FIBI

1
Fixed Income Instruments 5

• Zvi Wiener
• 02-588-3049
• mswiener_at_mscc.huji.ac.il

2
Fixed Income 5

• Mortgage loans
• Pass-through securities
• Prepayments
• Agencies
• MBS
• CMO
• ABS

3
Bonds with Embedded Options (14)

• Traditional yield analysis compares yields of
  bonds with yield of on-the-run similar
  Treasuries.
• The static spread is a measure of the spread that
  should be added to the zero curve (Treasuries) to
  get the market value of a bond.

4
Active Bond Portfolio Management (17)

• Basic steps of investment management
• Active versus passive strategies
• Market consensus
• Different types of active strategies
• Bullet, barbell and ladder strategies
• Limitations of duration and convexity
• How to use leveraging and repo market

5
Investment Management

• Setting goals, idea of ALM or benchmark
• GAAP, FAS 133, AIMR - reporting standards
• passive or active strategy - views, not
  transactions
• available indexes
• mixed strategies

6
Major risk factors

• level of interest rates
• shape of the yield curve
• changes in spreads
• changes in OAS
• performance of a specific sector/asset
• currency/linkage

7
Parallel shift
r
T
8
Twist
?r
T
9
Butterfly
?r
T
10
Yield curve strategies

• Bullet strategy Maturities of securities are
  concentrated at some point on the yield curve.
• Barbel strategy Maturities of securities are
  concentrated at two extreme maturities.
• Ladder strategy Maturities of securities are
  distributed uniformly on the yield curve.

11
Example

• bond coupon maturity yield duration convex.
• A 8.5 5 8.5 4.005 19.81 B 9.5 20 9.5 8.882
  124.17 C 9.25 10 9.25 6.434 55.45

Bullet portfolio 100 bond C Barbell portfolio
50.2 bond A, 49.8 bond B
12

• Dollar duration of barbell portfolio
• 0.5024.005 0.4988.882 6.434
• it has the same duration as bullet portfolio.
• Dollar convexity of barbell portfolio
• 0.50219.81 0.498124.17 71.78
• the convexity here is higher!
• Is this an arbitrage?

13

• The yield of the bullet portfolio is 9.25
• The yield of the barbell portfolio is 8.998
• This is the cost of convexity!

14
Leverage

• Risk is not proportional to investment!
• This can be achieved in many ways futures,
  options, repos (loans), etc.
• Duration of a levered portfolio is different form
  the average time of cashflow!
• Use of dollar duration!

15
Repo Market

• Repurachase agreement - a sale of a security with
  a commitment to buy the security back at a
  specified price at a specified date.
• Overnight repo (1 day) , term repo (longer).

16
Repo Example

• You are a dealer and you need 10M to purchase
  some security.
• Your customer has 10M in his account with no
  use. You can offer your customer to buy the
  security for you and you will repurchase the
  security from him tomorrow. Repo rate 6.5
• Then your customer will pay 9,998,195 for the
  security and you will return him 10M tomorrow.

17
Repo Example

• 9,998,195 0.065/360 1,805
• This is the profit of your customer for offering
  the loan.
• Note that there is almost no risk in the loan
  since you get a safe security in exchange.

18
Reverse Repo

• You can buy a security with an attached agreement
  to sell them back after some time at a fixed
  price.
• Repo margin - an additional collateral.
• The repo rate varies among transactions and may
  be high for some hot (special) securities.

19
Example

• You manage 1M of your client. You wish to buy
  for her account an adjustable rate passthrough
  security backed by Fannie Mae. The coupon rate is
  reset every month according to LIBOR1M 80 bp
  with a cap 9.
• A repo rate is LIBOR 10 bp and 5 margin is
  required. Then you can essentially borrow 19M
  and get 70 bp 19M.
• Is this risky?

20
Indexing

• The idea of a benchmark (liabilities, actuarial
  or artificial).
• Cellular approach, immunization, dynamic approach
• Tracking error
• Performance measurement, and attribution
• Optimization
• Risk measurement

21
Flattener
r
T
22
Example of a flattener

• sell short, say 1 year
• buy long, say 5 years
• what amounts?
• In order to be duration neutral you have to
  buy 20 of the amount sold and invest the
  proceedings into money market.
• Sell 5M, buy 1M and invest 4M into MM.

