% = 95.45
DL = -3(.01/1.10) = -.02727 = -2.727% = 87.54
New Balance SheetAssetsLiabilities and EquityAssets
95.45Liabilities 87.54Equity 7.91
Duration Gap Example
Duration and Repricing Gap Example