Bond Maths

Bond Mathematics

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Course coverage….

Day 1 Day 2Session 1 Session 1

Types of Treasury MarketsMoney MarketsGovt. Securities Markets

Valuation of floating rate bonds and bonds with embedded options

Types of risks Session 2 Session 2

Macro-economic analysis for treasury markets

Interest rate risk measurement

Session 3 Session 3Bond Mathematics Bond portfolio Planning and

Management Strategies Session 4 Session 4

Bond Mathematics continued…. Bond portfolio Planning and Management Strategies continued….

Session 1

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Session 1 covers….

•• Domestic and Forex Treasury Domestic and Forex Treasury

•• Money Market Money Market – Call Money Market, CBLO– T Bills, CPs and CDs – Liquidity Adjustment Facility– Fund Flow in the system – CRR and SLR– Interbank REPOs

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Session 1 covers….

•• Government Securities MarketGovernment Securities Market– Size & Products – Various types of G Secs. issued– Auction Mechanism & Role of Primary Dealers– SLR Securities– Trading & Settlement System

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Domestic and Forex Treasury

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Role and functions of treasury

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Financial System –Money and Capital Markets

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Financial System

Capital Market - Debt

- Equity

Money Market

Money & Capital market

Short term

Long term

Money Market

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Call/Notice Money Markets

• Market where funds are lent / borrowed for short term (less than one year).

• Funds are transacted on overnight basis in Market.

• Under market, funds are transacted for the period between 2 to 14 days.

call money

notice money

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Call/Notice Money Markets

Features of Call / Notice Money Markets

Participants Banks and Primary Dealers

Type of transactions Uncollateralized borrowing / lending

Maturity Overnight for call money2 to 14 days for notice money

Quotes / Rates Expressed annualized

Repayment of borrowing Both interest and principal are paid at maturity

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Market Participants Call / Notice Market

Borrowing:

• Commercial BanksAverage Fortnightly borrowing < 100% of capital funds (Tier I + II)

• Cooperative BanksDaily borrowing < 2% of aggregate deposits of previous financial year

• PDsAverage Fortnightly borrowing < 225% of net owned funds

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Lending:

• Commercial BanksAverage Fortnightly lending < 25% of capital funds

• Cooperative BanksNo limit

• PDsAverage Fortnightly borrowing < 25% of net owned funds

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• Following are not permitted in call/notice money markets :

Financial Institutions,

Insurance Companies, &

Mutual Funds

Call/Notice Market

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NDS- Call

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Collateralised Borrowing & Lending Obligation (CBLO)

• CBLO is a money market instrument and a product developed by CCIL for the benefit of the entities who :have either been phased out from inter bank call money market, or

have been given restricted participation in terms of ceiling on call borrowing & lending transactions and do not have access to the call market.

• CBLO is a discounted instrument issued in electronic book entry form for the maturity period ranging from one day to one year.

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CBLO Market

CBLO

Collateralized Borrowing and Lending Obligation is a part of money market and is open for all categories of participants such as banks, PDs, FIs, MFs, PFs, Corporates etc.

Need for CBLO

Since call money market is an inter-bank market, CBLO provides an excellent mechanism for banks as well as non-bank participants to manage short term liquidity.

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Collateralized Borrowing & Lending Obligation (CBLO)

• An obligation by the borrower to return the money borrowed, at a specified future date.

• An authority to the lender to receive money lent, at a specified future date with an option/privilege to transfer the authority to another person for value received.

• An underlying charge on securities held in custody (with CCIL) for the amount borrowed/lent.


• All Government Securities & T-bills, as & when notified by the CCIL.

• CBLOs are normally done for a minimum maturity period of one day.

• Interest to be calculated on an A/365 day year basis.

• CBLO bid, refers to lending of funds & borrowing of securities.

• CBLO offer, refers to borrowing of funds & lending of securities.

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CCIL asks for securities for margin maintenance :

For creating limits, securities are deposited with the CCIL, and credit against the same is arrived at on T+1 basis.

For withdrawing securities, CCIL gives credit on T+1 basis.

For substitution of securities, CCIL gives credit on T+1 basis.

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CBLO Market

Source : CCIL

(in Rs mn)

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Categorywise CBLO Activity

Source : CCIL

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Lending Borrowing

CBLO – Trading Platform

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Money Market Interest Rates

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Treasury Bills

• Treasury bills are short-term negotiable securities issued in their domestic money markets by Governments.

• They are used for short term funding as well as to control the money supply in the economy.

• They do not pay interest but are traded at discount to their par value or face value.

• They are the most liquid part of the money markets.

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Treasury bills

Treasury bills Auction day Notified Auction Amount

91-dayWednesday of every

week Rs. 5000 crores

182-day

Every alternate Wednesday (which is not a reporting week ) Rs. 1500 crores

364-day

Every alternate Wednesday (which is a

reporting week ) Rs. 1000 crores

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• Discount =

[(FV – IP)/FV]*100*365/No. of days to maturity

• Price =

Par Value*[(1 - Dis. *(t/365)]

• Yield =

[(FV – IP)/IP]*100*365/No. of days to maturity

Treasury Bills – Basic Calculations

Treasury bills Auction Data

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Auctions of 91 Day Government of India Treasury Bills (Amount in Rs. crore)Bids Accepted

Total Face Value

Date of Auction

Notified Amount Number

TotalIssue

Cut-off Price

ImplicitYield atCut-offPrice

(per cent)Com-petitiveNon-

Com-petitive6-May-09 8,000 49 8000 — 8,000 99.22 3.153213-May-09 5,000 58 5000 — 5,000 99.19 3.275420-May-09 5,000 35 5000 — 5,000 99.19 3.275427-May-09 5,000 41 5000 — 5,000 99.18 3.31623-Jun-09 4,500 39 4500 — 4,500 99.17 3.357010-Jun-09 5,000 22 5000 — 5,000 99.17 3.357017-Jun-09 5,000 42 5000 — 5,000 99.17 3.357024-Jun-09 5,000 19 5000 — 5,000 99.18 3.31621-Jul-09 2,000 1 2,000.00 — 2,000.00 99.23 3.11248-Jul-09 8,000 37 8,000.00 — 8,000.00 99.20 3.2347

