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Title: Weeks 048


1
Weeks 04-89Using Market-Value Accounting for
Changes in the PDV of the Banking Book to Manage
Interest-Volatility Risk
  • This segment uses Financial Engineering concepts
    to dig changes in economic value out of an FSFs
    banking book to show how to manage that
    value.
  • Tools consist of technical financial-valuation,
    risk-assessment skills, and hedging skills.

2
Metaphor Financial Company is a Portfolio of
Investment and Funding Vehicles
  • With tangible long positions in, e.g.,
  • Loans
  • Marketable Securities
  • Real Estate and Equipment
  • With tangible short positions in
  • Deposit-like accounts
  • Debt
  • Explicit commitments
  • With numerous harder-to-monitor intangible
    positions

3
Risk Comes from Opportunities to Lose Money from
Various Kinds of Adversity
  • Unexpected market moves.
  • Model risk a source of hedging errors.
  • Insufficient management oversight.
  • Carrying too much risk relative to capital.
  • Internal external fraud
  • Counterparty lawsuits.
  • Unsupportable Debts.

4
Bottom-Line Target Limiting Exposure of Equity
Capital to Fluctuations in Value
  • Risk management focuses on
  • Net Worth that Portfolio choices may Generate
  • the total volatility of Portfolio Return
  • understanding the risk exposure profile
  • Exposure to risk may be reduced
  • first and foremost, by asset diversification
  • by adjusting liability structure
  • by directly trading away risks via forward
    other derivative contracts

5
One Way to Partition FSF Risk Exposures
  • Credit Risks
  • loan and bond defaults
  • counterparty nonperformance in other contracts
  • Market Risks ( price volatilities and
    interrelated customer responses)
  • interest rates
  • foreign exchange
  • economic conditions
  • Strong Links between Market and Credit Risk
  • Operational Risk personnel systems flubs
    scams
  • Reputational and Regulatory Risk


6
Transition Weeks 3 through 7 focused on how to
identify and mitigate concentrations of default
risk that the economic structure of an FSFs
core customer base invite into its portfolio.
  • The next few weeks focus on
  • (1) how to measure concentrations of
    interest-rate risk (IRR),
  • (2) how to mitigate them, while recognizing that
    relationship customers invite IRR into an FSFs
    portfolio.

7
Review Strategies for Managing Credit Risk
  • Install systems that Measure Exposures (?RADAR)
  • Identify whether and how different exposures a
    deal creates can be
  • 1. Avoided or eliminated
  • 2. Transferred to another party
  • 3. Actively priced and kept within appropriate
    limits self-insured by Explicit or Implicit
    Reserve Accounts
  • Goal of Risk Mgr is to choose a profitable and
    comfortable combination of strategies

8
ASSUMPTIONS (1) investment vehicle of tangible
loans perpetuities (i.e., consols that have
no principal payment, but pay the same annual
coupon forever) and (2) funding vehicle
deposits that mature in one period, so that their
interest cost is reset at each renewal date.
IRR RISK-MANAGEMENT FOR A SIMPLE BUSINESS-PLAN
FSF
  • Here, assets are longer in maturity than
    liabilities. Exam questions can present balance
    sheets and income statements for FSFs where
    assets may be longer or shorter than their
    liabilities.

9
HYPOTHETICAL AND SIMPLIFIED
Tangible Balance Sheet
FSF
Possible Extensions in Accounting Tools BVA vs.
MVA mark-to-model value vs. strict
market-to-market on-balance-sheet vs. OBS
positions tangible vs. intangible assets
10
Initial Opportunity-Cost Balance Sheet
11
Accounting NW vs. 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.
  • Suppose a bank holding company has 50,000 shares
    outstanding and its current market price is 30
    per share S(30)50,000 1.5 million.