23
Use of futures to take position

• Assume that you would like to be longer then your
  benchmark.
• This means that you expect that interest rates in
  the future will move down more than predicted by
  the forward rates.
• One possible way of doing this is by taking a
  future position.
• How to do this?

24
Use of futures to take position

• Your benchmark is 3 years, your current portfolio
  has duration of 3 years as well and value of 1M.
  You would like to have duration of 3.5 years
  since your expectation regarding 3 year interest
  rates for the next 2 months are different from
  the market.
• Each future contract will allow you to buy 5
  years T-notes in 2 months for a fixed price.

25
Use of futures to take position

• Each future contract will allow you to buy 5
  years T-notes in 2 months for a fixed price.
• If you are right and the IR will go down
  (relative to forward rates) then the value of the
  bonds that you will receive will be higher then
  the price that you will have to pay and your
  portfolio will earn more than the benchmark.

26
Use of futures to take position
0 2M 3Y 5Y
-x(1r2M/6) (1r3Y)3 x(1r5Y)5

• One should chose x such that the resulting
  duration will be 3.5 years.

27
Bond Risk Management

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

28
Duration

• Macauley duration

Modified duration
Effective duration
Dollar duration
29
Fixed Income Risk

• Arises from potential movements in the level and
  volatility of bond yields.
• Factors affecting yields
• inflationary expectations
• term spread
• higher volatility of the low end of TS

30
Volatilities of IR/bond prices

• Price volatility in End 99 End 96
• Euro 30d 0.22 0.05
• Euro 180d 0.30 0.19
• Euro 360d 0.52 0.58
• Swap 2Y 1.57 1.57
• Swap 5Y 4.23 4.70
• Swap 10Y 8.47 9.82
• Zero 2Y 1.55 1.64
• Zero 5Y 4.07 4.67
• Zero 10Y 7.76 9.31
• Zero 30Y 20.75 23.53

31
Duration approximation

• What duration makes bond as volatile as FX?
• What duration makes bond as volatile as stocks?
• A 10 year bond has yearly price volatility of 8
  which is similar to major FX.
• 30-year bonds have volatility similar to equities
  (20).

32
Models of IR

• Normal model ?(?y) is normally distributed.
• Lognormal model ?(?y/y) is normally distributed.
• Note that

33
Principal component analysis

• level risk factor 94 of changes
• slope risk factor (twist) 4 of changes
• curvature (bend or butterfly)
• See book by Golub and Tilman.

34
Forwards and Futures

• The forward or futures price on a stock.
• e-rt the present value in the base currency.
• e-yt the cost of carry (dividend rate).
• For a discrete dividend (individual stock) we can
  write the right hand side as St- D, where D is
  the PV of the dividend.

35
Hedging Linear Risk

• Following Jorion 2001, Chapter 14
• Financial Risk Manager Handbook

36
Hedging

• Taking positions that lower the risk profile of
  the portfolio.
• Static hedging
• Dynamic hedging

37
Unit Hedging with Currencies

• A US exporter will receive Y125M in 7 months.
• The perfect hedge is to enter a 7-months forward
  contract.
• Such a contract is OTC and illiquid.
• Instead one can use traded futures.
• CME lists yen contract with face value Y12.5M and
  9 months to maturity.
• Sell 10 contracts and revert in 7 months.

38

• Market data 0 7m PL
• time to maturity 9 2
• US interest rate 6 6
• Yen interest rate 5 2
• Spot Y/ 125.00 150.00
• Futures Y/ 124.07 149.00

39

• Stacked hedge - to use a longer horizon and to
  revert the position at maturity.
• Strip hedge - rolling over short hedge.

40
Basis Risk

• Basis risk arises when the characteristics of the
  futures contract differ from those of the
  underlying.
• For example quality of agricultural product,
  types of oil, Cheapest to Deliver bond, etc.
• Basis Spot - Future

41
Cross hedging

• Hedging with a correlated (but different) asset.
• In order to hedge an exposure to Norwegian Krone
  one can use Euro futures.
• Hedging a portfolio of stocks with index future.

42
The Optimal Hedge Ratio

• ?S - change in value of the inventory
• ?F - change in value of the one futures
• N - number of futures you buy/sell

43
The Optimal Hedge Ratio
Minimum variance hedge ratio
44
Hedge Ratio as Regression Coefficient

• The optimal amount can also be derived as the
  slope coefficient of a regression ?s/s on ?f/f

45
Optimal Hedge

• One can measure the quality of the optimal hedge
  ratio in terms of the amount by which we have
  decreased the variance of the original portfolio.