15-Jul-09 8,000 61 8,000.00 — 8,000.00 99.19 3.275422-Jul-09 8,000 41 8,000.00 — 8,000.00 99.19 3.275429-Jul-09 8,000 19 8,000.00 — 8,000.00 99.20 3.2347

Treasury Bill Yields

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Auctions of T-Bills for the month of September

Date of Auction 91 Day T-Bills 182 Day T-

Bills364 Day T-

Bills Total

2-Sep-09 4,500 1,500 6,000

9-Sep-09 5,000 4,000 9,000

16-Sep-09 5,000 3,000 8,000

23-Sep-09 5,000 1,000 6,000

30-Sep-09 2,000 1,000

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Commercial paper

• Commercial Papers are negotiable short-term unsecured promissory notes with fixed maturities, issued by well rated companies generally sold on discount basis.

• These are basically instruments evidencing the liability of the issuer to pay the holder in due course a fixed amount (face value of the instrument) on the specified due date.

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Commercial paper

Features

• Commercial Papers when issued in Physical Form are negotiable by endorsement and delivery and hence highly flexible instruments

• Issued subject to minimum of Rs 5 lakhs and in the multiples of Rs. 5 Lac thereafter,

• Maturity is 7 days to 1 year

• Unsecured and backed by credit of the issuing company

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Commercial paper

Eligibility CriteriaAny private/public sector co. wishing to raise money through the

CP market has to meet the following requirements:

• Tangible net-worth not less than Rs 4 crore - as per last audited statement. Should have Working Capital limit sanctioned by a bank / FI.

• Credit Rating not lower than P2 or its equivalent - by Credit Rating Agency approved by Reserve Bank of India.

• Board resolution authorizing company to issue CPs

• PD and AIFIs can also issue Commercial Papers

Commercial paper

• The cash leg is settled through RTGS or high value. RBI cheques cannot be issued for settlement of CPs.

• The settlement of CPs is compulsorily in a demat mode.

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C Ps issued

Company name Issue date Amount Coupon rate Maturity

Tata Motors January 500 million 10.4% May

Tata Capital January Rs 2.5 Billion 11.1% March

Godrej & Boyce July Rs 200 million 4.35% Dec

Hindustan construction

companyJuly Rs 300 million 5.65% Oct

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Year:2009

Source: Reuters

Commercial Papers in India

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Certificate of Deposit

• Issued by banks for specified period of time and at a specified rate of interest.

• It is a time deposit with the bank.

• They offer a slightly higher yield than T-Bills because of higher default risk.

• In short, they are transferable, negotiable, short term, fixed interest bearing, maturity dated money market instruments.

Certificate of Deposit

• The maturity period of the CDs for banks varies between 15 days to 1 year. For FIs, the tenor of the maturity can extend upto 3 years.

• Here also, the cash leg is settled through RTGS or high value and RBI cheques cannot be issued for settlement.

• The settlement of CDs is compulsorily in a demat mode.

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Certificate of Deposits in India

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C Ds issued

Company name

Issue date Amount Coupon rate Maturity

United bank of India

August 09 Rs 1 billion 3.71% 3 months

Punjab National

Bank

December 08 Rs 1 billion 8.27% 6 months

IDBIDecember

08 Rs 1 billion 8.55% 1 year

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Source: Reuters

Government Securities Market

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GOI Securities

GOI securities

• Liabilities of GOI issued in the nature of Bonds

• Issued to finance budget deficits of GOI

• Available in both SGL & Physical form

• RBI acts the ‘Investment Banker’ ,‘Custodian’, ‘Registrar’ and market Regulator

• Primarily, wholesale in nature• Active trading & investment avenue

for Banks, PDs, FIs, Insurance Cos., PFs, MFs, etc

Auctions of GOI securities

Issues for 4th September 2009

Stock name Notified amount Method of issue

6.49% Govt stock, 2015 Rs 5000cr Price based auction using uniform price

auction method

6.90% Govt Stock, 2019 Rs 5000cr Price based auction using uniform price

auction method

8.24% Govt Stock, 2027 Rs 2000cr Price based auction using uniform price

auction method

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How GOI securities are issued ?

Auctions Price based auctionsYield based auctionsMultiple price auctions

Uniform price auctions

Which is widely used method?

Uniform ? / Multiple ?

What is non-competitive bidding ?

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Products

Issued by GOI

Dated Securities - Issued by GOI

– Bonds which include both coupon bearing and zero coupon

– Initial maturity > 1 year

Issued by state government

State Govt Loans - Bonds issued by State Govts.

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Some varieties of G Secs issued in past

Zero coupon Bonds – Four issuances between 1994 & 96

Floating Rate Bonds – Around 10 issues so far.

Capital Indexed Bond – Issued in 1997 with principal indexed to inflation, Lackluster response.

Issued in 2004 again with both principal andinterest indexed to WPI, Lacklusterresponse again.

Callable & Puttable G Secs. – Issued in 2002, only one issue so far.

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Some varieties of G Secs issued in past

When Issued Securities Takes place from issue announcement date till

one day prior to the issue.

NDS-OM

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Trading mechanism for G Secs.

• Negotiated dealing system (NDS) is an electronic platform for facilitating dealing introduced in 2002.

• NDS interfaces with CCIL for settlement of government securities transactions for both outright and repo trades.

• It remains as a reporting platform rather than a trading one.

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Trading mechanism for G Secs.

• In 2005, NDS-OM (order matching) system was introduced.

• It provides STP, is purely order driven and follows price / time priority. CCIL is the central counter party for all trades done through this platform.

• It facilitates better price discovery, liquidity, increase in operational efficiency & transparency.

• Settlement for G Secs. is done on T + 1 basis.

http://www.ccilindia.com/OMHome.aspx

NDS-OM

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Measures to improve liquidity

• Consolidation through re-issuance of existing securities.

• Intra-day short selling in G Secs. allowed for banks & PDs in 2006. Short selling now extended to 5 trading days.

