Weeks 064, 5,

1
Weeks 06-4, 5, 6

• The next few weeks focus on
• (1) how to measure concentrations of
  interest-rate risk (IRR) that might harm
  stockholders,
• (2) how to mitigate them, without giving up the
  opportunity to earn profits from allowing
  relationship customers to invite IRR into an
  FSFs portfolio.
• Objective To show how concepts of duration and
  convexity can be used to track and hedge
  concentrations of interest-rate risk that the
  economic structure of an FSFs core customer base
  invites into its portfolio.

2
Foundational Slide

• Stockholders should care about Total Return An
  institutions date-to-date change in economic NW.
  This is usually either more or less than either
  its date-to-date cash flow or its accounting
  income.
• Interest volatility impacts an FSFs economic
  net worth and therefore its total return, unless
  the effect of interest-rate swings on the prices
  of an FSFs assets happens to offset precisely
  the effect on the prices of its liabilities.
• To establish Interest-rate insensitivity requires
  the careful planning and execution of a
  offset-creating risk-management strategy.

3
IRR is also called Interest-Volatility Risk

• Volatility The root word is the Latin and modern
  Italian word, "volare" to fly. It is featured
  in a pop song recently reprised by the Gypsy
  Kings. The songs chorus goes as follows
  Vo-lare, o-o, cantare, o-o-o-o Nel blu dipinto
  di blu, felice di stare la su,
• Ironically the song emphasizes only the joy of
  flying high. The word volatility looks at
  exposure to the downs as well as the ups of
  return fluctuations. What soars high in the air
  cant stay there forever. The stock market
  teaches this lesson repeatedly.
• Returns that fly high in the air have to return
  to earth eventually. The word volatility is
  applied in finance to security prices, interest
  rates, currency values, and portfolio returns
  that alternate up and down swings over a wide
  range of values during successive business cycles.

4
Convenient to define a balance-sheet items
interest sensitivity ( IRR) as the change in
the items value associated with a hypothetical
change in the accretion factor (1R). This
definition produces a measure known as
Macauleys Duration, D. D is minus the
elasticity of Value (V) with respect to
(1R)D treats the percentage change in the
compound-interest factor (1R) as the initiating
force. This works because ?(1R) ?R itself.
Division by (1R) merely scales down the change
changes in would be much larger.
Measuring Interest Sensitivity
5
INTUITION OF Macaulays Concept of Duration (D)

• D averages the timing of flows, not the value
  of the flows. It is a weighted average of the
  futurity of all flows specified in the contract.
• D has the dimension of time t because the weights
  wt are pure numbers. The weights state
  percentage of the PDV of any item that accrues at
  date t.

6
Questions to Test Your Understanding of This
Definition

• What is the duration of a single-payment
  security?
• What are t and wt in a zero-coupon bond?
• What is the duration of each coupon stripped
  from different pieces of a multiperiod coupon
  bond?

7
To Calculate D and Total Return For Items in the
Banking Book, We Can Use the PDV Model to
Estimate Fair Values at Beginning and End Dates.

• a. Value of any fixed income stream may be
  expressed as a polynomial equation in the powers
  of a single variable x and written in standard
  implicit form
• b. The present discounted value (PDV) from
  purchasing any future stream of payments a1, ,
  anwhether coming from assets or liabilities is
  a polynomial equation in the discount factor
  .
• The factor may also be written as x(1R)-1.

8
Review Rule for Taking Derivatives of a PDV

• We note that the representative term in any
  polynomial has the form akxk. The first
  derivative is
• For a bond, this formula is simpler than it looks
  for n 2,

9
HOW CALCULUS SPAWNED THE CONCEPT OF D
Note is negative. Why? Calculate

10
D is easier to remember as the absolute value of
an elasticity. An elasticity is a double-log
derivative. D establishes a log-linear
approximation to the true effects that changes in
the accretion factor (1r) have on the P of a
position or instrument.EP, (1r)
lt 0.
D is an Elasticity
11
Usefulness of the Word Futurity

• 1. "Futurity" expresses the distance in time
  until a payment is received. Although an
  artificial term, it is a foundational one.
• Macaulays Duration and maturity both tell us
  about the average futurity of payments due in
  contractual sequence
• 2. Maturity interval until the final payment
• Maturity neglects the futurity of any interim
  cash flows Example of how to define the average
  life of a T-period Coupon Bond

12
Three Intuition-Building Interpretations of D

• D represents a price-weighted measure of the
  futurity (t or x) of a stream of projected cash
  flows. The weight given each futurity is the
  percentage of the price (i.e., the PDV) of a
  position attributable to the claim to payments
  scheduled at t.
• D expresses the sensitivity of the PDV of an
  earnings stream to changes in market interest
  rates.
• D tells us the asset or position being examined
  has the same futurity as a single-payment bond
  whose maturity is D.

13
Risk Officers Must Understand that the PDV Model
Implies That Duration is Merely a First-Order
Measure of the Interest Sensitivity of Fair Values

• It predicts the change in PDV of anything that
  occurs when R goes from R0 to R0?R as long as ?R
  is small.
• PDV(R) can be expressed as a Taylor Series
  expanded around RR0. This provides a way of
  using first and higher-order derivatives of the
  PDV polynomial to express Interest Sensitivity
  with respect to ?R as precisely as one wants.

14
HIGHER-ORDER MEASURES CAN BE EXTRACTED FROM
TAYLOR SERIES REPRESENTATION FOR ? PDV

• Stopping at the first derivative is only a
  first-order approximation to expanded
  around RR0
• Second derivative is called convexity It
  describes the extent to which the graph of true
  interest-rate sensitivity would bend inward if
  f(Ro) is positive or outward (if it is
  negative) relative to line through the origin
  with the slope given by the first derivative.

15
CONVEXITY
Duration (D) is a Benchmark measure that itself
is apt to change as interest rates change.

• f ''(R) gt 0 implies Duration i.e., the negative
  f '(R) is increasing with R Graph of f(R)
  flattens out as R rises.
• f ''(R) lt 0 implies f '(R) is decreasing with R
  Graph of f(R) steepens as R rises.