If R is low the hedge is not effective!
46
Optimal Hedge

• At the optimum the variance is

47
Example

• Airline company needs to purchase 10,000 tons of
  jet fuel in 3 months. One can use heating oil
  futures traded on NYMEX. Notional for each
  contract is 42,000 gallons. We need to check
  whether this hedge can be efficient.

48
Example

• Spot price of jet fuel 277/ton.
• Futures price of heating oil 0.6903/gallon.
• The standard deviation of jet fuel price rate of
  changes over 3 months is 21.17, that of futures
  18.59, and the correlation is 0.8243.

49
Compute

• The notional and standard deviation f the
  unhedged fuel cost in .
• The optimal number of futures contracts to
  buy/sell, rounded to the closest integer.
• The standard deviation of the hedged fuel cost
  in dollars.

50
Solution

• The notional is Qs2,770,000, the SD in is
• ?(?s/s)sQs0.2117?277 ?10,000 586,409
• the SD of one futures contract is
• ?(?f/f)fQf0.1859?0.6903?42,000 5,390
• with a futures notional
• fQf 0.6903?42,000 28,993.

51
Solution

• The cash position corresponds to a liability
  (payment), hence we have to buy futures as a
  protection.
• ?sf 0.8243 ? 0.2117/0.1859 0.9387
• ?sf 0.8243 ? 0.2117 ? 0.1859 0.03244
• The optimal hedge ratio is
• N ?sf Qs?s/Qf?f 89.7, or 90 contracts.

52
Solution

• ?2unhedged (586,409)2 343,875,515,281
• - ?2SF/ ?2F -(2,605,268,452/5,390)2
• ?hedged 331,997
• The hedge has reduced the SD from 586,409 to
  331,997.
• R2 67.95 ( 0.82432)

53
Duration Hedging
54
Duration Hedging
If we have a target duration DV we can get it by
using
55
Example 1

• A portfolio manager has a bond portfolio worth
  10M with a modified duration of 6.8 years, to be
  hedged for 3 months. The current futures prices
  is 93-02, with a notional of 100,000. We assume
  that the duration can be measured by CTD, which
  is 9.2 years.
• Compute
• a. The notional of the futures contract
• b.The number of contracts to by/sell for optimal
  protection.

56
Example 1

• The notional is
• (932/32)/100?100,000 93,062.5
• The optimal number to sell is

Note that DVBP of the futures is
9.2?93,062?0.0185
57
Example 2

• On February 2, a corporate treasurer wants to
  hedge a July 17 issue of 5M of CP with a
  maturity of 180 days, leading to anticipated
  proceeds of 4.52M. The September Eurodollar
  futures trades at 92, and has a notional amount
  of 1M.
• Compute
• a. The current dollar value of the futures
  contract.
• b. The number of futures to buy/sell for optimal
  hedge.

58
Example 2

• The current dollar value is given by
• 10,000?(100-0.25(100-92)) 980,000
• Note that duration of futures is 3 months, since
  this contract refers to 3-month LIBOR.

59
Example 2

• If Rates increase, the cost of borrowing will be
  higher. We need to offset this by a gain, or a
  short position in the futures. The optimal
  number of contracts is

Note that DVBP of the futures is
0.25?1,000,000?0.0125
60
Beta Hedging

• ? represents the systematic risk, ? - the
  intercept (not a source of risk) and ? - residual.

A stock index futures contract
61
Beta Hedging
The optimal N is
The optimal hedge with a stock index futures is
given by beta of the cash position times its
value divided by the notional of the futures
contract.
62
Example

• A portfolio manager holds a stock portfolio worth
  10M, with a beta of 1.5 relative to SP500. The
  current SP index futures price is 1400, with a
  multiplier of 250.
• Compute
• a. The notional of the futures contract
• b. The optimal number of contracts for hedge.

63
Example

• The notional of the futures contract is
• 250?1,400 350,000
• The optimal number of contracts for hedge is

The quality of the hedge will depend on the size
of the residual risk in the portfolio.
64

• A typical US stock has correlation of 50 with
  SP.
• Using the regression effectiveness we find that
  the volatility of the hedged portfolio is still
  about
• (1-0.52)0.5 87 of the unhedged volatility for
  a typical stock.
• If we wish to hedge an industry index with SP
  futures, the correlation is about 75 and the
  unhedged volatility is 66 of its original level.
• The lower number shows that stock market hedging
  is more effective for diversified portfolios.