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Primary dealers

The role of Primary Dealers is to :

(i) participate as Principals in Government of India issues through bidding in auctions

(ii) provide underwriting services (3% of issue size is MUC, 3% to 30% is competitive AUC)

(iii)offer firm buy - sell / bid ask quotes for T-Bills & dated securities

(iv)Develop Secondary Debt Market

Primary dealers

• Deutsche Securities (India) Pvt. Ltd.

• ABN AMRO Bank N.V.

• ICICI Securities Primary Dealership Limited

• Bank of America

• STCI Primary Dealer Limited

• Bank Of Baroda

• IDBI GIlts Ltd.

• Kotak Mahindra Bank Ltd.

• Canara Bank

• SBI DFHI Ltd

• Citibank N.A

• PNB Gilts Ltd.

• Corporation Bank

• HDFC Bank Ltd.

• Hongkong and Shanghai Banking Corpn. Ltd.(HSBC)

• JP Morgan Chase Bank N.A, Mumbai Branch

• Standard Chartered Bank

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Issue Devolvement

• When the undersubscription of a security issue forces the underwriting investment bank to purchase unsold securities during an offering.

• Devolvement is often an indication that the market currently has negative sentiments toward the issue.

• This negative sentiment can have a significant impact on subsequent demand.

• Devolvement poses substantial risk for the underwriting investment bank.

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Issue Devolvement

• When it is required to purchase unsubscribed shares of an issue, it will often purchase the stock at a higher-than-market-value price.

• Because demand is lower than anticipated, there are few buyers for the security at its issued value.

• Typically, the investment bank will not hold onto the floundering issue for too long and will usually liquidate the shares in the market, often causing a financial loss.

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RBI Guidelines for Issue Devolvement

• PDs, under current RBI guidelines, are expected to underwrite a minimum of 50 per cent of the notified amount under the Minimum Underwriting Commitment (MUC).

• PDs will invariably bid at all auctions at least to the extent of their underwriting commitment as accepted by the Reserve Bank.

• PDs would be allowed to set-off the accepted bids in the auction against their underwriting commitment accepted by the Bank, in case of devolvement.

• Devolvement of securities, if any, on Primary Dealers will take place on pro-rata basis, depending upon the amount of underwriting obligationof each Primary Dealer after setting off the successful bids in the auction.

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For the week ended October 2, 2009

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Source : CCIL

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SLR Securities

• Dated securities and T bills issued by the Government of India

• Dated securities issued by State Government

• Other “approved” Securities

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Investors Mandated to invest in GSecs.

Banks 24% of NDTL

Life Insurance Companies

25% of Controlled Funds

General Insurance Companies

30% of Total Assets

Pension & General Annuity Businesses

20% of Total Assets

NBFCs accepting public deposits

15% in liquid assets, not less than 10% in approved securities

Size of G-Sec Markets in India

Government Securities Market in India – A ProfileAmount in Rs. crore; ratios in percent

Indicator 1991-92 1995-96 2000-01 2004-05 2005-06 2006-07 2007-08Outstanding stock (in Rs. crore)

76,908 1,69,526

4,53,668

8,24,612

9,29,612

10,32,296 13,32,435

Outstanding stock/GDP (%)

11.8 14.3 21.5 26.2 26 26.3 28.3

Average maturity of securities issued during the year (in years)

.. 5.7 10.6 14.1 14.1 16.9 14.9

Weighted average cost of securities issued during the year (in per cent)

11.8 13.8 11 6.1 7.3 7.9 8.1

Source : RBI

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Government Bond Yields

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Source : RBI

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Source : RBI

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Understanding Macro Economic Variables

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Important macro economic variables

• A number of macro economic variables affect the liquidity in thesystem which in turn affects bond markets.

• Some of these can be considered as long term while some cause changes in short term interest rates.

• Let us review some factors affecting long term as well as short term liquidity in the system.

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Long – Term Factors

• GDP Growth

• Credit growth

• Interest Rates

• Inflation

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Growth Indicators

GDP growth The monetary value of all the finished goods and services produced

within a country's borders in a specific time period, though GDP is usually calculated on an annual basis.

GDP = C + G + I + (Exports-imports)

where:

"C" is equal to all private consumption, or consumer spending, in a nation's economy;"G" is the sum of government spending;"I" is the sum of all the country's businesses spending on capital & the nation's total net exports, calculated as total exports minus total imports.

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Growth Indicators

• Higher GDP or higher national output indicates greater aggregate demand for coming years.

• Investment decisions are based on growth rate of economy / sectors.

• Investments are funded by domestic savings (domestic household + private sector + public sector) and external sources.

• Which policy initiative is taken to support real GDP growth?

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Interest Rates

Interest rates

• Cost of Money in the economy, determined by demand and supply of money.

• Factors creating demand for liquidity

• Factors creating supply of liquidity

• Nominal Vs. real interest rates

Interest rates

20092006-07

2007-08

2008-09 Apr May June

Cash Reserve Ratio (%) 6.5 7.5 5 5 5 5Bank Rate (% p.a.) 6 6 6 6 6 6

Inter-bank Call Money Rate (Mumbai) (%)

0.50-4.90

6.15-9.30

2.50-5.75

1.75-3.40

1.00-3.30

1.25-3.35

Deposit Rate (%)

(a) 30 days and 1 year 3.00-9.50

3.00-7.50

3.25-8.00

3.00-7.00

2.50-7.00

2.50-7.00

(b) 1 year and above 7.50-9.60

8.25-9.00

8.00-8.50

7.00-8.50

6.50-8.25

6.50-8.00

Prime Lending Rate (%) 12.25-12.50

12.25-12.75

11.50-12.50

11.50-12.25

11.00-12.25

11.00-12.25

Source : RBI73

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Inflation

Inflation

• The rate at which the general level of prices for goods and services is rising, and, subsequently, purchasing power is falling.

• Poses a major risk to fixed income markets.

CPI

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WPI Inflation

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WPI – Major groups

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Commodity prices

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Credit and Liquidity

• Credit – Food and Non-food credit

• LiquidityDemand for and supply of fundsShortage and surplus of credit

• Impacts (growth rate, prices, interest rates)

Assessing short term liquidity

• Trends in Money Markets

• Liquidity Adjustment Facility (LAF)

• Bank Rate

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Trends in Money Markets

• Call Money Market is a market for overnight borrowing & lending.