12
Ignoring Stock-Market errors in valuation (V), a
banks market cap S equals the sum of MVTNW
plus unobservable components values of
enterprise-contributed NW and government-contribut
ed NW.
S Estimates MVTNW EI FG
  • This means MV of TNW EI FG?
  • ANS S-V.
  • In this example, if V0, S 10EIFG.

13
Marking to Model In loan bond markets, four
potential "observables" exist the instruments
maturity, price Pt PDV, coupon rate C, and
promised principal payment F.
  • Given values for n, F, and C, the PDV model
    solves for R, the yield to maturity that makes Pt
    PDV.
  • We overcome the nonobservability of the price of
    assets that dont actively trade by applying a
    reference rate R that is observable in a
    comparable market. This generates a marked to
    model or fair value Pfair.

14
Selecting and Logically Justifying A Comparable
Market for Each Position and an Appropriate
Reference Rate is an Art
Target ROE at most banks today is 15 to 20
15
IllustrationThe Misplaced Trust Co.
  • On the day it opens, borrowers convince the bank
    to make 100 million in three-year loans, while
    all depositors are asked to put their money in
    one-year certificate accounts.
  • Principal and interest on all loans is compounded
    at 8 percent and due at maturity.
  • Implicit deposit interest is 1.5 and along with
    explicit interest is assumed to be paid at the
    end of each year.
  • The bank holds no other assets and funds itself
    with 80 million in deposits issued at a 5
    percent explicit yield, obtaining the rest of its
    financing from stockholders.

16
Exercise
  • A. Neglecting LLRs, find the banks tangible
    balance sheet and construct (I.e., accrue) the
    banks GAAP income statement on these positions
    for the next year.
  • B. Can we predict its economic profits?
  • C. Warning Calculations presume unrealistically
    that the banks interest-rate risk and default
    risk are not correlated.

17
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.

18
Answer Key
  • A. Tangible Balance Sheet
  • Loans 100 Dep. 80
  • NW 20
  • Accrual Income Statement for the First Year (in
    mil.)
  • GAAP Net Income is Loan Revenue Minus Total
    Interest Expense
  • 8 (.05 .015) 80 8 - 5.2 2.8
  • First-year ROE 2.8/20 14

19
Answer Key (cont.)
  • B. The accrual assumption that is used to impute
    accounting income on the yet-to-be-repaid
    three-year loans is tenuous economically.
    Opportunity-cost income would differ by any
    change in loan value that might occur over the
    year. Economic income would be reduced if
    interest rates rise the projected default rate
    rises or the salvage rate recoverable on default
    declined.
  • C. If rates rise because the economy is strong,
    the present value of the contractual cash flows
    would fall. However, estimates of default
    probability (p) might also fall and the projected
    recovery rate (s) on defaulted loans might
    improve. Contrariwise, lower interest rates
    brought about by a weak economy promise to raise
    balance-sheet values but hurt p and s.

20
To test your understanding of MVA, exam questions
would pose alternate assumptions about the banks
investment and funding vehicles. Examples of
different assumptions
  • EI might be 5 rather than zero.
  • Deposits or Assets might be 2-period or 3-period
    instruments.
  • The Deposit Rate might be set by a different
    formula.

21
Reconciling Fair Value with BVA
  • 1. If changes in market interest rates can
    impair the values of loans and debt
    instruments, GAAP rules ought to reserve for this
    danger and charge off the value of impairments
    that occur.
  • 2. Conscientious outside analysts must impute
    these reserves and market-induced revaluations of
    bank positions because few banks fully report
    them.
  • 3. Imputations 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.

22
Summary opportunity-cost accounting is another
name for Fair-Value and market-value accounting
(MVA).
  • Market-value accountants enter 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.

23
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.

24
HOW RISK CAN BE MANAGED
25
Risk Management Seeks to Limit Possible
Fluctuations in Economic Profits or 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.

26
IRR for A simple business-plan intermediary
  • 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.

27
Just like Credit Risk, Interest-Volatility Risk
(i.e., revaluation risk) can be insured, traded
away, or hedged.
Reminder A financial hedge seeks to erect
counterbalanced positions to keep particular
classes of risks out of one's portfolio and to
control against "wealth erosion" from the
selected sources.

