[5] Convexity Properties of convexity

Limitations of Duration 1) Where required yield increases, convexity decreases. => +ve convexity
Duration measure does not capture the change in bond prices well for Recall:
large changes in yields. => underestimates actual price
• Overestimate price decline with an increase in interest rates
• Underestimate price increase with a decrease in interest rates

Convexity
Captures second order effect. (Duration only captures the linear effect)
($! )
𝐜𝐨𝐧𝐯𝐞𝐱𝐢𝐭𝐲 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 =
(& $ ! 2) For a given yield and maturity, lower coupon rates will have greater
, * *") - , ,") !.&
For standard bonds: ∑ *+) + /𝑃 convexity.
)"& %!$ )"& &!$
$- ) $-, &(&"))(!.&# )
)
)
Note: ZCBs have the highest convexity
*
OR 1− − + .
&' ()"&)& &$ )"& &!( )"& &!$ !

Adjustment to annual (if there are multiple periods):

convexity measure in 𝑚 period per year

convexity measure in year = Intuitively,
𝑚$
If r decreases to 0, the PV of c=1 would converge to $1. (1 discounted by
($! 1). Where interest rates increase, the discount factor kicks in and the
Dollar (money) convexity = = 𝐜𝐨𝐧𝐯𝐞𝐱𝐢𝐭𝐲 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 ×𝑃 further out the cashflows, the bigger the discount rate hence the bigger
(& $
the convexity. => This is only roughly since it does not take into
(! ) $
% 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒: = 𝐜𝐨𝐧𝐯𝐞𝐱𝐢𝐭𝐲 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 𝑑𝑟 consideration of the second order effect
! $
=> The lower the c, the greater the weight is placed on the later year
Duration with Convexity bonds (e.g. 5 year ZCB). Hence the greater the convexity.
(! ) $
% change in price: = −𝐷1 𝑑𝑟 + convexity measure× 𝑑𝑟 3) For a given yield and modified duration, lower coupon rates will have
! $
smaller convexity.
Value of convexity => Coupon effect and maturity effect. But the maturity effect dominates.
Main driver.

Effective convexity (approximation)

!! "!" #$!#
𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑐𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 =
!# ∆& $
Note: 𝑃" − 𝑃' − 𝑃# − 𝑃' = 𝑃" + 𝑃# − 2𝑃'
Comparing to Bond A, Bond B increases in price more when interest rates
decline and decreases in price less when interest rates rises. Hence, Bond Yield volatility
B is preferred. => AKA more convexity is always preferred. In practice, bond price changes are products of two factors:
Note: 1) The impact per basis-point change in the yield-to-maturity (Duration /
- Two bonds can have the same duration but different convexities. convexity) => sensitivity
- Convexity is preferred when there is a large expected change in yields 2): the number of basis points in the yield-to-maturity change (Yield
- If you think a bond is overpriced due to convexity, buy the bond with volatility) => magnitude
lower convexity and sell the one with higher convexity => arbitrage. Note: Non-parallel shift in the yield curve occurs frequently in reality.
Hence, sensitivity analysis alone is not enough.
[5] Immunization Duration and immunization strategy Immunization rebalancing (slide 23)
Investment of assets in such a way that the existing business is immune to Recall: Dynamic strategy that involves periodic portfolio rebalancing in response
a general change in the rate of interest. Where investment horizon = Macaulay duration, to changes in interest rates and liability or portfolio payments
=> Strategy used by pension funds, life insurance companies and banks Gain from Reinvestment income is offset by the loss from price income Note: Necessary to rebalance the immunization portfolio in response to
and vice versa. As such, future value remains the same. changes in interest rates and liability or portfolio payments
Example:
Given an insurance company that guarantees interest rate of 6.25% every Immunization of a Portfolio to Satisfy a Single Liability Rebalance as soon as a significant discrepancy in durations between
six months for 5.5 years. Also suppose that the payment made by the 𝐷2 = 𝐷3 liabilities and the immunization portfolio occurs due to
policyholder is $8,820,262. Where D = modified duration. • changes in interest rates
Amount guaranteed to policy holder 5.5 years from now As such, both assets and liabilities will have the same price sensitivity to • payments made by immunization securities
= $8,820,262 (1.06252)11 = $17,183,033 changes in interest rates. • liabilities been paid off
Therefore, the present value of the cash flow from the portfolio equals the
Where a portfolio manager buys a $8,820,262 par value of a bond selling present value of the future liability. (Value of A and L moves together) Immunization limitations
at par with a 12.5% yield to maturity that matures in 5.5 years, the target 1) Immunization risk
amount is never guaranteed. Note: For bonds with given yield and modified duration, the one with • Immunization matches duration, which assumes a flat yield curve
lower coupon rate will have a lower convexity. As such the value for the • Immunization only protects against parallel yield curve shifts
immunised strategy is always higher (convexity effect)
2) Call Risk
Calculating weights of bonds to use to immunize a portfolio of liabilities • When the universe of acceptable issues includes corporate bonds, the
(W1 x m1) + [(1-W2) x m2) = mod dur target yield may be jeopardized if a callable issue is included that is
Find Weights of bonds to purchase to immunize subsequently called.