16

• Duration is only a Semi-Sophisticated Way to
  Measure Interest-Rate Risk. But one has to
  measure something before one can truly be able to
  manage it.
• Basic finance courses teach students to apply D
  to individual instruments this course studies
  how to apply D to a portfolio of asset and
  liability positions.
• From an algebraic perspective, balance sheets
  treat liabilities as assets that carry a negative
  sign.

17
As a foil by which to understand the difference
between maturity and duration, it is instructive
to define a concept of futurity that occupies a
logically intermediate position between maturity
and duration the so-called Weighted life (WL) of
a bond. Reminder A foil is a character that
is put into novel or a play to help the
audience to understand another (often more
important) character by contrast.
UNDERSTANDING D MORE FULLY
18
Intuition-Builder WL expresses the idea that, at
a given market yield and maturity, it is
reasonable to regard the cash flows from a
high-coupon bond as shorter in futurity than a
low-coupon one. Why? Compare a bond whose C
1 with a bond whose C 20. More of the bonds
present value is returned early when C20.
19
WL corresponds to what is called the maturity
buckets approach to blending the different
futurities of an instruments various cash flows.
It weights the futurity of the cash flows
scheduled for each designated bucket (or window
of time) by the percentage of the total of
undiscounted cash flows the contract promises to
accrue in that bucket.

• Let vt of an instruments total cash flows
  scheduled for date t.

The variable t is averaged vt are the weights.
20
Example of WL for a 5-year 20 coupon bond with
face F.

• Total cash flow is 2F. Weight of each coupon
  is 1/10.
• But v5 .6. Why?
• For a portfolio, WL is calculated by allocating
  post-dated cash inflows and outflows across
  designated timing segments that are envisioned
  as maturity buckets.

21
Suppose F was 100?

• WL ?
• WL depends only on the timing structure of the
  contract, not the size of the principal involved.
  (F washes out of the calculation. Only the ratio
  of C to F matters).

22
Macaulay's concept of Duration (D) resembles WL
more closely than it resembles maturity.
COMPARISON OF WL AND D

• Two Similarities
• Both D and WL average the timing of flows, not
  the value of the flows. Each is a weighted
  average of the futurity of all flows specified in
  the contract.
• Both WL and D have the dimension of time t
  because the weights wt are pure numbers.

23
Duration differs from WL by using weights wt that
represent the percentage --not of the bonds cash
flows-- but of the bond's present value (i.e.,
its price P) that is due at each date t. Let Ct
the cash flow from bond at t.wt (1Rt)-t
Ct/P discounted value of t-period cash flow
sum of discounted
values of all flows

• The denominator of each weight in D is the PDV
  (i.e., equilibrium price) of the complete
  contract.

24
WL vs. D of COUPON INSTRUMENTS

• Ignoring single-payment securities, why must
  weighted average life always be greater than
  duration? ANS. The WL calculation ignores the
  "time value of money." Makes no use of PDV
  concepts. This neglect overweights distant flows
  relative to nearby ones.
• The unit period in some bonds and mortgages is
  less than a year. There are handy formulas for
  securities that pay coupons f times a year. The
  basic idea is the unit R becomes R/f.

25

• -Let f no. of times payments are made in a year.
  Duration of a dollar f times per year forever at
  the current interest rate R is
• Queries What if f1? Ans.
  Is D a function of R? What is the WL of a
  perpetuity? Ans. Its indefinitely large.
• What is D for a perpetuity for which R .10 and
  f 1?
• Does D double if R falls to .05?

26
Example If a perpetuitys R rises from .10 to
.125, P falls from 10 to 8.
Queries

• a. What happens to the bondholders wealth if R
  rises? The holder loses value issuer gains.
• b. If R falls? Holder gains. What happens to
  the liability owed by the bond issuer?
• c. Why ought students of FSF management learn to
  look at both sides of the transaction?

27
Duration Falls with Yield to Maturity for
Maturity and Coupon Rate Fixed
The relation between ?D and ? n varies with
coupon, maturity, and R, because ceteris
paribus D varies when any of these variables
changes.
Do WL and maturity also change with yield? No!!!
28
FIRST SET OF DURATION EXERCISES

• Question No. 1 Find the duration of a 10-year,
  6-percent annual coupon-bond, when the market
  yield is 10 percent.
• We must first find the price of this hypothetical
  bond, which need not par.
• We can reinterpret the bond synthetically as a
  position in 10 separate zero-coupon securities.
  The duration formula tells us that when the
  denominator of the weights is adjusted to give
  the relative NW the position places in each
  security, weighted durations add across
  component positions in a portfolio.

29
Here is a model WORKSHEET
30
Question No. 2 Weighted average life, WL.
WORKSHEET

• Compared to D, WL increases the relative weights
  of distant receipts.

31
Intuition-Building Questions

• a. How much shorter is D than M and WL for this
  security? D7.422 WL8.31 M10.
• b. Would duration be higher or lower if the
  coupon rate on the bond was higher (say 10
  percent)? Lower. Why? Distant flows have less
  proportionate value.
• c. What if the coupon rate 0? ANS. For a
  single-payment bond, DWLmaturity.
• stripping concept. Synthetic repackaging and
  assembly CAN create zero-coupon securities

32
Test Your Intuition
a. Why would the duration of the 10-year annuity
be less than the duration of a perpetuity? ANS.
The intuition is that the annuity has no value
that accrues after n10. b. Why would the
duration of a 10-year coupon bond be larger than
that of a 10-year annuity? ANS. More of the
coupon bonds value occurs at n 10.

• Breaking bond down synthetically, we can show
• a. Duration of coupon stream is
• b. Duration of return of principal F raises the
  duration of the position

33
REVERSE ENGINEERING VIEWING D SYNTHETICALLY

• The duration of any set of fixed n-period cash
  flows may also be calculated as the sum of the
  present-value-weighted futurities of the
  individual pieces.
• The first term in the equation is the duration of
  the unit perpetuity.
• The second term subtracts off the duration of the
  cash flows that have to be surrendered at date n.
  Hence,
• Dn .

34
Duration, if only it were so simple.
35
WE CAN ONLY APPROXIMATE PERCENTAGE PDV CHANGES
WITH DURATION
True Relationship (is convex to origin)
Queries Why do both curves go through the
origin? In what quadrants does D overstate
change? Where does it understate it?
36

• If is not zero, duration must be
  understood to be DD(r).
• The existence of a nonzero derivative of the D(r)
  function introduces curvature CX (convexity
  toward the origin) into the true graph of
  against . CX
• Convexity reduces or increases the magnitude of
  gains and losses relative to predictions made
  from the linear model of percentage-price change.