• Since Indian call money markets are very active, trends in call money and other short term money markets provide important information about liquidity in the system.

• Increase in call money rates indicates tightness of liquidity in the short term which if continues may spread to term money market as well

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Fund flow in the system - Call money rates

Range of Rates Weighted Average RatesAs on

Borrowings Lendings Borrowings LendingsJuly 2, 2009 1.25 — 3.30 1.25 — 3.30 3.18 3.18July 3, 2009 1.25 — 3.30 1.25 — 3.30 3.13 3.13July 4, 2009 1.25 — 3.30 1.25 — 3.30 3.22 3.22July 6, 2009 1.25 — 3.30 1.25 — 3.30 3.17 3.17July 7, 2009 1.25 — 3.30 1.25 — 3.30 3.13 3.13July 8, 2009 1.25 — 3.30 1.25 — 3.30 3.16 3.16July 9, 2009 1.25 — 3.30 1.25 — 3.30 3.17 3.17July 10, 2009 1.25 — 3.30 1.25 — 3.30 3.21 3.21July 11, 2009 3.20 — 3.35 3.20 — 3.35 3.25 3.25July 13, 2009 2.00 — 3.35 2.00 — 3.35 3.23 3.23July 14, 2009 2.00 — 3.30 2.00 — 3.30 3.23 3.23July 15, 2009 2.00 — 3.30 2.00 — 3.30 3.22 3.22

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Indicator for Term Money Markets

• Since in India, term money market is not liquid, a rupee term money rate is arrived at by using USD Libor and USD / INR premium. This rate is essentially MIFOR and is an important benchmark for fixed income derivatives.

MIFOR = ((1 + $ LIBOR)*(1 + Forward Premia)) - 1

• MIFOR calculation assumes that interest rate parity holds in India.

• Interest rate parity says, forward premium is a function of interest rate differential.

• However in India, forward premium is a function of demand and supply by exporters, importers, borrowers and lenders rather than interest rate parity since there are capital restrictions.

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Forward Premium

To hedge against forward To hedge against forward

Contracts entered into with contracts entered into with

exporters importers

Banks sell spot and enter Banks buy spot and enters

into buy sell swap (receive into sell buy swap (pay

premium) premium)

Thus demand and supply determined forward premium.

This in turn affects the MIFOR curve.

Liquidity Adjustment Facility (LAF)

RBI Reverse Repo rate

• The RBI borrows at reverse repo rate from the market on a daily basis.

RBI Repo rate

• The RBI lends at repo rate on a daily basis.

• Only banks, PDs & FIs can participate in RBI LAF.

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CROMS - REPO

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Source : CCIL

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Source : RBI

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Source : RBI

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Inter-bank Repos

• Banks, PDs, Mutual funds, Insurance cos etc can borrow/lend through inter-bank repo/reverse repo transactions.

• The maturity can be upto 1 year.

• It is OTC & free to be determined between 2 counterparties.

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• The dealers lends funds to the RBI directly through NDS at the RBI Reverse Repo rate.

• The RBI intimates the amount accepted at the rate on the NDS.

• The bids are submitted electronically through the NDS.

• There are 2 LAFs – bidding by 1030 hrs (result by 1200 hrs) & bidding by 1530 hrs (result by 1630 hrs).

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Bank Rate

• 2 key bank re-finance rates are directly linked to - the bank/PD refinance levels & the export re-finance to the banks

• Provides the basic liquidity to the banking sector

• Is used as a interest rate policy signal tool by the RBI

• Lays down the basic interest rate structure

Liquidity Management CRR and SLR

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CRR & SLR

• CRR & SLR are liquidity management tools.

• CRR is used by RBI in recent times to manage the liquidity in the system directly

• Increase in CRR absorbs banking liquidity and indicates possible tightening of short term interest rates

Maintenance of Cash Reserve Ratio (CRR)

Demand Liabilities include all liabilities which are payable on demand, like :•Current deposits & demand liabilities portion of savings bank deposits.

•Margins held against letters of credit/guarantees.

•Money at call and short notice from outside the banking system.

Time liabilities are those which are payable otherwise than on demand, like :•Fixed deposits, cash certificates, cumulative and recurring deposits, time liabilities portion of savings bank deposits.

•Deposits held as securities for advances which are not payable on demand.

•Gold deposits.

CRR is to be maintained on Net Demand & Time Liabilities (NDTL) of a bank.

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• Loans/borrowings from abroad by banks in India will be considered as ‘liabilities to others’ and will be subject to reserve requirements.

• Other demand and time liabilities include

Interest accrued on deposits,

Bills payable,

Unpaid dividends,

Any amounts due to the banking system which are not in the nature of deposits or borrowing.

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• Banks are required to maintain the prescribed CRR based on their NDTL as on the last Friday of the second preceding fortnight.

• Currently the CRR is 5%.

• Interest on CRR balances is determined by RBI from time to time.

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Process of CRR Maintenance

• Assume that the NDTL on which the CRR is to be maintained is Rs 10,000cr.

• At 5%, the total CRR balance that has to be maintained per day is Rs 500cr; which means a total fortnight product of Rs 7000cr over the reporting fortnight.

• Of this, the bank has to maintain a minimum of 70% of the balance on any day of the reporting fortnight. The remaining may be adjusted over the rest of the days of the fortnight.

• Hence, a bank may choose to maintain only Rs 350cr (70% of the daily balance requirement) for the first 7 days of the fortnightand then maintain the remaining Rs 650cr on the remaining 7 daysof the fortnight such that the total CRR product is Rs 7000cr.

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Statutory Liquidity Ratio (SLR)

• SLR is to be maintained on Demand & Time Liabilities (DTL) of a bank.

• All banks are required to maintain a uniform SLR of 24% of the total of their demand and time liabilities in India as on the last Friday of the second preceding fortnight.

• Here there are no special concessions for inter-bank liabilities.