Find (if not given) price of available bonds 3) Credit Risk and the Target Yield
• The target yield may not be achieved if any of the bonds in the
To immunize, find PV of bonds to be fulfilled given the PV of liability: portfolio default or decrease in value because of credit quality
PV(1) = PV of liability x W1 deterioration.
Note: Coupon interest is unaffected (This is the simple interest) PV(2) = PV of liability x W2
- With new rates, interest on interest will be different => reinvestment Zero-Coupon Bonds and Immunization
income (higher rates => higher reinvestment income and vice versa) Number of 1 to buy = PV(1) / Price of 1 Alternative approach to immunizing a portfolio against changes in the
- Since held to maturity, we are not concerned with the price of the bond Number of 2 to buy = PV(2) / Price of 2 market yield is to invest in zero-coupon bonds with a maturity equal to the
(capital gains) investment horizon.
- Hence total return will be affected and the target value of 17,183,033 is More immunization examples: (slide 18-23)
not guaranteed. Yield changes the next day after you chose the immunization portfolio This is consistent with the basic principle of immunization, because the
duration of a zero-coupon bond is equal to the liability’s duration.
- Also, buying a longer bond does not promise/guarantee anything either.
However, in practice, the yield on zero-coupon bonds is typically lower
than the yield on coupon bonds making this strategy more costly. (aka
Yield changes right before your liability is due more expensive to buy => since high demand)
[6] Corporate Bonds and Credit Risk General principals/guidelines on green bonds (From ICMA) 1) Credit spread risk
- Proceeds: proceeds should be used on eligible green projects with clear Risk that bond price will decline due to an increase in credit spread (above
Corporate Bonds – Debt obligations issued by corporations environmental benefits. a benchmark e.g. treasury yields)
Provisions for paying off bonds prior to maturity - Communicate clearly to investors the objectives of the projects and the Credit spread widens when:
(1) Traditional Call – call provision, call price, deferred call, currently process by which the issuer determine that the project fit within the - Issuer’s creditworthiness declines
callable issue eligible green project categories. - Increase in headline risk
Note: Noncallable-for-life issues are referred to as bullet bonds. - Management of the proceeds - Increase in market liquidity risk
- Reporting
(2) Refunding a bond issue means redeeming bonds with funds obtained => Green bonds have similar structure and risk/return profiles as 2) Corporate downgrade risk
through the sale of a new bond issue, often at a lower interest. => issue conventional bonds. Risk that one or more of an issue’s debt obligations will be downgraded.
new bond at new lower rate and use the proceeds to redeem previous (Rating transition matrix => specifies this information)
bond Challenges: 3) Headline risk
Note: A noncallable bond still provides greater assurance against - “greenwashing,”-- issuers misrepresenting the positive environmental Corporate announcement results in an adverse impact on the credit
premature and unwanted redemption than does refunding protection. impact of bond proceeds. Greenwashing can occur due to the relatively spread, but does not result in an immediate downgrade of debt.
broad criteria for what constitutes a green bond and lack of formal
(3) Make-whole call provision: the payment when the issuer calls a bond is issuance guidelines in many emerging markets. 4) Market Liquidity Risk
determined by the present value of the remaining payments discounted at - price, fungibility, austerity, identification, and ear-marking Risk that the price at which investors can actually transact may differ from
a small spread over a maturity-matched Treasury yield. - Added costs of disclosure and high demand -- More expensive? Few the price indicated in the market (bid-ask spread => liquidity is affected). It
Note: This provision is less likely to be exercised than a traditional call. evidence so far. depends on:
=> usually thrown in with the issue as a sweetener for investors - Relatively small market compared to regular bonds, potential liquidity - The size of the issuer
concerns. - The credit quality of the issuer
(4) Sinking fund requirement : A corporate bond issue may require the Note: During times of financial stress or crisis, market liquidity can decline
issuer to retire a specified portion of an issue each year. => Social Bonds, Sustainability bonds, Sustainability-linked Bonds sharply, causing yield spreads to widen
The design of the sinking fund provision is to reduce credit risk.
Bankruptcy and creditor rights Seniority ranking
Special Structures for High-Yield Corporate Bonds - Debtors who are unable to meet their debt obligations are set to be Refers to the priority of payment, with the most senior debt having the
Deferred coupon structures => Issued by firms involved in LBOs (avoid either liquidated or reorganized. first claim on the cash flows and assets of the issuer.
heavy interest payments for 3 to 7 years) - Chapter 7 => liquidation => once the senior one is paid, remainder goes to the next class