37
For an FSF, convexity becomes more important in
two circumstances (1) the larger the change in r
becomes and (2) the more imbedded options
customers enjoy.

• In practice, FSF managers often take the
  value for CX used in the corresponding
  Taylor-series representation and add the
  following second-order term to the equation
  1/2 CX(?r)2.
• Managing Macauleys Duration cannot fully protect
  against IRR. It does so only in the absence of
  optionality and for parallel infinitesimal shifts
  in the yield curve.
• Nonparallel shifts are addressed by a burgeoning
  software optionality is tougher to model
  precisely.

38
Why is Convexity Helpful?

• 1. Strategically Using it helps to immunize Net
  worth against large movements in r.
• Substantively In two positions, the
  algebraically more convex position becomes
  shorter faster for a given rise in r and becomes
  longer faster for a given fall in r.

Query For what movement in interest rates are
liability positions more convex than asset
positions and vice versa?
39
Could Anything Be Even More Useful than D?

• It is conceptually simpler to focus on a concept
  known as Modified Duration, D.
• D
• The fundamental equation of IRR tells us that the
  percentage price of any income stream responds to
  a tiny change in yield as follows

40

• D formulation expresses the price sensitivity of
  a position whose value is P to a given
  basis-point change in the yield.
• Setting ?R.01.0001, -PD ?R gives the
  marginal price value of a basis point pvbp.
• pvbp -PD(.0001)
• Some websites now report pvbp, D, and convexity
  as a characteristic of individual bonds
• But stakeholders ought to concern themselves with
  pvbp, D, and convexity of FSF NW.

41
New TopicII. Algebra of Synthetic Replication

• Replication means to reproduce exactly, as
  (e.g.) in cloning animals. Finance theory uses
  Replication As a Valuation Aid. Logically, we
  are free to calculate the value of any payment
  stream by looking at the value of an
  easier-to-calculate substitute stream that
  replicates its particular payments. The economic
  justification for this substitution is the Law
  of One Price.
• We can replicate the cash flow generated by a
  finite annuity of maturity n as a contract
  written on the difference between two
  perpetuities one starting its payments now and
  the other starting at the maturity date tn.

42
We can Replicate the Cash Flows of any Unit
Annuity as the Difference Between Two Unit
Perpetuities
Line 1 Line 2 Line 3
Query How to replicate C1 and C2 of a 2-period
Bond ?
43
A structured derivative is a tradeable claim that
can be extracted from elements imbedded in a
standard financial contract.
REVIEW

• Financial engineering can strip and recombine
  n 1 derivative instruments from a n-period
  coupon bond F, C1, Cn. Alternatively, a
  portfolio of annuities and zero-coupon bonds
  (i.e., bullet payments) can be constructed
  synthetically to be equivalent to the bond C
  units of a n-year unit annuity plus a n-year
  zero-coupon bond of principal F.
• Synthesizing debtor (shorts) vs creditor
  positions?

44
Example Stripping Coupons From a Two-Period Bond
45
Synthetic Way to Fair Value n-Period Assets
that Generate a Fixed Annual Cash Flow

• Need valuation skill Knowing how to find the
  value of a unit annuity paying one dollar per
  year for n years (V1,n).
• V1,n depends only on the current market interest
  rate, R and the date of the endpoint of the
  stream, tn
• V1,n (1R)-1(1R)-2 ...(1R)-n
• Value of the one-dollar annuity (i.e., the
  bracketed sum of component values) is the sum of
  a geometric series.

46
The sum of such a geometric series in k is
Substituting (1R) for k gives value of unit
annuityNo problem on a calculator (for
reasonable n). ? With a PERPETUITY, the (1R)-n
term vanishes
UNDERLYING ALGEBRA
47
FAMOUS REVERSE-ENGINEERING FORMULA A

• Cash flows from an n-year unit annuity may be
  replicated by a contract written as buying
  perpetuities and selling them forward.
• The Law of One Price tells us that the value of
  the unit annuity at t0 has the same value as the
  value of a replicating portfolio of
  perpetuities

Formula a

• Interpretation The first term in the sum is the
  value of the perpetuity at t0 the second term
  is the value given up in synthetically selling
  the instrument forward at date n at rate R. The
  n-year annuity generates the same cash flows as
  buying a perpetuity today and simultaneously
  contracting to sell the perpetuity forward in n
  years at the current interest rate R.

48
VALUE GIVEN UP IN THE FORWARD SALE
The cash flows for the perpetuity whose first
payment begins at tn1 is worth less at t than
it will be worth at tn
Value Today of Perpetuitys Cash Flows Due After
n
49
Famous Formula a embodies three synthetic
megacepts that help to interpret the value
V1,n.

• 1) concept of a forward transaction (the sale of
  a perpetuity today for delayed delivery at tn)
• 2) concept of a replicating portfolio of assets
  and liabilities (the spot purchase and forward
  sale)
• 3) concept of a synthetic derivative instrument
  (the value of the synthetic replicating portfolio
  derives from the value of tradable underlying
  securities as the difference between the payoffs
  of two standard instruments)

50
Important Financial-Engineering Perspectives on
Replication

• One can treat formula a as describing a contract
  or deal that establishes a synthetic incremental
  balance sheet and proceed to calculate the
  contracts net worth. The formula tells us that
  the interest-volatility risk of holding the
  perpetuity as an asset is reduced by accepting
  the interest-volatility risk of the liability
  that constitutes the forward sale.
• Every multiperiod instrument may be valued as a
  series of forward item values.
• Formula a corresponds also to a financial
  intermediary whose business plan is to lend via
  perpetuities and to issue a forward liability
  to finance some of the asset value.

51
By the Law of One Price, all replicating
portfolios have the same PDV.

• We can show this by brute force for any n Take
  n1
• Using formula a, V1,? for R.10 is

Double-Checking Must the direct-discounting
value .
52
First term in Formula a is the value (V1,?) of a
unit perpetuity. V1,? is severely exposed to
interest-rate risk. Value changes as interest
rates rise and fall.