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Maintenance of Statutory Liquidity Ratio (SLR)

• SLR can be maintained in either of the following forms :

Cash,

Gold,

Approved securities (GOI securities / SDLs)

• So, in this case, if the DTL of a bank is Rs 10,000cr, the SLR it has to maintain is Rs 2400cr worth of securities or any other form as allowed by the RBI.

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Fixed Income Mathematics

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Time value of money

• Cash inflows or cash outflows occurring at different points of time have different time value.

• Hence to compare these cash flows to arrive at a particular decision, the cash flows have to be adjusted for time value.

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Time value of money

• For example, a company wants to take a decision whether to expand its operations or not. It expects the following cash flows from the project for next four years.

Year 0 1 2 3 4

Cash flows (1000) 200 300 500 600

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Time value of money

• For comparing these cash flows, we have to adjust them for time value.

• This can be done either by compounding or discounting.

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Time value of money

Compounding

(1000) 200 300 500 600

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Time value of money

Compounding The formula is,FV = PV (1 + r)n

Here FV = Future Value, PV = Present ValueR = rate of interest

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Time value of money

Future value of annuity

• Annuity is the term used to describe a series of period flows ofequal amounts. Future value annuity is used to find future value of series of payments made.

FVA = A [(1 + k)n -1] / k

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Time value of money

Discounting

(1000) 200 300 500 600

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Time value of money

Discounting

The formula is

PV = FV / (1 + r)n

Present value of annuity

• Annuity is the term used to describe a series of period flows ofequal amounts.

PVA = A [(1 + k)n -1] / k (1 + k)n

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Time value of money

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Understanding Yield To Maturity (YTM)

Basic Bond terminologies

• Par Value or Face Value of bond

• Coupon Rate

• Maturity Date

Yield Concepts

• Current Yield = Annual Coupon / Current market price

• Yield To Maturity = Discount rate that makes sum of present values of all cash flows from the bond equal to its market price.

Bond Valuation

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Bond Valuation

• Valuation of bond involves discounting the future cash flows at an appropriate rate to arrive at the present value of bonds.

• This principle is applicable to all types of bonds, whether zero coupon or coupon bearing bonds.

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Coupon rate, Discount Rate & Price

• Discount rate used by the market is nothing but expected yield from the bond

Relationship between coupon rate, yield and bond’s price

• When Coupon rate < yield required by the market, bond trades at a

• When Coupon rate > yield required by the market, bond trades at a

• For Par bonds, Coupon rate yield required

discount

premium=

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Coupon rate, Discount Rate & Price

• As maturity approaches, price of discount bond

• As maturity approaches, price of premium bond

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Valuing bonds with Semi-annual cash flows

Adjustments required

1. No. of semi annual periods = Annual Periods * 2

2. Yield per period = Annual Yield/2

3. Coupon per period = Coupon per period/2

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Valuing bonds between coupon payments

• The bond price, between any two interest payment dates, will have an element of accrued interest till the date.

• Dirty prices (or full price) are bond prices including the interest accrued.

• Clean prices are bond prices excluding the accrued interest.

• Market always quotes a clean price. But the seller of the bond receives dirty price

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Valuing bonds

• Long Term Vs. Short Term bonds– Which bonds are more volatile?

• High Coupon Vs. Low Coupon Bonds– Which bonds are preferred in volatile times?

• Capital Gains Vs. Capital Losses– Capital Gains and Losses arising from equal basis point change in

interest rates are same or not?

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Day Count Conventions

• Various markets use different day count conventions to calculate the interest rate:

– Actual / Actual Actual no. of days between two payment dates are divided by actual no. of days in a year. For a normal year, no. of days are 365, leap year they are 366.

– 30 / 360

Each month is considered to have only 30 days. Hence the entire year is of 360 days.

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Day count and quoting conventions for Government securities

– Actual / 360

Actual no. of days between two payment dates are divided by 360 days.

– Actual / 365Actual no. of days between two payment dates are divided by 365 days.

– In Indian markets, Government Securities use 30/360 convention while all other products use Act / 365.

Revisiting YTM and ZCYC

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Shortcomings of YTM

YTM is based on following assumptions,

The intermediate cash flows to the investor can be reinvested at a rate equal to the yield-to-maturity and

The investor holds the bond till maturity

• The first assumption can rarely hold true due to the dynamic nature of interest rates. Similarly the investor may not hold the bond till maturity.

• Hence the realized yield may be quite different from YTM, giving rise to risk.

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Zero Coupon Yield Curve

• The relationship between yield and maturity is called yield curve

• Yield curve is developed for Zero Coupon Treasury Securities

• The yield curve is developed from On The Run Treasury securities. These are the securities which are recently auctioned and hence may be in good demand.

• Let us say, the recently auctioned issues are for 3-month, 6 month, 1 Y, 2Y, 3Y, 5Y, 10 Y and 30Y.

• How will the yields be calculated for say 4 Y or 12 Y period?

• These are calculated by interpolation.

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Zero Curve Generation - Bootstrapping

• Except the short term securities, the yields on long term securities are not zero coupon yields.

• Hence a process is required to calculate zero coupon yields from coupon yields.

• The procedure used for calculating the zero coupon yield curve is called Bootstrapping.

• Bootstrapping is a process which determines an appropriate discount rate associated with a unique maturity.

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Bootstrapping

• The zero coupon yield curve is generated by plotting the zero coupon yields on a time scale.

• Derive the implied spot rates from the prices and yields of coupon bonds.

• Treat each coupon as a mini-zero coupon bond, i.e. think of a coupon bond as a portfolio of zeros.

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Bootstrapping

• Separate the cash flow, each with its own discount (spot) rate

• Use bonds of progressively longer maturities, starting from T-bills

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Zero Coupon & YTM Arbitrage

• Since zero coupon & YTM methods would give different answer for the bond at same par yield curves, there can be an arbitrage.

• One can buy a bond at a price derived from YTM method, strip every cash flow of the same (coupons & principal) as separately traded securities & sell individual STRIPs.

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Zero Coupon & YTM Arbitrage

• Each of such STRIP will be a single cash flow & hence a zero coupon price.

• Hence, if the buyer of the bond sells all the STRIP securities individually, the total realisation should technically be equal to the purchase price – otherwise there will be an arbitrage.