28
  • 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.

29
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.

30
  • Chapter 11 of the Text introduces the crude
    low-tech idea of tabling the time-to-repricing
    of the assets and liabilities on an FSFs
    banking book across of series of repricing
    buckets to calculate the FSFs distribution of
    repricing gaps
  • What bias exists in letting distant () gaps
    offset shorter (-) gaps?

31
(No Transcript)
32
The Text also Explains how an ALCO might read and
respond to the gaps or mismatches revealed by
this table.
  • Stress-Scenario Analysis Suppose all interest
    rates move permanently up by X . How big a
    reduction will the FSF experience in accounting
    earnings on its existing positions over the next
    4 quarters and the following two years?

33
FSF profits and NW may be 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. Usefulness of
T-accounts let us slog through numerical
examples of how interest volatility affects each
of Economic Income and Net Worth
34
Does Bucket Analysis or GAAP income statement
record ?MVTNW? No. Each records only changes
in selected revenues and expenses
Revenue (Deposit Interest Cost) GAAP Profit
35
However, Economic-Risk Managers Internally
Reprice the Banking Book Immediately
  • a. Value of any fixed income stream may be
    expressed as a polynomial equation in the single
    variable x and written in standard implicit
    form
  • b. The present discounted value (PDV) of 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.
  • c. From an algebraic perspective, balance sheets
    treat liabilities as assets that carry a negative
    sign.

36
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 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 the
    downs as well as the ups of return
    fluctuations. What soars high in the air cant
    stay there forever.
  • 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.

37
Key Slide
  • An institutions date-to-date ?NW plus its
    date-to-date cash flow constitutes its
    date-to-date opportunity-cost total return.
  • Interest volatility impacts an FSFs income and
    net worth unless the effect of interest-rate
    swings on the prices of an FSFs assets happens
    to offset the effect on its liabilities.
  • To establish Interest-rate insensitivity requires
    the careful planning and execution of a
    offset-creating risk-management strategy.

38
Risk Officers Must Recognize that PDV also Tells
us How to Measure Interest Sensitivity of Fair
Values
  • Calculate the change in PDV that occurs when R
    goes from R0 to R0?R
  • PDV 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

39
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 or
    outward relative to line through the origin with
    the slope given by the first derivative.

40
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 the elasticity
of V with respect to (1R)D treats the
percentage change in the compound-interest factor
(1R) as the initiating force. With discrete
compounding, (1R) is the rate of value
accretion over time.
41
If a perpetuitys R rises from .10 to .125
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?

42
Financial Engineering leads us to view an FSFs
NW as a weighted average of the firms nonzero
balance-sheet positions, with -of-NW portfolio
shares as weights. Similarly, the IRR of net
worth (or indeed any balance-sheet position) is a
weighted average of the IRR of the positions
or - component instruments.
Summary Insight
43
Valuable Insight For Value Creation and Risk
Mangement
  • This perspective portrays an FSFs NW as a
    synthetic or derivative contract written on the
    sum of the numerous value differences
    (interest, expenses, fees, and price changes)
    that generate a portfolios net cash flows.

44
  • This 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. (Parallels to bookies and used-car
    dealers)

45
Concept Interest-Rate Risk (IRR) for a
financial institution is best understood as a
market revaluation risk. When standard
historical-cost book-value accounting (BVA) is
used, interest-rate changes have no immediate
effect on accounting NW, but impacts on BVA net
worth flow through to affect accounting profits
after a time lag.
46
Flowing projected income through a Balance Sheet
is a skill we worked on in class one.
  • When R changes, historical-cost accounting
    implicitly smoothes the data. It amortizes
    into an institutions future earnings the
    immediate change in the PDV or opportunity-cost
    value of the asset.
  • Amortization raises or lowers the net realized
    rate of return that accrues on the unchanged book
    value of the asset relative to the opportunity
    cost that the market offers on new funds.