• When R .10, a unit perpetuity
• is worth 10.
• Suppose R rises to .125? V1,? 1/R?
• Suppose R falls to .08 ? V1,? ?

8
12
53
It is easy to calculate how Fluctuations in R
change the value of an n-year unit annuity by
calculating V1,n for three benchmark values of
R Suppose n 2

• R .10? First term is same as in the one-year
  calculation (10) only the value of the forward
  sale changes
• V1,2 10 - (1.10)-2 10
• V1,2 10 10(.826) 10 - 8.26
• we can doublecheck by discounting the annuitys
  scheduled cash flows directly
• 1/1.1 1/1.21 .91 .83

1.74
1.74
54
-- R .125? First term in the pricing formula
is now 8. The discount factor in the second term
is now (1.125)-2 0.79, so that

• V1,2 8 - (.79)8
• 8 - 6.32 1.68 (which lies below the R10
  value).
• -- R .083? First term is now 12. (1.083)-2
  0.852,
• V1,2 1.78 (above the R10 value of 1.74).

55

• The value of the underlying asset and the
  synthetic forward liability move simultaneously
  but in an opposite direction. The forward sale
  trades away much of the assets IRR to the
  synthetic partner .
• 1) As R rises 2.5 percentage points from R10,
  the negative forward position improves by 1.94.
• 2) As R falls 12/3 percentage points, the
  negative forward position deteriorates by 1.96.

56
Managerial tools consist of technical
financial-valuation skills, risk-assessment
skills, and hedging concepts.Duration (D) is a
benchmark that can establish accountability to
stakeholders for managing IRR.
III. HOW INTEREST-RATE RISK CAN BE HEDGED
57
Risk Management Seeks to Price Risk Accurately
and to Limit Possible Fluctuations in Total
Return and Net Worth

• In general, value fluctuations depend on
• volatility of individual assets and liabilities
• correlations among individual positions
• explicit use of risk-management instruments
• Some positions within a portfolio are natural
  risk offsets (or internal hedges) for others
• for example, increased interest rates reduce
  values of both asset and liabilities. This Hedge
  is imperfect because the two effects are seldom
  equal.

58
Managerial Perspective on IRR

• Opportunities to earn net interest income change
  not just because of changes in a borrowers
  default prospectsbut also because over the
  business cycle market-determined yields on the
  intermediarys deposits may change faster or
  slower than yields on its loans and investments.
• Exposure to risk may be reduced
• first and foremost, by asset diversification
• by adjusting liability structure
• by directly trading away risks via insurance,
  forward, or other derivative contracts

59
Overview Strategies for Managing Risk

• Install systems that Measure Exposures (?RADAR)
• Identify whether and how different exposures a
  deal creates can be
• 1. Avoided
• 2. Transferred to another party or Diversified
  (Insured or Hedged)
• 3. Actively priced and kept within appropriate
  limits self-insured by Explicit or Implicit
  Reserve Accounts
• Goal of Risk Manager is to choose a profitable
  and comfortable combination of strategies

60
Exploring the Metaphor The word hedge means a
protective position.

• The root meaning of "hedge" is that of a
  carefully maintained natural "barrier,"
  "enclosure," or "screen" built up from a group of
  shrubs or small trees planted in close rows.
• As a barrier, a hedge provides decent protection
  from dogs, but imperfect protection from smaller
  pests such as squirrels and crows.
• When intended mainly as ornamental boundary
  markers, hedges serve further functions as fences
  that impede unwanted guests or eyes and control
  against soil erosion.
• Query How do the screening, enclosure, and
  anti-erosion functions parallel the portfolio
  services performed by a "financial hedge?"

61
Hedging is opaque and usually imperfect. A
financial hedge seeks to erect one or more
counterbalanced positions to keep particular
classes of risks out of one's portfolio and to
control against "wealth erosion," especially
from large moves in recognized sources of risk.
62
Definition A hedged position is a
counterbalanced exposure to a risk. A second
transaction is developed to offset an initial
position so as to generate a counterbalanced
balance sheet.

• Mixed metaphors abound in hedging vocabulary a
  hedge is presumed to stand on the two legs of
  its counterbalanced positions. The offsetting
  positions are described by analogy as "legs" that
  keep one's wealth standing when it is rocked by a
  potentially upsetting force.
• It is unusual for the hedging leg to wholly
  match the preexisting portfolio of uncertain
  future cash flows.

63
LEGS OF A HEDGE

• Opening a second leg helps to keep an
  institution's profits from tumbling below water
  at the slightest push.
• Hedging Contrasts with metaphorical plunging
  jumping into an untested pool of water with both
  feet.

64

• Risk Management means taking steps to price and
  to control the impact of individual-position
  risks on an institutions targeted bottom line.
• Whether one welcomes or fears an upward or
  downward movement in a given interest rate varies
  with algebraic sign of one's portfolio "position"
  in the associated contract.
• On any balance sheet, Assets have negative IRR
  exposure. Liabilities have positive exposure to
  interest-rate increases.

65
Managing Interest-Volatility Risk If
the future course of interest rates were known in
advance, IRR would not exist. Financial risk
comes from the dispersion of possible outcomes
due to unpredictable movements in financial
variables (here R). Worry attaches to
unpleasant eventsdownside possibilities.

• For outstanding fixed-rate debt, increases in R
  harm a creditor and benefit a debtor. For debt
  whose terms are still being negotiated, the
  reverse holds.
• Decreases in R have opposite effects.

66
For an FSF, IRR comes from the possibility of
adverse market revaluations of NW due to a ?R
Duration of NW captures the exposure to changes
in PDV valuation of every item on an FSFs
balance sheet and income statement

• Portfolio revaluation is rooted in different
  positions different timings for their
  interest-rate resets.
• Flows from maturing or prepaid positions must
  be put to work at fresh reinvestment interest
  rates. Sometimes this is good news, but
  sometimes it is bad.

67

• Management needs to make IRR measurements not
  just for individual instruments, but for
  portfolio positions.
• Measures of Portfolio Duration can be built up
  from synthetic instruments whose durations are
  straightforward to compute.

68
No appraisal method is an exact method. Users
must make allowances for errors in relying on
either MVA approach. PDV is only a valuation
model and should include an error term.