3 types: - Chapter 11 => reorganization Secured debt: the debt holder has a direct claim on certain assets and
1) Deferred interest bonds – sold at deep discount, no interest for initial their associated cash flows.
period A company that files for protection under the bankruptcy act generally Unsecured debt (debenture): unsecured bondholders have only a general
2) Step-up bonds – pay low coupon for initial period and steps up becomes a debtor in possession (DIP), and continues to operate its claim on an issuer’s assets and cash flow
3) Payment-in-kind(PIK) bonds – option for issuer to pay coupons in cash business under the supervision of the court. Note: All creditors at the same level of the capital structure (seniority
or a similar bond ranking) are treated as one class => pari passu
Credit risks
Green Bonds Risk of loss resulting from the borrower (issuer of debt) failing to make full
Fixed income instruments that fund environment improvement projects and timely payments of interest and/or principal.
Issued by: Govt, Corps and Supernations (worldbank, IMF etc) - Default risk (default probability) — the probability that a borrower
defaults
Benefits: - Loss severity (loss given default) — the portion of a bond’s value
- Do good for environment (including unpaid interest) an investor loses.
- Better diversification Expected loss
- Better signalling of issuer = Default probability × Loss severity given default Priority claim is not absolute
- Larger investor base => higher demand; lower borrowing cost = Default probability × (1 – Recovery rate) Lower ranked debt may have claim over more senior ranked debt
[6 CONT] Corporate Yields and Spread Special Considerations on High-Yield Debts
Recovery Rate => portion of debt value that’s eventually recovered Yield on corporate bond = Real risk-free interest rate + Expected inflation Liquidity: Having cash and/or the ability to raise cash (turn assets in cash)
Defaulted debt usually continues to be traded based on the assessment rate + Maturity premium + Liquidity premium + Credit spread is crucial to High-yield companies
that through liquidation or reorganisation, the corporate bonds will have Yield spread = Liquidity premium + Credit spread Issuer liquidity is a key focus of high-yield analysis. Sources of liquidity:
recovery value. Note: Spread and Liquidity premium increases during times of distress (strongest to weakest)
- Varies across industries, business cycles, seniority ranking (by design, Cash on the balance sheet => Working capital => Operating cash flow =>
recovery rates are linked to seniority) Spread Risk: Effect on prices and returns from the change in spreads Bank credit facilities => Equity issuance => Asset sales (fire sale)
Note: Recovery rates are averages => Using Duration and Convexity analysis for change in spreads: Financial Projections:
Return impact ≈ –(MDur × ∆Spread) + ½Cvx × (∆Spread)2 - Scenario analyses on future earnings and cash flows
Example: Promised yield vs Expected return Debt Structure:
Promised return Issuer vs Issue Ratings - Tend to have many layers of debt
Given a high yield bond: FV = $100, c = 10% p.a., 6 months to maturity, P = Issuer credit rating is to address an obligator’s overall creditworthiness. - Too much secured bank debt (“top-heavy”) lowers its debt capacity in
$90 The issuer credit rating usually applied to its senior unsecured debt. the future. => lowers capacity to raise more funds in the future
Corporate Structure:
YTM/2 = 16.67% Issue ratings refer to specific financial obligations of an issuer and takes - Holding companies– parent/subsidiaries structure may lead to lower
Promised yield (YTM) = 16.67% x 2 = 33.33% into consideration such factors as ranking in the capital structure. recovery rate => does the debt sit with the SPV or parent
- Need to know where an issuer’s debt resides (parent versus subsidiaries)
This is the return on condition of no default which may be unlikely as it is a Cross-Default provisions: events of default on one bond trigger default on and how cash can move from subsidiary to parent (“upstream”) and vice
high yield (risk) bond. all outstanding debt — the same default probability for all issues. versa (“downstream”)
Covenant Analysis: important for high-yield debts because of reduced
Expected return Notching – Where specific issues may be assigned different credit ratings margin of errors.
Given CAPM holds and markets are efficient, based on factors other than default probability: - Change of control put – force issuer to buy back bonds
Bond beta = 0.5, MRP = 5%, Rf = 3% => Seniority ranking - Restricted payments – restricting payment to shareholders
=> Expected loss severity - Limitations on liens – protecting unsecured creditors (Recall: Liens are
Expected annual return 3% + 0.5(5%) = 5.5% => Structural subordination secured debt)
6M rate of return = 5.5%/2 = 2.75% - Restricted versus unrestricted subsidiaries
Risks in Relying on Rating Agencies - Restricted subsidiaries offer guarantees to their parent holding company
Where the probability of financial distress (default) is 50% in 6months: Credit ratings are dynamic and hence they can move up/down over time debt
4'% )''"4 " 4'% 6789:*9( &9:;<9&= - Ratings agencies are not infallible => GFC - Also need to know the covenants in an issuer’s bank credit agreements.