65
FRM-GARP type question

• Zvi Wiener

66
FRM-GARP 9818

• A portfolio consists of two positions One
  position is long 100M of a two year bond priced
  at 101 with duration of 1.7, the other position
  is short 50M of a five year bond priced at 99
  with a duration of 4.1. What is the duration of
  the portfolio?

67
FRM-GARP 9818

• The dollar duration is sum of dollar durations,
  so
• 100M 101/100 1.7 171.7M
• -50M 99/100 4.1 -202.95M
• total dollar duration is -31.25M, portfolios
  value is 50M, thus its duration is -0.61.

68
Cap and Floor

• Cap MaxiT-iC, 0
• Floor MaxiF-iT, 0
• What is Long Cap and Short Floor position?
• Cap - Floor
• MaxiT-iC, 0 - MaxiC-iT, 0 iT-iC
• pay fixed swap

69
FRM-GARP 9850

• A hedge fund leverages its 100M of investor
  capital by a factor of 3 and invests it into a
  portfolio of junk bonds yielding 14. If its
  borrowing costs are 8, what is the yield on
  investor capital?

70
FRM-GARP 9850

• 300M invested at 14 yield 42M, borrowing costs
  are 200 at 8 or 16M, the difference of 26M
  provides 26 yield on equity of 100M.

71
FRM-GARP 9851

• A portfolio consists of two long assets 100
  each. The probability of default over the next
  year is 10 for the first asset, 20 for the
  second asset, and the joint probability of
  default is 3. What is the expected loss on this
  portfolio due to credit risk over the next year
  assuming 40 recovery rate for both assets.

72
FRM-GARP 9851

• 0.1?(1-0.2) - default probability of A
• 0.2?(1-0.1) - default probability of B
• 0.03 - default probability of both
• Expected losses are
• 0.1?(1-0.2)?100?(1-0.4)
• 0.2?(1-0.1)?100?(1-0.4)
• 0.03?200?(1-0.4)
• 4.8 10.8 3.6 19.2M

73
Example

• Assume a 1-year US Treasury yield is 5.5 and a
  Eurodollar deposit rate is 6. What is the
  probability of the Eurodollar deposit to default
  (assuming zero recovery rate)?

74
FRM-GARP 9724

• Assume the 1-year US Treasury yield is 5.5 and a
  default probability of a one year Commercial
  Paper is 1. What should be the yield on the CP
  assuming 50 recovery ratio?

75
FRM-GARP 0047

• Which one of the following deals has the largest
  credit exposure for a 1,000,000 deal size.
  Assume that the counterparty in each deal is a
  AAA-rated bank and there is no settlement risk.
• A. Pay fixed in an interest rate swap for 1 year
• B. Sell USD against DEM in a 1 year forward
  contract.
• C. Sell a 1-year DEM Cap
• D. Purchase a 1-year Certificate of Deposit

76
FRM-GARP 0047

• Which one of the following deals has the largest
  credit exposure for a 1,000,000 deal size.
  Assume that the counterparty in each deal is a
  AAA-rated bank and there is no settlement risk.
• A. Pay fixed in an interest rate swap for 1 year
• B. Sell USD against DEM in a 1 year forward
  contract.
• C. Sell a 1-year DEM Cap
• D. Purchase a 1-year Certificate of Deposit

77
FRM-GARP 98

• A step-up coupon bond pays LIBOR for 2 years,
  2?LIBOR for the next two years and 3?LIBOR for
  the last two years. The principal amount is paid
  at the end of year 6. Prices of zero coupon bonds
  maturing in 2, 4, and 6 years are Z2, Z4, Z6.
  What is the price of the step-up bond?

78
FRM-GARP 98

• 0 1 2 3 4 5 6
• ? L L 2L 2L 3L 3L100

79

• 0 1 2 3 4 5 6
• ? L L 2L 2L 3L 3L100
• 100 L L L L L L100
• 300 3L 3L 3L 3L 3L 3L300
• 100 L L L L100
• 100 L L100
• Z2 100
• Z4 100
• 2Z6 200
• ? 300 - 100 - 100 Z2 Z4- 2Z6