• Marking to PDV value relies on hypothetical
  projections of returns and a procedure for
  selecting a discount rate that prices the
  uncertainty of the projections.
• Model error can come from inserting either the
  wrong projections or the wrong interest rates
  into the PDV equation.

69
Dependence of D on the Ability to Make and
Support Appropriate Assumptions is the Linchpin
of PDV
Selecting and Logically Justifying Projections
and an Appropriate Reference Rate are Arts
70

• ALCOs that have less faith in PDV employ a
  lower-tech way of measuring IRR. They table the
  time-to-repricing of the assets and liabilities
  on an FSFs banking book across of series of
  repricing buckets or windows also called
  timing segments to calculate the FSFs
  distribution of repricing gaps
• Crudeness Compared to D and Convexity Bias is
  generated by letting distant () gaps offset
  shorter (-) gaps on a dollar-for-dollar basis.

71
If staff is careful, an ALCO can analyze
intelligently the mismatches revealed by such a
table.

• Stress Tests and Scenario Analysis Seek to
  determine (e.g.) what happens if all interest
  rates move permanently up by X .
• Might ask ALCO staff might to calculate how big a
  reduction the FSF will experience in accounting
  earnings on its existing positions over the next
  3 quarters and the following two years?

72
IV. Ways to Re-Engineer the Duration of Net
Worth

• FSF profits and NW are better conceived as
  financially engineered stochastic payoffs on
  synthetic contracts for exchanging value
  differences. An intermediary promises to
  allocate to its owners the difference between the
  putatively observable difference between asset
  returns and deposit costs.
• Financial Engineering creates synthetic
  contracts by recombining selected pieces of
  actual contracts.
• Synthesization makes use of algebraic identities
  between PDV formulas for perpetuities, annuities,
  and coupon bonds. These formulas transform
  intuitive ideas of stripping and forward
  sales into algebraic operations.

73
Synthetic Perspective Offered by Financial
Engineering leads us to view an FSFs total
return on its NW as a weighted average of returns
on the firms portfolio of individualized nonzero
tangible and intangible balance-sheet positions,
with -of-NW portfolio shares as weights.
Similarly, the IRR of net worth (or indeed of
any balance-sheet position) is a weighted average
of the IRR of the positions or - component
instruments.
Insight
74
This Focus Clarifies that Value Creation and Risk
Management are Enterprisewide Affairs

• This perspective uses an infinite Taylor series
  (i.e., a power series) to represent an FSFs NW
  as a synthetic or derivative contract. This
  contract is written on the sum of the numerous
  value differences (interest receipts, expenses,
  service fees, and value changes) that generate a
  portfolios net cash flows in all conceivable
  future periods.

75

• This focus clarifies the contribution that each
  position makes to the firms overall profit and
  IRR. It also clarifies that adding
  offset-generating positions (hedging) can lay
  off some or all of the IRR in the flow of deals
  that an FSFs customers bring through its
  portals. (Laying off unwanted positions
  parallels the behavior of bookies and used-car
  dealers)

76
Essence of IRR management is Matching and
Mismatching the Durations and Convexities of the
Two Sides of an FSF Balance Sheet. Hypothetical
objective is to Control the Value of Owners
Equity.
MEASURING THE IRR OF AN FSFS BALANCE SHEET

• Duration is useful because
• (1r) is the accretion or compounding factor in
  interest accumulation.
• P is the value of a particular instrument or
  balance-sheet position. Note that the minus sign
  recognizes that the relation between P and (1r)
  is inverse.

77
Why must ? ANS. The
Difference Rule of basic calculus tells us

• Query Can 0? An Institution for
  which would be zero for all possible
  changes in R might be said to be completely
  protected or "hedged" against "interest
  volatility risk."
• Query Is zero IRR what risk managers should aim
  for? No! Such immunization is not necessarily a
  desirable target for an FSF because giving up
  risk usually implies giving up some return.

78
Why is interest volatility important to FSFs?

• Net worth is the difference in the aggregate
  values of each individual asset and liability
  across the balance sheet.
• Variation in interest rates affects the value of
  every asset and liability in a banks portfolio
  that is not a par floater altering the
  profitability of almost every deal that was made
  in the past but has not yet matured.
• Applying PDV to accounting statements gives us a
  disciplined way to aggregate whether and how
  interest-rate movements affect an FSFs income
  and NW bottom lines.
• Changes in fair value of an institution's net
  worth (i.e., its ownership capital) arise as
  the sum of all changes in properly signed item
  values. S may be understood as a synthetically
  engineered contract whose item values fluctuate
  with changes in market yields.

79
When R Moves, FSF Staff Better Know What the
Duration of Net Worth is
80
As long as component portfolio positions are
PDV-weighted properly, Durations add across
portfolio positions to get DN
HOW TO CALCULATE DN

• Let DAthe average futurity of the asset position
  and DLthe average futurity of an institutions
  debts
• Formulas are true weighted averages because
• WA and WL so that WA WL
  1 .
• (These formulas are central to Week 5 and 6
  exercises)

81
Two Ways to Approximate when
1.
2.
Where rE is the opportunity cost of equity
capital.
82
Calculating pvbp of N
?P P(-D)?R

• pvbp formula is the ?P formula with ? R set
  .0001 and P set N.
• Why are the duration and pvbp of an FSFs
  stockholder-contributed net worth deserving of
  managerial attention? In what sense does DN and
  pvbp (N) 0 mean no net interest-risk exposure?

83
Simplest illustration of DN is to look at the IRR
of an all-equity one-asset bank. Suppose bank
assets are perpetuities paying 100,000 a year
when R.10.

• What happens to banks NW when R rises to .125?
  We did the central calculation as a valuation
  exercise. Tangible NW falls to 800,000 a 20
  loss in NW.
• Alternate Cases What if the bank had initially
  financed itself by taking a leveraged position
  consisting of 200K in equity and 800K in
  deposits?
• By itself, leverage tends to magnify an owners
  IRR, but even zero-duration deposit contracts
  shift some IRR to depositors unless repayment is
  perfectly guaranteed.