Expected 6m return = = 1+ 2.75%
>' - Idiosyncratic risk/events hard to predict Equity-like approach to high-yield analysis:
Therefore; Expected Recovery = $79.95
- Ratings tend to lag market pricing of credit - View high-yield debt as the mixture of investment grade bond and equity
) " 6789:*9( ?9*@&, ∗ !&B:9 – )#8&;D ∗ :"8.&
Note: Odds are higher to go from a higher level to a lower level => hence
Expected Recovery ($) = the concern is being downgraded Credit Analysis on Sovereign Debts
8&;D
Key issues of sovereign analysis:
Recall: Price = (1-prob)x(c+par) + [prob X expected recovery] 4Cs of Credit Analysis - a government’s ability to pay
Note: This is the combined expected payoff Capacity – ability of borrower to make debt payments on time - its willingness to pay
=> Industry structure (porters 5 forces)
Ratings Agencies and Credit Ratings Collateral – quality & value of assets supporting issuer’s indebtedness Non-sovereign government bonds
Rating Categories => Industry & company fundamentals (cyclical/non-cyclical & growth General obligation bonds (GO) – unsecured bonds issued with the full faith
Investment grade: prospects): ratio analysis and credit of the issuing non-sovereign government. These bonds are
Aaa, Aa, A, Baa by Moody’s ratings Covenants – T&Cs of lending agreements issuer must comply with supported by the taxing authority of the issuer.
AAA, AA, A, BBB by S&P/Fitch ratings => negative and affirmative
Speculative grade or “Junk” bonds: Character – quality of management Revenue bonds – issued for specific project financing
Rated below Baa by Moody’s and BBB by S&P/Fitch The bottom line: Factors that makes the firm go out of business easily = Have a higher degree of risk than GO bonds because they are dependent
higher credit risk on a single source of revenue.
[7] Mortgage-Backed and Asset-Backed Securities Mortgage calculation (Cont) Cash Flows of Pass-Throughs
Basics features of the mortgage market Remaining mortgage balance at the end of any month t: Cash flows from MBSs are generated from the cash flows from the
Mortgage - loan collateralized with real estate (e.g. house/commercial ()"B)& #()"B)% underlying pool of mortgages, minus servicing and other fees.
𝑀𝐵* = 𝑀𝐵'
property). ()"B)& #) - Weighted Average Coupon Rate (WAC)
Foreclosure - The mortgage gives the lender the right to sell the Portion of the monthly mortgage payment that is the scheduled principal - Weighted Average Maturity (WAM)
mortgaged property to recover funds toward satisfying the debt obligation payment for a month: - Pass-through Rate (PT Rate): Interest on MBS; lower than WAC (aka
B()"B)%"(
if the borrower defaults. 𝑆𝑃* = 𝑀𝐵' , Also 𝑀𝑃 = 𝑆𝑃 + 𝑖 𝑀𝐵*#) spread)
()"B)& #)
• Recourse loan – lender has a claim against the borrower for the Prepayment and Cashflow
NOTE: For ARMs, the monthly mortgage payment adjusts periodically
shortfall between the amount of the outstanding mortgage balance Cash flows of a pass-through security are unknown due to prepayment
based on the new rate. (recasting the loan)
and the proceed received from the sale of the property. => Models used to estimate prepayment (prepayment speed aka speed)
• Nonrecourse loan – lender does not have claim and can look only to Securitization based on:
the property to recover the outstanding mortgage balance – “strategic - Refinancing incentive
Process in which the assets of a corporation or financial institution are
default”. - Seasoning
pooled into a package of securities backed by the assets – Assets-Backed
Note: Residential mortgages are non-recourse loans in many states of US, Securities (ABS) - Monthly factors
but recourse loans in most European countries.
=> An originator that sells the assets (e.g. mortgages, AR) to an issuer
Public Securities Association (PSA) model - based on seasoning
Common types of ABS: secured by mortgages (largest aka MBS),
Types of Residential Mortgage Loans (Credit Classification) Prepayment tends to be greater during the early part of the loan, then
automobile loans, credit card receivables, and home equity loans.
Prime loan – originated loan where the borrower has a high credit quality stabilize after about three years
Subprime loan – where borrower is of a lower credit quality/loan is not Annualised prepayment speed => conditional prepayment rate (CPR)
Mortgage Backed Securities (MBS) / Pass-throughs
first lien
Asset backed securities formed with mortgages • PSA Model: Note:
Note: First lien refers to where the lender would have the first call on the Created by pooling mortgage loans and issuing certificates entitling the - Coeff. Of PSA measures the
proceeds in the case of liquidation and repossession of the property.
investor to receive a pro rata share in the cash flows of the specific pool of CPR (%)
speed of prepayment
mortgage loans that serves as the collateral for the security. - Payment rate increases till
Important ratios => Only one class of bondholders, sometimes referred to as single-class
9.0 • 150 PSA
6.0 • 100 PSA
approx. 30 months where it
Loan to value ratio (LTV) – loan amt/mkt value of property (most impt 3.0 • 50 PSA
MBS. stabilizes
indicator in predicting default) e.g. 90% LTV = 10% downpayment 0.2
Month - 100 PSA => standard
Front ratio - total monthly payments/borrower’s monthly pretax income Agency Pass-throughs