47
New Unit Measurement
  • One has to measure something before one can
    truly be able to manage it.
  • Duration (D) is a Benchmark measure that can
    Establish Accountability for Managing
    Interest-Rate Risk
  • In basic finance course, D is applied only to
    individual instruments

48
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.

49
Maturity Buckets Only Provide a Crude Measure
of IRR
  • Management needs to make IRR measurements not
    just for individual instruments, but for
    portfolio positions.
  • Duration measures can be built up from synthetic
    instruments whose durations are straightforward
    to compute.

50
Vocabulary
  • 1. "Futurity" expresses the distance in time
    until a payment is received.
  • 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

51
Let me remind you that 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. 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.
52
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.
53
WL corresponds to the buckets approach. It
weights the futurity of all cash flows by their
percentage of the total of undiscounted cash
flows the contract promises.
  • Let vt of an instruments total cash flows
    scheduled for date t.

The variable t is averaged vt are the weights.
54
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.

55
Macaulay's concept of Duration (D) resembles WL
more closely than maturity.
INTUITION OF D
  • 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.

56
Questions to Test Your Understanding
  • 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?
  • How could an FSF benefit from pooling same-dated
    coupons into a derivative security?

57
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 is the PDV (i.e.,
    equilibrium price) of the contract.

58
Duration Falls with Yield to Maturity for
Maturity and Coupon Rate Fixed
The numerical relation between ?D and ? n varies
with coupon, maturity, and R, because D varies
with these variables.
Do WL and maturity also change with yield?
59
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. Are there handy formulas for
    these cases?
  • -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?

60
Three Intuition-Building Interpretations of D
  • D represents a price-weighted measure of the
    futurity of a stream of projected cash flows. The
    weight given each futurity is the percentage of
    the price 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 bond has the same futurity as a
    single-payment bond whose maturity is D.

61
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 bond, which
    need not par.
  • We can reinterpret the bond 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.

62
Here is a model WORKSHEET
63
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 10 percent? Lower.
    Why? Distant flows have less proportionate value.
  • c. What if the coupon rate 0? ANS.
    DWLmaturity.
  • stripping concept. Synthetic repackaging and
    assembly CAN create zero-coupon securities

64
Question No. 2 Weighted average life, WL.
WORKSHEET
  • Compared to D, WL increases the relative weights
    of distant receipts.

65
Duration, if only it were so simple.
66
WE CAN ONLY APPROXIMATE PERCENTAGE BOND-PRICE
CHANGES WITH DURATION
True Relationship (is convex to origin)
67
  • 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.

68
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 calculate a
    value for CX 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.

69
Review Transition
  • 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 mathematical operations.

70
  • The duration of any set of fixed periodic 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 .

71
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.
New Unit 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.

72
a.Why would the duration of a 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
    n10.
  • a. Duration of coupon stream is
  • b. Duration of return of principal F raises the
    duration of the position

73
The textbook contrasts Duration Management with
this less high-tech approach. Buckets-based
strategies partition the time line of future
business. It treats contract timing segments
in which net cash flows from repayments of (A-L)
are scheduled as metaphorical buckets in which
inflows of liquidity may accumulate.
  • Idea is for managers to calculate the algebraic
    sign and amount of scheduled net profit flows
    across a parallel grid of maturity classes for
    fixed-rate instruments and a parallel grid of
    time-to-repricing segments for adjustable-rate
    instruments.

74
Why must ? ANS. ?NW
replicates the Difference Rule
  • 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? Such immunization is not necessarily a
    desirable target for an FSF because giving up
    risk usually implies giving up some return.