84
ANY PAR FLOATER HAS ZERO D
Suppose deposits were par floaters whose
economic value is completely interest-insensitive.
The initial balance sheet has a substantial gap
between the D of its righthand and lefthand sides

• Queries The D of deposits is zero and 5.
  If R rises to 12.50, the leveraged NW falls to
  zero. This 100 decline is five times as large
  as the 20 decline observed in the all-equity
  case.
• Suppose DdepDA. What would be new value of DN?
  ANS DA. Effect of leverage is neutralized, but
  IRR remains.

85
V. NUMERICAL EXERCISES AND ISSUES FOR CLASSROOM
DICUSSION

• Begin by Reviewing the Value of Envisioning IRR
  Management as an Exercise in Financial
  Engineering (Review and Exercises)

86

• Every Multiperiod Instrument Establishes a
  Position in a Series of Forward Contracts
• An algebraically negative relation between R and
  PDV of a security holds term by term for elements
  of coupon bonds of any maturity. Value reduction
  caused in any promised cash flow by rising rates
  is all the greater the further out in time a
  payment is due.
• This simple insight tells us how we can explore
  the effects of ?R on the value of an FSFs
  positions and not just effects on the value of
  individual instruments.

87

• In practice, risk managers treat fluctuations in
  a reference rate called the interest rate R as
  driving the yields on every instrument on their
  balance sheet.
• The reference rate R is usually the yield on a
  particular n-period Treasury coupon bond.
• However, when appropriate, R could be the yield
  on a mortgage or zero-coupon instrument.

88
Review Fair value of any instrument or
portfolio is a polynomial in (1R)-1 PDV

• Changes in value due to changes in R are the sum
  of the effects of ?R on the values of each
  time-dated scheduled cash flow ak that is
  imbedded in the PDV of the portfolio.
• PDV conception expresses two kinds of Value
  Additivity
• 1) The value of a PDV sum is the sum of the value
  of the component pieces.
• 2) The change in the value of the PDV sum is the
  sum of the changes in the value of its pieces.

89

• Changes in interest rates change the PDV of
  individual balance-sheet positions. If interest
  changes and/or cash flows are unpredictable, so
  are the changes in position value.
• The Replication Principle Relies on the Law of
  One Price Each dollar raised and invested at
  stale yields can be neither more nor less
  valuable than a freshly funded investment that
  replicates the same promised cash flows.
• Analogy PDV acts like a time machine it tells
  us how the promised value of future payments and
  receipts must be discounted algebraically at the
  appropriate rate R to translate them back to an
  "appropriate(fair) present value.

90
Question 3 of Week 4 Exercise Illustrates This By
Asking You To View Annuities as Synthetics
Constructed From Long and Short Positions in a
Perpetuity

• a. PDV of the perpetuity beginning today.
• b. PDV of a perpetuity whose
  payments begin at date n1 if r is to remain
  unchanged. Also, its expected delivery price
  at t n.
• c. Given that the market rate r is the contract
  rate, the synthetic forward contract is neither
  in nor out of the money today.
• PV of a synthetic unit annuity of long and short
  positions

91
Exercises on Portfolio Duration and Net Worth
Problem Number 1 Suppose a newly chartered bank
acquires perpetuities paying 30 million once a
year. a. Assuming the bank initially funds its
assets entirely by owners equity and the market
interest rate on perpetuities of the risk class
held by the bank is rA .10. Find the duration
and pvbp of the banks tangible net worth.
92
Answer to All-Equity Problem
Assuming 1 on fresh positions, the
governing formula expresses DN as a weighted
difference formula needed a. In
no-leverage case,
years. What is DL? 11
years?
93
b. Suppose the bank sells 250 million of
uninsured deposits on which principal and a
payment of 8 interest are due in one year.
Assume that the cash is immediately distributed
to stockholders (perhaps via a share repurchase
program), r remains at .10 and re .20. Show
what this transaction does to the duration and
pvbp of the banks tangible net worth.
94

• Answer to Part b
• Why not 30?MVTNW
  A - L 300 - 250 50In this case, the asset
  position is leveraged.
• Why not
  divide by 30?
• 66-561 years.
• N.B. the -5 years measures the extent to which
  the leveraged IRR in the asset is passed on to
  creditors.

95
Stress Test Could This Bank Survive a 200 bp
increase in R?

• Suppose we had asked instead, how large a ?R
  would be required to drive MVTNW to zero,
  assuming the banks cost of equity capital was
  .20 per annum?
• First find pvbp
• pvbp -DP(.0001) (50m)(.0001)
  -254,200.
• Then divide initial MVTNW by pvbp 196.7
  basis points

96
c. What would DN be if 290 million of the 8
one-year uninsured deposits had been issued
instead? Note that DN increases with the extent
of deposit funding because the added leverage
increases its loss exposure.
97
Answer to Part c MVTNW 300 - 290 10. What
is DL now? 301 years This is an
outrageously large IRR exposure. DN is usually
kept between 2 and -2 years nowadays. Natural
Stress Test for depositors to apply How large a
move in R would wipe out N? ANS
98
d. Suppose instead the bank had issued 290
million in federally insured perpetual deposits
paying 8 interest once a year. Find DN. Why is
this value less than case c? Why is it even less
than case a?
99

• Answer to Part d
• Given the banks high leverage, insurance is
  needed to make low rL credible. Why?
• DN
• 330 - 391.5 -61.5 years
• What does a negative DN mean
  intuitively?
• case d is much less than case c because the
  duration of the deposit funding goes from being
  much lower to even higher than the duration of
  the assets.
• case d is less than case a because FSF managers
  have gone beyond hedging. They have reversed
  stockholders interest-rate risk exposure at the
  same time that they leveraged their investment in
  the FSF.

100
APPLYING DN TO MARKET CAPITALIZATION, S

• Same approach can be used to approximate the IRR
  of stock-market capitalization. Our valuation
  method assumes that balance-sheet positions will
  be held forever and that the market rate of
  return on bank equity is always RE .20.
• a. Find the market value of S the banks total
  net-worth position for case 1.d.
• b. Find the value of the banks tangible net
  worth (TNW) for this same case.
• c. Suppose enterprise-contributed intangible net
  worth has the value EI 10. Find the value of
  government-contributed net worth, FG.

101
Accounting (i.e., GAAP) NW is an Imperfect
Estimate of Stock-Market Value S

• The market capitalization (S) of a firm may be
  defined in two ways (1) as the product of its
  share price times the number of shares
  outstanding and (2) as the bottom line of its
  economic balance sheet.