0 1 30 360
(benchmark)
Back ratio – similar to front ratio but with other loan payments added to
MBSs created by the following agencies are referred to as agency pass- *
the numerator 𝐶𝑃𝑅 = 0.06 , if 𝑡 ≤ 30,
throughs. E'
- Government National Mortgage Association (GNMA) 𝐶𝑃𝑅 = 0.06, if 𝑡 > 30.
Interest rate - Federal National Mortgage Association (Fannie) Note: Steady state at t = 30 months
Referred to as the note rate: fixed or change over the life of the loan.
- Federal Home Loan Mortgage Corporation (Freddie)
Fixed-rate mortgage (FRM) - interest rate is set at the closing of the loan
Note: Guaranteed by the agencies, and the loans they purchase must be Monthly prepayment rate (Single Monthly Mortality Rate (SMM))
and remains unchanged over the life of the loan.
conforming loans, meaning they meet their underwriting standards. 𝑆𝑀𝑀 = 1 − 1 − 𝐶𝑃𝑅 )/)$
Adjustable-rate mortgage (ARM) - note rate is based on the index Hence, credit risk is minimal as compared to other types
(reference) rate, and a margin over the index
Prepayment for the month t:
Conventional Pass-Throughs (non agency pass-throughs) SMM × (beginning mortgage balance for month t – scheduled principal
Calculation of Mortgage Payments (for fully amortising fixed rate loans) Sold by commercial banks, savings and loans, other thrifts, and mortgage
B()"B)& payment for month t)
𝑀𝑃 = 𝑀𝐵' (Financial calculator can be used to solve for PMT) bankers.
()"B)& #)
Where 𝑀𝑃 = monthly mortgage payment ($), 𝑀𝐵 0 = original mortgage AKA private labels, are often formed with nonconforming mortgages; that Cashflow from a mortgage portfolio:
balance ($), is, mortgages that fail to meet size limits and other requirements placed CF = Interest + Scheduled principal + Prepaid principal
𝑖 = note rate divided by 12, and 𝑛 = number of months of the mortgage on agency pass-throughs.
loan. Therefore, the credit risk is higher for conventional than agency.
*Recall: formula for solving the constant payment in an ordinary annuity.
[7 CONT] CMO cont Nonagency Residential Mortgage-Backed Securities (RMBS)
Cashflow yield (for pass-throughs) => makes it comparable to other bond Sequential-Pay Tranches – each class of bond would be retired Typically backed by nonconforming mortgages. => more credit risk than
Semiannual cash flow yield = (1 + yM)6 – 1, where yM is the monthly int rate sequentially. (sequential-pay CMOs) Agency MBS
Bond equivalent yield = 2[(1 + yM)6 – 1] Sequential payment rules Credit enhancement : Securities without a government guarantee or a GSE
- Each tranche receives periodic coupon interest payments based on the guarantee must be structured with additional credit support to receive an
Average life (of a MBS) outstanding balance at the beginning of the month. investment-grade rating.
The average time to receipt of principal payments (scheduled principal - Each tranche not entitled to receive principal until the entire principal of
payments and projected prepayments), weighted by the amount of the preceding tranche has been paid off. Four forms of credit enhancement:
principal expected. Note: Tranches can have average lives that are both shorter and longer 1)senior-subordinated structure
* HIJKLJHMN IOLOJPOQ MR RJSO R than the collateral => investors of different preferences than the collateral 2) excess spread
average life = ∑ G*+) T = number of months
)$ (RTRMN HIJKLJHMN)
3) overcollateralization
Note: The average life depends on the PSA assumption (e.g. 100, 150 PSA) Accrual Bonds (aka Z bond/class => like a ZCB) 4) monoline insurance
Does not receive interest – interest is accrued and added to principal bal.
Prepayment risk and negative convexity
=> No upfront payment, essentially lengthening average life. Structural Credit Enhancement (aka credit trenching)
Negative convexity when interest rates are low => The interest not paid used to speed up prepayment of other classes. The redistribution of credit risks among the bond classes comprising the
When interest rates decrease, bond prices increase. However, just like a structure in such a way as to provide credit enhancements by one bond
callable bond, for a MBS, prepayment risk increases, hence cashflows
Planned Amortization Class (PAC) bonds: class to the other bond classes in the structure.
received have to be invested at a lower rate. Therefore, a negative effect
Scheduled principal repayment => more predictable cashflow => Achieved by creating bond classes with different priorities of cashflow
on price. This effect goes against the effect of the increase in price. Hence, Virtually no prepayment risk => comes at the expense of non-pac class (aka senior-subordinated structure)
prices don’t increase as fast and convexity becomes negative.
Note: Duration is not negative but rate of change in duration is negative.
Interest Only (IO) and Principal Only (PO) tranches => Senior bond class: bond class with highest rating
Recall: When interest rates decrease, you want a higher duration and Stripped mortgage-backed securities are created by paying all the => Subordinated bond classes: Bond classes below the senior class
higher convexity in all scenarios
principal to one bond class and all the interest to another bond class. Note: Cashflows are known as waterfall
(STRIPS) Mystery MBS 1 Mystery MBS 2