75
As long as component portfolio positions are
PDV-weighted properly, Durations add across
portfolio positions to get 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 7 and 8
    exercises)

76
You Guys Better Know What the Duration of Our Net
Worth is
77
D is easier to remember as the absolute value of
an elasticity. An elasticity is a double-log
derivative. It 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.
78
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

79
Two Ways to Approximate when
where
80
  • 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 pvpb and D as a
    characteristic of individual bonds

81
  • pvbp formula is the ?P formula with ? R set
    .0001.
  • Why is the duration of an FSFs
    stockholder-contributed net worth deserving of
    managerial attention? In what sense does DN 0
    mean no net interest-risk exposure?
  • What do the formulas tell us about the hedging
    effectiveness of duration matching? Two
    intuitive exercises, Interpret DN 1. Next
    suppose DA DL 1 and A/N 20. Is DN 0?
    No. Why not? Leverage
  • How many months longer than DA 1 would DL have
    to be to make DN 0? Assuming rA r L, let us
    solve and see 0 20 - (1x) 19. 1/19x.

82
Simplest illustration of DN is to look at the IRR
of an all-equity simple business-plan 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 already solved this problem in 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 the terms of the deposit
    contracts shift some IRR to depositors.

83
Suppose deposits were par floaters whose 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?
    DA. The effect of leverage is neutralized.

84
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.

85
Algebra of Synthetic Replication
New Topic Self-Study Calculation Aids ( slides
85 to 102)
  • 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.

86
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 ?
87
Stripping a Two-Period Bond
88
A structured derivative is a tradeable claim that
can be extracted from elements imbedded in a
standard financial contract.
  • Financial engineering can extract or strip
    three derivative instruments from a 2-period
    coupon bond F, C1, C2. 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 2-year unit annuity plus a 2-year
    zero-coupon bond of principal F.
  • Synthesizing debtor vs creditor positions?

89
Finding Fair Value of n-Period Assets with a
Fixed Annual Cash Flow
  • Knowing how to find the value of a unit annuity
    paying one dollar per year for n years (V1,n) is
    a useful building block. V1,n depends only on
    the current interest rate, R and the endpoint of
    the stream, date 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.

90
The sum of 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
91
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 of Perpetuitys Cash Flows After n
92
  • 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 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.

93
Famous Formula a embodies three course
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)

94
By the Law of One Price, all replicating
portfolios have the same PDV.
  • Lets show this by brute force for n1
  • Using formula a, V1,? for R.10 is

(the direct-discounting value
.
95
The value (V1,?) of a unit perpetuity is exposed
to interest-rate risk. Value changes as interest
rates rise and fall.
  • Suppose R .10. A unit perpetuity would be
    worth
  • Suppose R rises to .125? V1,? 1/R?
  • Suppose R falls to .08 ? V1,? ?

10.
8
12
96
Suppose future values of R are unpredictable
(i.e., risky or volatile), but bounded by
R12.5 and R81/3. (N.B. midpoint of interval
is not 10)
  • Suppose all values of R were equally likely
    between .083 and .125? E(R) .0831/2(.042)

  • .083 .021 10.4

97
Calculate for yourselves how Fluctuations in R
change the value of a 2-year unit annuity by
calculating V1,2 for three benchmark values of R
  • 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
98
-- 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).

99
  • The value of the underlying asset and the forward
    liability move simultaneously but in an opposite
    direction. The forward sale trades away much of
    the assets IRR to another party.
  • 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.

100
Important Financial-Engineering Perspectives on
Replication
  • One can treat formula a as describing a contract
    or deal that establishes a synthetic balance
    sheet and proceed to calculate the contracts net
    worth. The formula tells us that
    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. The previous
    slide explains this by interpreting ?V1,2
    component by component.
  • 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.

101
  • 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 the yield on a particular
    n-period coupon bond.
  • However, when appropriate, R could be the yield
    on a mortgage or zero-coupon instrument.

102
Fair value of any bond is algebraically 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 equation.
  • 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.