• This ensures the pricing parity.

Day 2 – Session 1 Valuation of Floating Rate Bonds

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Indian Scenario for FRBs

• Both Government as well as corporates issue FRBs.

• G FRBs have been issued in the past with varying spreads over 364 day T Bill rates

• Corporates have to provide spreads depending upon the credit quality

• Trading in FRBs is extremely limited. Hence market prices may not be available.

• For valuation purposes, FIMMDA gives prices of these bonds.

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Valuation of floating rate bonds

• Floating rate bonds pay coupon which is benchmarked against a particular interest rate.

• The coupon will be decided at each reset date and may have a spread over the benchmark rate.

Following terms are crucial to the concept :

• Benchmark rate : a market determined interest rate used for the calculation of coupon rate from time to time.

• Reset frequency : the frequency at which a coupon rate is reset

• Coupon payment frequency : the frequency at which coupon payment takes place.

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Methodology

• The standard method of valuing a bond is to discount all the cash flows of the bond at a rate based on the yield available on a comparable instrument in the market.

• In simple terms, the cash flows to be discounted in case of floating rate bond is

Next coupon to be received which is known with certaintyValue of the bond at the next coupon date

• A view can be taken that the bond will reset at par. In this case the valuation of bond on any given day will be simple.

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Issues

• FRBs having Zero Spread over reference rateValue this by taking only one cashflow (i.e. next coupon & principle)Value by taking multiple cash flows (all coupons and principle)

• FRBs having a spread over reference rateAssuming the spread has not changed since issue of this bond

Value this by taking only one cashflow (i.e. next coupon & principle)Value by taking multiple cash flows (all coupons and principle)

Assuming that the spread has changed since the issue of this bond (for corporate bonds)

Value this by taking only one cashflow (i.e. next coupon & principle)Value by taking multiple cash flows (all coupons and principle)

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Valuation of bonds with embedded options

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Bonds with embedded options

• Callable or redeemable bonds are bonds that can be redeemed or paid off at the option of the issuer prior to the bond’s maturity date. (embedded call option)

• Puttable bonds are bonds that can be redeemed or paid off at the option of the investor prior to the bond’s maturity date. (embedded put option)

• Investors normally calculate the yield to call (or yield to put)in addition to yield to maturity and the lower of the two (called yield to worst) is used by conservative investors to take their decision.

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Valuation of Bonds with Embedded Options

• The potential investor in a callable bond must be compensated for the risk that the issuer will call the bond prior to the stated maturity date.

• Two disadvantages faced by an investor in callable bonds are reinvestment risk and truncated price appreciation when yields decline.

• The traditional approach to valuing callable bonds has been in terms of the yield to worst.

• Yield to worst is calculated by calculating the YTM and the yield to call for every call date and then selecting the lowest of all the calculated yields.

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Contd…

• At present the valuation of callable bonds in India happens on Yield to Worst basis and Valuation of puttable options is done on yield to best basis.

• Thus the optionality element is not considered.

• Let us see if this element is considered how the bond pricing can be done.

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Contd…

To value a bond with an embedded option it is necessary to understand that the bond can be decomposed into an option-free component and an option component.

• Value of a callable bond = Value of non-callable bond - Call option premium

• Value of a putable bond = Value of non-putable bond + Long a put option

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Valuation Methodologies for Bonds with Embedded Options

• Binomial Method

• Monte Carlo Method

Understanding Types of Risks

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Interest Rate and Reinvestment Risk

As a result of the above shortfalls, the investor is exposed to the following risks

• Interest Rate Risk : If the investor does not hold bond till maturity, an increase in future interest rates could lead to a capital loss when the bond is sold in the secondary market.

• Reinvestment Risk : The assumption of the intermediate cashflows being reinvested at the yield-to-maturity, exposes the investor to the risk that the future reinvestment rates would be less than the yield-to-maturity.

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Interest Rate and Reinvestment Risk

Interest Rate Risk Reinvestment Risk

If interest rate goes upIf interest

rate goes down

Hence the valuation of bonds can be done by usingZero Coupon Yield Curve

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Other risks from bond investing

Call & prepayment risk

• The bond may have a call option for the issuer. Creates uncertainty of cash flows. Yield to Call instead of Yield to Maturity may be relevant yield.

Yield curve risk

• What is a yield curve?

• It is a relationship between yield and term to maturity.

• Types of yield curve :

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Yield curve risk

• Yield curve risk arises for a portfolio of bonds. In a portfolio, many bonds with different maturities exit. Yield curve shift may not be parallel and hence each bond’s price will change differently. This is yield curve risk.

Credit risk

• Default risk

• Credit spread risk

• Downgrade risk

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Liquidity risk

• Risk of having to sell the bond below its indicated value.

• Bid ask spread indicate the level of liquidity risk.

Exchange-rate risk

• Arises from investing in foreign currency denominated bonds.

Volatility risk

Inflation risk

Sovereign risk

• Risk arising out of investing in bonds issued by foreign governments.

Day 2 – Session 2 Interest Rate Risk Measurement

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Measuring Interest Rate Risk

Full Valuation Approach

• For a change in interest rates, the changed values of bonds can be calculated to arrive at changed value of the portfolio.

Duration / Convexity approach

• In this case, the duration and or convexity of bonds / portfolios is calculated to arrive at approximate estimate of change in the portfolio value due to change in interest rates.

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Macaulay & Modified Duration

• Macaulay & Modified duration are two measures of duration

• Modified Duration is the weighted average time to the present value of cash flows

• Modified Duration directly gives the percentage change in price with a unit change in yield. This is shown in the following slides.

• DMod = [DMac / 1+(ytm/f)]

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Modified Duration as Interest Rate Sensitivity

• Modified Duration is used as a measure of interest rate sensitivity.

• Using the formula for duration

dP/dy = -(MD)*P

• This can be written as

MD = -(dP/P)/dy

• It can be seen from the above that MD is equal to minus of percentage change in price for a unit change in the yield.

• Hence, MD can directly be used as an interest rate risk measure for bonds.