1200 4500
1100 4000

IO 1000 3500
PO

price
price
900
3000
800
2500
700
600 2000
0.04 0.09 0.04 0.09

interest rate interest rate

Contraction And Extension Risk (same concept)

Contraction risk – Rise in price (for a passthrough) will not be as large as
=> Special case of positive relationship between bond prices and interest
that of an option-free bond as a drop in interest rates will give the
rates. (unlike regular bonds)
borrower an incentive to prepay the loan. (Think: Prepayment risk) For a decrease in interest rates:
=> Shortens the average life
IO => Price increases when rates decrease but cashflow is now negative Note: For non-agency MBS, prepayment risk typically redistributed only at
since less future interest payments as principal is getting paid up. the senior bond class level.
Extension risk – Higher rates tend to slow down the rate of repayment,
Therefore, price goes up initially but decreases when homebuyers start to
increasing the amount invested at the coupon rate (which is lower than aggressively payoff the principal amount Shifting interest mechanism
the market rate)
PO => Price increases; since principal gets paid up earlier as interest rates Redirects prepayments disproportionately from the subordinated bond
=> Lengthens the average life
decrease. class to the senior bond class according to a specified schedule. => usually
done in senior-subordinated structure backed by subprime loans
(Agency) Collateralized Mortgage Obligations (CMOs) (MBS)
Adding them together;
Bond classes created by redirecting cashflows of mortgage-related Just becomes a regular => Scheduled principal payments are allocated based on the senior
products in order to mitigate prepayment risk for some investors. MBS with a negative percentage (interest)
- Tranches (Bond classes)
convexity at lower rates senior prepayment percentage
- Principal payments from underlying collateral used to retire tranches on
= 𝑆𝑒𝑛𝑖𝑜𝑟 interest+𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 × 𝑆𝑢𝑏𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
a priority basis. (prepayment risk is therefore not completely eliminated)
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
[7 CONT] Key Role of the Special Purpose Entity in securitization
Commercial Mortgage-Backed Securities 1) potential for reducing funding cost (as compared to a corporate bond)
Backed by a pool of commercial mortgages on income-producing property => Credit rating of ABS(MBS) is independent of the main co, tied to SPV
such as office building, etc. Hence, some classes may have a credit rating better than the main co
Indicators of credit performance: 2) decouple the credit risk of the entity needing funds from the bond
Loan-to-value ratio (LTV) and Debt-service ratio (DSR) classes issued by the SPV => keeps main company safe
=> Creditors cannot go after the main co to recover loans
Considerable call protection => Sequential pay tranche, prepayment
lockout etc Benefits of securitization
For banks:
HOWEVER, BALLOON RISK - Increase the amount of funds available to lend. => frees up cash on its
balance sheet
Non-Mortgage Asset-Backed Securities - Improve profitability by increasing loan origination and the related fees.
Auto Loan ABS
Backed by auto loans and lease receivables For investors:
Internal Credit Enhancement: - Enables financial innovation, which allows investors to access securities
- A senior/subordinated structure with profiles that match their risk, return and maturity needs that are
- Overcollateralization otherwise not directly available.
- A Reserve account – an excess spread account
For financial market:
Credit Card Receivables ABS - Allows for the creation of tradable securities with better liquidity than
- Cash flows include finance charges, fees, and principal repayments that of the original loans on the bank’s balance sheet.
- Non-amortizing loans
For company:
Securitisation - ABS provide an alternative (and cheaper) means of funding operations
Step 1: Setup a separate legal entity called Medical Equipment Trust that can be considered alongside bond, preferred equity, and common
(MET). This legal entity is called as special purpose entity/vehicle (SPE or equity issuance.
SPV).
=> SPV is normally considered bankruptcy remote from the seller of the Additional Risks of Securitization
loans. Adverse selection:
- Are lowest quality loans securitized?
Step 2: Mediquip sells US$200million of loans to MET (SPV), and receives
from MET US$200 million in cash. Moral hazard:
- Do banks and other originators monitor borrowers less if they sell the