103
IRR APPLICATIONS
  • Envisioning IRR Management as an Exercise in
    Financial Engineering (Review and Exercises)

104
  • Changes in interest rates change the PDV of
    individual balance-sheet positions. If interest
    changes 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.
  • 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.

105
  • 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.

106
Review Rule for Taking Derivativesof a PDV
  • We note that the representative term in any
    polynomial has the form akxk. The first
    derivative is
  • This formula is simpler than it looks
  • For n 2,


107
Five-Part Financial Engineering Question Using
Perpetuities
  • 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
  • c. Given that the market rate r is the contract
    rate, the synthetic forward contract is neither
    in nor out of the money today.
  • d. PV of a combined long and short position in
    perpetuity

108
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 finances 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.
109
Answer to Problem a
Assuming 1 on fresh positions, the
governing formula expresses DN as a weighted
difference a. In this case,
years. What is DL? 11
years?
110
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.
111
  • Answer to 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.

112
c. What would DN be if 290 million of the 8
one-year uninsured deposits had been issued
instead? Why would DN increase with the extent
of deposit funding?
113
Answer to c MVTNW 300 - 290 10. What is DL
now? 301 years DN increased because
the small increase in the amount of outstanding
deposits is greatly outweighed by the expanded
extent to which stockholders investment is being
leveraged.
114
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?
115
  • Answer to d
  • Given the banks high leverage, insurance is
    needed to make low rL credible.
  • DN 330 - 29(13.5)
  • 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.

116
2. Same approach works for stock-market
capitalization. Let us project 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 going-concern
value EI 10. Find the value of
government-contributed net worth, FG.
117
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.
118
3. This question applies 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?
119
b. 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 ?
120
c. 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
121
d. 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

122
4. Let us debate the issue raised in the
following claims a. A bank cannot protect the
market value of its net worth and expect to show
earnings, too. b. The hypothetical liquidation
value of a financial institution's tangible
assets is irrelevant as long as it remains a
going concern. c. In purchasing or analyzing the
value of a going financial-services concern, one
looks at projections of its future earnings, not
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. 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. Nor is the real estate being sold in
the current market.
123
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124
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125
Banc One case frames IRR control in Language that
Embodies two alternative perspectives on what
to target in an IRR management program.
Next Section of Slideshow is Self-Study for the
Banc One Case
  • Earnings and net-worth perspectives differ in how
    to measure and therefore how to control
    mismatching in the durations of the asset and
    liability sides of an FSFs balance sheet.
  • Income Statement Perspective Looks at the Rate
    Sensitivity of the Net Income an FSF receives
    from the positions (or subportfolios) on its
    economic Balance-Sheet.
  • Balance-Sheet Perspective Looks at the Gap or
    Mismatch inherent in subportfolios that imply an
    uneven or imbalanced balance sheet.

126
Measuring the Rate Sensitivity of Income
  • Numerically, the "rate sensitivity" of income on
    each side of the balance sheet is opposite to its
    price sensitivity.
  • Rate sensitivity concerns itself with an
    application of duration to flows of accounting
    income. The less price-sensitive (i.e.,
    shorter-duration) side of the balance sheet is
    less volatile in PDV precisely because the cash
    flows from the position are more
    interest-sensitive.

Measuring Balance-Sheet Mismatch Looks at
whether and how much an FSFs assets or
liabilities are longer in duration.
127
Linguistic Reconciliation of the two perspectives
  • For earnings to be liability sensitive, net
    income flows must be more sensitive to movements
    in market yields on liabilities than returns on
    assets are. This means an FSFs balance sheet
    must be longer in duration on the asset side than
    on the liability side.
  • For earnings to be asset sensitive requires the
    Reverse condition To make net income more
    sensitive to movements in market yields on assets
    requires DA lt DL.