102
Ignoring Stock-Market errors in valuation (v), a
banks market cap S equals the sum of MVTNW
plus unobservable component values of
enterprise-contributed NW and government-contribut
ed NW.
S Estimates MVTNW EI FG ( v)
103
Tangible vs. Intangible Positions (Review)

• Tangibility relates to the ease with which an
  asset can be valued separately and sold to
  another FSF. (Tangibility changes with
  information and contracting technologies).
• An intangible asset is either a right conferred
  by a government or other corporation or it is a
  source of value that does not have a separate
  physical existence from the other assets of a
  firm.

104
Questions 2 and 3 apply the concept of duration
to a banks market capitalization (S). a. What
important categories of enterprise-contributed
intangible positions contribute to a banks
market capitalization (S)? What
government-contributed intangible asset would an
economist perceive to constitute a potentially
important, additional part of S?
105
Problem 2
a. Projected Annual Income 30 - (.08) 290
30 - 23.2 6.8 million If assets and
liabilities are projected not to change over
time, S may be modelled as a perpetuity S(N)
34 million b. MV of TNW 300 -
290 10 million c. FG S-MVTNW -EI 34 -20
14 million.
106

• Please express the duration of a banks market
  capitalization (DS) as a function of the
  durations of its tangible assets (DA), its
  tangible liabilities (DL), and its
  enterprise-contributed and government-contributed
  net intangible positions (DE and DG).

Answer DS wADA wLDL wEDE wGDG wA wL
wE wG 1 WA MVA/S WL ? WE ? WG ?
107

• Assume that, at current interest rates, DL 0 and
  DE DA10 years. Assume that the market value of
  corresponding balance-sheet positions are
  SFG10 and (AE) 100. For the banks
  market capitalization to be insensitive to
  increases in interest rates, what would be the
  implied duration of the government-contributed
  equity position?

Ans DS 0 iff. DF -100
108
Policies for Controlling FG Reducing the Amount
of IRR Shifted to the FDIC

• Return to case where FG 14 mil. In what two
  ways could the government make its exposure to
  interest-rate risk actuarially fairer to
  taxpayers?

• Charge an appropriate risk-based fee for
  guarantees (i.e., raise explicit charges)
• Force bank to change its Balance Sheet
• a. Lower DE and DA
• b. Raise DL
• c. Reduce leverage

109
Gambling vs. Hedging at Fannie and Freddie?
110
SWINGING THE BET
111

• Potential True/False (Explain) Questions
• a. A bank cannot maximize the market value of its
  net worth and minimize its IRR at the same time.
• b. Authorities can safely ignore the liquidation
  value of a financial institution's tangible
  assets as long as it remains a going concern.
• c. In purchasing or analyzing the value of a
  going financial-services concern, one must look
  at projections of its future earnings and at its
  current balance sheet.
• d. Market-value accounting for deposit
  institutions is unnecessary and too impractical,
  expensive, and dangerous for authorities to
  consider seriously.
• e. The maturity of many deposits and loans are at
  the option of the customers. These customer
  options mean that increases in R lower values of
  borrower-callable claims but leave values of
  depositor-puttable claims more or less unchanged.
• f. There is no reason to insist that current
  values of real estate accepted as collateral on
  loans have a bearing on a banks loan-loss
  reserve. The lenders do not want the real estate
  sold in the weak market and the houses are
  certainly not going anywhere.

112
Because the durations of instruments and
portfolios vary as interest rates change,
static control strategies are inadequate.
VI. Dynamic Hedging

• It is a disastrous mistake to suppose that the
  duration or convexity of an institutions net
  worth is a constant that managers may set once a
  quarter or so and then forget about.
• Recall landscaping metaphor IRR management
  may be likened to a householders trimming the
  hedges on the boundaries of its yard. The
  managerial implications of this metaphor turn on
  the need to expect rain (adversity or good
  luck) and to plan to respond to the amount of
  rain that falls. This trimming must be planned
  to occur from time to time, and to occur more
  frequently the more it rains.

113
Regular readjustment of hedges resembles
"trimming." Continual realignment of the
durations of the two sides of an FSFs balance
sheet is called dynamic hedging.

• Dynamic hedging seeks to offset changes in
  durations of A and L as R changes and options are
  exercised. Contrast with mindless
  once-and-for-all or passive static strategies
  for coping with interest-rate change.
• Difference lies between totally controlling and
  partially and temporarily managing interest-rate
  risk exposure.
• Dynamic Hedging is indispensable because an
  institution cannot afford not to sell hedges
  and options to its customers.

114
Both Voluntary vs. Involuntary Portfolio
Mismatching Occur

• Customers routinely hold valuable imbedded
  options that let them retime loan and deposit
  contracts e.g., for early loan repayment, for
  refinancing, for early deposit withdrawal, and
  for activating credit lines for taking down
  annuities or policy loans in life insurance
  contracts. The values and durations of these
  options change with interest rates.
• Customers pay handsomely for optionality.
  Compensation paid for optionality is a profitable
  part of almost every deal a bank writes.
• Customers often fail to appreciate the cost of
  these options.
• FSFs often make options troublesome to exercise.

115
The best proprietary FSF models of IRR
internally adjust interest spreads to price
the optionality that imbedded puts and calls
conveys to its customers.

• Maturity of a passbook account is not really
  zero. Maturity is entirely at the customers
  option. The speed of customer deposit runoff
  caused by rising market interest is not fixed.
  It depends on how quickly an FSF resets its
  yields when and as market interest rates rise.
• Correspondingly, on the other side of an FSF
  balance sheet, prepayments are also an endogenous
  variable in the system. The SPEED of loan
  runoffs rises when interest rates fall.

116
OPTIONALITY IS AN IMPORTANT SOURCE OF CONVEXITY

• Whichever way interest rates move, one or another
  set of customers finds its imbedded option in the
  money and truncate scheduled FSF contract
  benefits.
• FSF managers and regulators know that DA and DL
  change with interest rates, not just because the
  PDV of given positions change, but because
  interest-sensitive customers can alter these
  positions by prepayments, deposit flows, and loan
  requests that activate implicit or explicit
  credit lines.
• Long-run survival in a repeat business forces
  FSFs to accept customer-initiated variation in
  the timing of funds flows.