Step 3: MET (SPV) issues and sells securities that are backed by the pool of loans?
securitized loans and receives cash.
=> These securities are the ABS, and the US$200 million of loans represent Complex structures are difficult to price, and very illiquid.
collateral.
=> The periodic cash payments made by Mediquip’s customers are used to
make the periodic cash payments to the ABS investors.
[8] Derivative markets and instruments Futures contract 3-month Eurodollar Futures Contract (cont)
Derivatives Agreement to buy or sell an asset for a certain price at a certain time One tick change = 0.01% (1bp)
Financial instrument that derives its performance from performance of an Note: Similar to a forward but traded on an exchange Hence for every 0.01 (1bp) change in the 90 day LIBOR rate:
underlying asset (underlying) => can be stock, bonds, commodities etc $25 = $1,000,000(.0001) x 90/360 gets transferred between margin
=> Usually transforms the performance of the underlying asset Unique features: accounts
- Gains and losses of buyers and sellers are settled daily (Mark-To-Market) Note: Each contract is for 3 months (90 days), with a 360-day convention.
Forward commitments: - Deposit account – margin account
Where two parties agree (required) to transact on a future date at a Initial margin: Required deposit at initiation of contract E.g. Given a 3m Eurodollar futures maturing in next march quoted at 97.35
predetermined price (e.g. forwards, futures, swaps) Maintenance margin: Amount needed to be maintained after trade is
initiated (usually lower than initial margin) => However, once fall below
Contingent claims: (aka margin called) top up back to initial margin
Where one party has the right but not the obligation to buy/sell to
another party on a future date, at a predetermined price, contingent on
some outcome. (e.g. options)
//FORWARD COMMITMENTS//
Structure of derivative markets
Exchange-traded derivatives markets: Hedging interest rate risk (lock-in rates) with Eurodollar futures contract
- Standardised contracts 1. Do you take a long or short position? What rate do you lock in?
- Exchange verifies execution and record (clears) identity of both parties => Where you are borrowing and think that rates are going up, you would
- Tranfers payments from one party to another (settlement) Note: Some futures have rules on price limits (upper & lower). Once hit, want to lock in the rate today. Hence, SHORT. (When price of bonds go
=> Provides liquidity, Guarantees against defaults (by posting margin) and trading stops down interest rates go up => sell the contract so that immunized when
provides transparency (but lacks privacy and flexibility) prices indeed go up)
Open interest
Over-the-counter markets (OTC) Number of contracts outstanding at any given time – total number of long 2. Given initial quote at 5.85%, where interest rates indeed go up to e.g.
- Customised contracts or equivalently short positions (not both) 6.25%.
- Informal network of dealers => Indicates market interest in the product as well as the trend Gain/Loss = (625 - 585)bps x $25 = $1000
- Less regulated (lacks transparency but more privacy) Note: Beware of double counting
3. Effective borrowing rate: solve for r
Forward contracts Cash from selling contract x [1 + r*(90/360)] = Cash paid to lender
OTC contract between two parties to buy (long) or to sell (short) an Where:
underlying asset, at a specified future time (expiration date), at a fixed Cash from selling contract = $1,000,000+[(625-585)$25] = $1,001,000.
price agreed upon today (forward price/delivery price). Cash paid to lender = $1,000,000(1 + 0.0625(90)/360) = $1,015,625.
Note:
- No initial cash transaction 3 month SOFR futures
- Can be physical delivery or cash settled 3-month Eurodollar Futures Contract Secured interbank overnight interest rate collateralized by Treasury
- Each contract is for $1mm face value of Eurodollar three-month time securities. Quoted 100 - R, where R is the 3m implied SOFR for the ref qtr
Profit/Loss for forwards, F 0(T) = initial price agreed upon deposits paying the rate set in the futures contract. Contract months => Mar/Jun/Jul/Sep (39 qtrs.)
Long: Profit= ST - F 0(T) Short: Profit= F 0(T) - ST - Delivery months: March, June, September and December plus the four Note: Delivery month is 3 months after the contract month unlike 3m
nearest months. Eurodollar (same as contract month) => profit is realised 3 months later
- The contracts are cash-settled (delivery never takes place)
- Quoted in terms of simple interest rates, converted to an index value.
The index equals 100 –𝑖, where 𝑖 is the annualized LIBOR rate.