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Price Value of a Basis Point

• The Price Value of a Basis Point (PVBP) is the price change of a security for a one basis point change in yield.

• It is equal to Dollar Duration divided by 100.

PVBP = $D/100

• For example, the 5 year bond, has

Dollar duration of 3.31

PVBP of 3.31 / 100 = 0.0331

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Drawbacks of Using Duration for Hedging

Yield

Price

Y0 Y1 Y2

Actual Price

Price Predicted by Duration

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Convexity

• Convexity measures the change in duration when the interest rate changes.

• The relationship between market value and interest rate is not linear.

• While duration measures the market value weighted average maturity of future flows, convexity is a function of dispersion of flows around that average.

• Zero Coupon bonds will have lowest convexity, other things being equal.

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Convexity

• Duration is an accurate measure only for small yield changes.

• Convexity combined with duration allows us to do better approximation of price than using duration alone.

• Convexity = ∑{t*(t+1)*CF/(1+r)^t}/P

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Convexity

• Convexity measures how duration changes with interest rates.

• It is the second derivative of price with respect to yield.

1 d2P

C = --- ---------

P dy2

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Change in Price Due to Convexity

• Consider a bond with price P and convexity C. If the yield on the bond changes by dy, the change in the price of the bond will be given by

= ½ * C * (∆y)2

• It can be seen that the change in price due to the property of convexity is always positive.

• Because of convexity the bond price rises at a faster rate and falls at lower rate with changes in the yield.

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Total Change in Price of a Bond with Yield

• The total change in the price of a bond due to a change in yield is the sum of two componentsChange in price due to durationChange in price due to convexity

Bond Portfolio Planning & Management Strategies

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Alternative Bond Portfolio Strategies

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Passive Portfolio Strategies

• Passive strategies emphasize buy-and-hold, low energy management.

• Try to earn the market return rather than beat the market return.

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• Buy and hold

• Indexing

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Buy and hold

• Buy a portfolio of bonds and hold them to maturity

• Can by modified by trading into more desirable positions

Indexing

• Match performance of a selected bond index

• Performance analysis involves examining tracking error for differences between portfolio performance and index performance

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Active Management Strategies

• Active management strategies attempt to beat the market

• Mostly the success or failure is going to come from the ability to accurately forecast future interest rates

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• Interest Rate Anticipation

• Valuation Analysis

• Credit Analysis

• Yield Spread Analysis

• Bond Swaps

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Interest-rate anticipation

• Risky strategy relying on uncertain forecasts of future interest rates, adjusting portfolio duration.

• Ladder strategy staggers maturities.

• Barbell strategy splits funds between short duration and long duration securities.

Valuation analysis

• A form of fundamental analysis, this strategy selects bonds that are thought to be priced below their estimated intrinsic value.

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Barbell Strategy

• Barbells are a strategy for buying short-term and long-term bonds, but not intermediate-term bonds.

• The long-term end of the barbell allows you to lock into attractive long-term interest rates, while the short-term end insures that you will have the opportunity to invest elsewhere if the bond market takes a downturn.

• Essentially, the barbell strategy is built around the concept of focusing on the maturities of the securities that are part of the portfolio and making sure that the maturity dates are either very close or at a distant date.

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Barbell Strategy

This is how it works :

• When you see appealing long-term interest rates, you buy two long-term bonds. You also buy two short-term bonds. When the short-term bonds mature, you receive the principal and have the opportunity to reinvest it.

• This means that two blocs or groups are created within the portfolio, rather than having securities that mature consistently from one period to the next.

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Barbell Strategy

• Part of the purpose of the barbell strategy is to allow for a quick turnover of a significant amount of the assets in the portfolio at one time.

• For example, attention should be paid to the bloc of short-term investments, so they can all be rolled over into new short-term investments as they reach maturity.

• Typically, this leads to an increase in the value of the investments that are turned over, thus increasing the overall value of the investment portfolio.

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Credit analysis

• Determines expected changes in default risk

• Try to predict rating changes and trade accordinglyBuy bonds with expected upgradesSell bonds with expected downgrades

• Credit analysis models such as Altman’s Z-score model may be useful for predicting changes in ratings

• High yield bonds may warrant special attention

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Yield-spread analysis

• Monitor spread within and across sectors, bond ratings, or industries

• Trade in anticipation of changing spreads

Bond swaps

• Selling one bond (S) and buying another (P) simultaneously

• Swaps to increase current yield or YTM, take advantage of shifts in interest rates or realignment of yield spreads, improve quality of portfolio, or for tax purposes.

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Bond Swaps

• Pure yield pickup swapSwapping low-coupon bonds into higher coupon bonds

• Substitution swapSwapping a seemingly identical bond for one that is currently thought to be undervalued

• Tax swapSwap in order to manage tax liability

• Swap strategies and market-efficiencyBond swaps by their nature suggest market inefficiency

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Core-Plus Management Strategy

• Core-plus bond-portfolio management is a combination approach, which is beyond a pure passive policy or one of the several active portfolio management styles.

• The idea here is to have a significant (core) part of the portfolio (say 70 to 75%) managed passively in a widely recognized sectorsuch as the U.S. Aggregate Sector or the U.S. Government/Corporate sector.

• The difference between these 2 sectors is that the aggregate includes the rapidly growing mortgage-backed and asset-backed sectors, which forms the “core” part of the portfolio and is managed passively.

• It needs to be managed passively because these segments of the bond market are quite efficient and hence it is not worth the time and cost to attempt to derive excess returns within these sectors.

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Core-Plus Management Strategy

• The rest of the portfolio would be managed actively in one or several additional “plus” sectors, include high-yield bonds, foreign bonds, and emerging-market debt.

• Here, it is felt that there is a higher probability of achieving positive abnormal rates of return because of potential inefficiencies.

• Hence, these are considered good avenues for active management since they generally experience above-average rates of return.

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Matched-Funding Techniques


• Classical (“pure”) immunization

• Dedicated portfolio

• Horizon Matching

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• Classical (“pure”) immunization strategies attempt to earn a specified rate of return regardless of changes in interest rates.