117

• Customer optionality is increasingly recognized
  as creating by itself a negative asset
  convexity and positive liability convexity.
• Increases in R shorten maturities of liabilities
  and lengthen those of assets lower values of
  borrower pre-payment options but increase values
  of depositor-puts.
• Decreases in R do what to same values?
• 1) increase value of borrower calls
  shortening asset durations
• 2) reduce value of early withdrawal options
  lengthening liabilities

118
A. Macro HedgingB. Interest-Rate Swaps and Swap
Duration C. Calculating Duration of Swaps D.
Background for Banc One Case
VII. Swap-Based Synthetic Balance-Sheet Surgery

• Edward J. Kane
• Boston College

119
Hedgers Must Use Market-Value Accounting (MVA).

• Market-value accounting enters asset and
  liability items at synthetic values that are
  either derived from hypothetical models or
  taken from market values observed on comparable
  substitute instruments.
• Using comparables to assign item values is a
  strict form of marking to market. But this
  requires a market in which trading can be
  observed.
• Carrying out a model-based revaluation of items
  across the balance sheet may be described as
  generating fair values via a marking to
  model.

120
Improving on Historical Costs

• 1. If increases in market interest rates can
  impair the values of loans and debt
  instruments, GAAP numbers could be adjusted by
  reserving for this danger and charging off the
  value of impairments that occur.
• 2. Conscientious analysts could impute these
  reserves and market-induced revaluations of bank
  positions even if few banks wish to fully report
  them.
• 3. Imputations could combine two kinds of
  evidence
• Implications of movements in stock prices.
• Opportunity-cost reworkings of the bottom lines
  of accounting reports of the bank and its major
  customer or customer groups.

121
A. Macro Hedging

• This segment focuses on Macro Hedging
  Possibility of using as few as one transaction to
  hedge a portfolio of positions rather than
  hedging items one-by-one.
• Financial engineering provides methods for
  integrating the effects of different risk
  exposures on the capital an FSF needs to support
  them.

122
Fully Integrated View of FSF Risk Management
Risk Analysis
Risk Adjustment
IRR Credit Risk Appraisal
Originating structuring deals
Info.
Services
123
B. Swap A particular kind of forward contract
between two counterparties

• Interest-Rate Swap An agreement to exchange the
  coupon interest flows from two different
  hypothetical or notional instruments over a
  series of future settlement dates.
• Each coupon-swap agreement creates synthetically
  a financial instrument that could not otherwise
  be traded in financial markets. Aim is
• to reduce borrowing costs
• to hedge interest-rate risk, or
• to speculate (i.e., gamble) on the future
  course of interest rates.

124
Vocabulary The maturity of a swap iscalled its
tenor.

• The usage traces to tenors Latin meaning as a
  course of continuous or uninterrupted progress.
• Each subobligation has its own maturity.
  Tenor stresses that the final item is no
  different from earlier items.
• Two nonfinancial meanings of tenor Besides its
  additional meaning as a voice quality, tenor
  can also mean the subject of a metaphor.

125
Other Swaps Terminology

• Notional Value assumed face amount P used by
  contract in translating contract interest rates
  into cash flows
• Half Swaps Fixed Half vs. Variable Half
• Selling on market or at-the-market both halves
  have equal value no premium or discount
• Selling off-market instances where obligations
  imposed by the two halves of a swap are not
  equally valuable at current interest rates.
  E.g., value of the PF and RF halves might be
  substantial, when PV and RV are not.

126
As an incremental balance sheet, every
interest-rate swap has two parts a pay half
and a receive half. Each side accepts an
obligation and receives a claim to something
valuable

• -- RVPF Receive Variable Rate, Pay Fixed Rate
• -- RFPV Receive Fixed Rate, Pay Variable Rate

? Concept of a half-swap Receive
half ? an on-balance-sheet asset Pay
half ? an on-balance-sheet liability
127
Underlying notional instruments on which swaps
are based have an exact or rough cash-market
counterpart. The different notional instruments
have the same maturity and differ as to whether
the contract interest rate on the instrument is
fixed (Rt) or floating ( ).

• The party that is long the fixed-rate obligation
  is usually of higher credit standing than the
  short. This side Pays Fixed, Receives Variable
  PFRV or RVPF.
• RxPz Notation mimics the order of a balance
  sheet.
• Cash Settlement only the net cash-flow
  difference is paid on any settlement date. Why
  is this efficient?

128

• Let P be the notional principal that is used to
  calculate the interim cash flows. P need not be
  precisely the same as the principal on the
  underlying cash instruments. Rather in an OTC
  market it is a negotiated variable that can be
  used to equalize the initial market value of the
  two sides of the swap.
• At each settlement date (typically every six
  months), the floating rate is ordinarily
  reset, although the settlement and reset dates
  are in principle contracting variables. The
  reset makes the payment due at the next
  reset date knowable in advance nonstochastic.
  
• The check written for the difference at each
  settlement date is called the difference check.
  The payline equals the absolute value of ( -
  R) P.
• At some dates, the payoffs go from the fixed side
  to the floating side at other dates, funds flow
  the other way.

129
A Replication Perspective Swaps are synthetic
substitutes for financial intermediation which is
an ancient financial-engineering substitute for
direct finance.

• Opportunity Costs of undertaking a swap parallel
  the transaction costs we identified in comparing
  the costs of direct and indirect finance.
• 1) Expenses of shopping for best deal
• 2) Expenses of Due Diligence
• 3) Contracting and Enforcement Expense

130
Standard RFPV Swaps Example

• July 1, 1998
• initiate fixed-for-floating interest rate swap
  with notional principal of 1,000,000 and tenor
  of 2 years
• January 1, 1999
• RECEIVE 5 per annum fixed rate
• PAY OUT July 1, 1998 six-month LIBOR rate (4.5)
• July 1, 1999
• RECEIVE 5 per annum fixed rate
• PAY OUT six-month LIBOR rate set January 1, 1999
• January 1, 2000
• RECEIVE 5 per annum fixed rate
• PAY OUT six-month LIBOR rate set July 1, 1999
• July 1, 2000
• RECEIVE 5