128
Exercise Please Fill in the Following Table
129
Short-Funding implies DA gt wDL.
  • Typical of the average SL.
  • Short-funding allows substantial profits to be
    recorded if the yield curve is positively sloped
    and slope is relatively constant.
  • Lack of symmetry in FSF exposure to interest-rate
    risk comes from imbedded options that customers
    enjoy and must be expected to exercise adversely
    to FSF profits.
  • Catch-22 for thrifts
  • mortgage prepayments and refinancings truncate
    FSF capital gains when interest rates fall.
  • homeowners resort to seller financing of home
    sales that slow down the speed of prepayment from
    its normal value when fresh rates rise
    substantially above contract yields.
  • Lower mortgage rates in 1998 and early 1999 led
    to massive prepayments and refinancings. A
    became dramatically shorter and refinanced assets
    became effectively longer.

130
Review 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?"

131
Review 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.

132
  • 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.

133
Hedging analysis begins with an identification of
the specific risks a manager wants to scale back
(the hedgeable item) and estimates of the costs
and benefits of alternative strategies for
reducing this risk.
Hedging Strategy
134
IRR Management Whatever is explicitly targeted,
programs to manage the interest sensitivity must
control or offset mismatching of durations across
the asset and liability side of the balance
sheet.
  • Four principal Targets Whose Duration could be
    Managed on an unweighted vs. weighted basis.
  • 1) BVA net worth
  • 2) MVA net worth (? market capitalization S)
  • 3) accounting income
  • 4) net economic income.
  • Each of the four targets has a gap to be
    controlled
  • Sometimes immunization targets are stated on a
    percentage basis NW/A Income/A.

135
Estimates of Sensitivity of Returns on an
Institutions Stock to interest rates and other
potentially relevant risk factors can be
expressed as slope coefficients in a Market
Model of Firms Stock return.
The Stock Market as an Accounting Mechanism
  • Models of Stock-Market Assessment of FSF
    Riskiness
  • 1. Market Model for Return on Stock of Firm j at
    Time t Rjt aj bjRMt ujt, j 1, 2, ...,
    N.
  • 2. Two-Index Model of Same Return Rjt aj
    bjRMt cjRIt vjt.

136
  • Why might slope estimates of stock market
    sensitivity be useful to a manager? Two Reasons
  • First, the hypothesis that markets are
    informationally efficient implies that the
    stock price will estimate and respond to net
    effects of managers efforts to hedge exposures
    to particular classes of risk. Provides a check
    on Internal hedging models used by a banks
    management. Managers may focus too narrowly on
    controlling imperfect model-based measures of
    default or interest-rate risk. (Banc One Model
    errors in 1993-1994).
  • Second, a strong ethical case can be made that
    managers should concern themselves with the
    long-run value of an institutions stock price.
    Calculating the fair value of stockholder's
    bottom line provides a way to integrate measures
    of the firms accounting performance and earnings
    risk exposure.

137
Stock-market fluctuations indicate that in U.S.
most FSF stock prices are in fact sensitive to
interest rates, business-cycle developments, and
news about management character.
  • Ceteris paribus, most FSFs S rises when interest
    rates fall and falls when interest rates rise or
    threaten to rise. In late-1999 and early 2000,
    large-bank stocks responded strongly and
    inversely to hints that Fed policies would raise
    interest rates.
  • Subsequently, evidence of macro slowdown
    sometimes muted this effect and sometimes added
    increases in nonperforming loans into the mix.
  • In July and October 2002, JPM and Citi stocks
    fell sharply as their role in the Enron and
    WorldCom accounting scandals was more fully
    revealed.

138
JPM Stock Price(through June 25, 2003)
139
Extensions of Classroom Illustration Lets
calculate the simple business-plan banks
realized economic (i.e., opportunity-cost) income
on its first-day deals. Over any period, revenue
has two parts ?L and accrued interest.
Rest of Show is Self-Study
140
Suppose no changes in interest rates occur during
year one.
  • At origination, Lt
  • Lt1
  • Query Why did the exponent in the numerator
    remain unchanged, while the exponent in the
    denominator fell by one? ANS. Promised payment at
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