Must balance the components of interest rate riskPrice risk: problem with rising interest rates

Reinvestment risk: problem with falling interest rates

Immunize a portfolio from interest rate risk by keeping the portfolio duration equal to the investment horizon

Duration strategy superior to a strategy based only a maturity since duration considers both sources of interest rate risk

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Immunization Strategies

• Difficulties in Maintaining Immunization Strategy

Rebalancing required as duration declines more slowly than term to maturity

Modified duration changes with a change in market interest rates

Yield curves shift

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Dedicated portfolios• Designing portfolios that will service liabilities• Different types:

Exact cash matchConservative strategy, matching portfolio cash flows to needs for cashUseful for sinking funds and maturing principal payments

Dedication with reinvestmentDoes not require exact cash flow match with liability streamGreat choices, flexibility can aid in generating higher returns with lower costs

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Horizon matching

• Combination of cash-matching and immunization

• With multiple cash needs over specified time periods, can duration-match for the time periods, while cash-matching within each time period.

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Contingent Procedures

• Contingent procedures are a form of structured active management.

• The procedure here is referred to as contingent immunization.

• It provides an opportunity to the portfolio manager to actively manage the portfolio with a structure that constrains the portfolio manager if he/she is unsuccessful.

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Contingent Procedures

Contingent Immunization

• It allows a bond portfolio manager to pursue the highest returnsavailable through active strategies, while relying on classical bond immunization techniques to ensure a given minimal return over the investment horizon.

• In simpler terms, it allows active portfolio management with a safety net provided by classical immunization.

• This technique requires the client to willingly accept a potential return below the current market return, referred to as a “cushion spread” - i.e. the difference between the current market return and some floor rate.

• This cushion spread in required yield provides flexibility for the portfolio manager to engage in active portfolio strategies.

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Bond analytics

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Rich-Cheap Analysis of Bonds

• Rich and cheap refers to the pricing of a security relative to comparable securities in the secondary market.

• Rich, or overvalued bonds, have lower yields than bonds with similar terms and credit ratings.

• Cheap, or undervalued bonds, have higher yields than paper with similar maturity and credit risk.

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Rich-Cheap Analysis of Bonds

This is how it works :

• Firstly, construct the adequate current zero-coupon yield curve using data for assets with the same characteristics in terms of liquidity and risk.

• Then compute a theoretical price for each asset to obtain the spread between the market yield to maturity and the theoretical yield to maturity.

• For each asset, implement a Z-score analysis so as to distinguish actual inefficiencies from abnormal yields. This statistical analysis provides signals of short or long positions to take in the market.

• Short and long positions are unwound according to a criterion that is defined a priori.

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Curve Flatteners/Steepeners

Bull Flattener

• A yield-rate environment in which long-term rates are decreasing at a rate faster than short-term rates.

• This causes the yield curve to flatten as the short-term and long-term rates start to converge.

• If the yield curve is exhibiting bull flattener behavior, the spread between the long-term rate and the short-term rate is getting smaller because long-term rates are decreasing as short-term rates are increasing.

• This could occur as more investors choose long-term bonds relative to short-term bonds, which drives long-term bond prices up and reduces yields.

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Bear Flattener

• A yield-rate environment in which short-term interest rates are increasing at a faster rate than long-term interest rates.

• This causes the yield curve to flatten as short-term and long-term rates start to converge.

• If the curve is flattening, the spread between long-term rates and short-term rates is narrowing.

• A bear flattener often occurs when the government raises interest rates in the short term.

• Increasing interest rates drives short-term bond prices down, increasing their yields rapidly in the short term, relative to long-term securities.

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Bull Steepener

• A change in the yield curve caused by short-term rates falling faster than long-term rates, resulting in a higher spread between the two rates.

• When the yield curve is said to be a bull steepener it means that the higher spread is caused by the short-term rates, not long-term rates.

• A steepener differs from a flattener in that a steepener widens the yield curve while a flattener causes long-term and short-term rates to move closer together.

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Bear Steepener

• A widening of the yield curve caused by long-term rates increasing at a faster rate then short-term rates.

• This causes a larger spread between the two rates as the long-term rate moves further away from the short-term rate.

• This widening yield curve is similar to a bull steepener except with a bear steepener this is driven by the changes in long-term rates, compared to a bull steepener where short-term rates have a greater effect on the yield curve.

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Rolling down the yield curve

• Fixed-interest managers can also seek extra return with a bond investment strategy known as riding the yield curve, or rolling down the yield curve.

• When the yield curve slopes upward, as a bond approaches maturity or "rolls down the yield curve," it is valued at successively lower yields and higher prices.

• Using this strategy, a bond is held for a period of time as it appreciates in price and is sold before maturity to realises the gain.

• As long as the yield curve remains normal, or in an upward slope, this strategy can continuously add to total return on a bond portfolio.

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Rolling down the yield curve

• The purpose of rolling down the yield curve is to benefit from certain interest rate environments.

• For instance, when the yield curve is relatively steep and interest rates are relatively stable, the manager will benefit by rolling down the curve versus a buy-and-hold of the short-maturity instrument.

• However, there are risks to riding the yield curve, most obviously the greater interest rate risk associated with the riding strategy (as reflected by its higher duration).

• Thus, if one is riding and yields rise substantially, the investor will incur a capital loss on the riding position.

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Spread Analytics

Zero Volatility Spread (Z-Spread)

• It is a tool used in the analysis of an asset swap that uses thezero-coupon yield curve to calculate the spread.

• The Z-spread is the number of basis points that would have to be added to the spot yield curve so that the bond's discounted cash flows equal the bond's present value.

• Each cash flow is discounted using its maturity and the spot rate for that maturity term, so each cash flow has its own zero-coupon rate.

• The spread is calculated iteratively and provides a more accurate reflection of value than other measures as it uses the entire yield curve to value the cash flows.

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Spread Analytics

Option adjusted spread

• Option adjusted spread (OAS) is the flat spread over the treasury yield curve required to discount a security payment to match its market price.

• This concept can be applied to mortgage-backed security (MBS), Options, Bonds and any other interest-rate Derivative.

• The OAS describes the market premium over a model including two types of volatility:

Variable interest rates

Variable prepayment rates.

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Bond Maths