index price = 𝑃 = 100 − (100×𝑖)

Swaps Options (cont) Principle of no arbitrage
An OTC agreement to exchange cash flows between two parties at Law of one price: Two goods that are perfect substitutes must sell for the
specified future times according to certain specified rules. Payoffs & profit (at expiration) same current price
=> One party pays variable, other party pays fixed/variable payments Long Call: cT = max(0, S T – X) Long Put: p T = max(0, X – S T) => If two assets have the same payoff in ALL states, they must have the
Profit = max(0, S T – X) – c0 Profit = max(0, X – S T) - p 0 same price
E.g. B/E = X + c0 B/E = X - p 0
Interest rate swaps: Fixed for float, floating for float
Currency Swaps: Yen for Euro, Dollar for Pound
Asset or Commodity Swaps
Credit Default Swaps
The derivative price must be determined by the price of the underlying.
Interest rate swaps Otherwise arbitrage opportunity!
Plain vanilla interest rate swap is a widely used interest rate swap.
Short Call: cT = – max(0, S T – X) Short Put: p T = – max(0, X – S T) Pricing of forwards
5% Profit = – max(0, ST – X) + c0 Profit = – max(0, X – S T) + p 0
Party A Party B The initial value of the forward contract is zero: no initial cash outlay. It
LIBOR B/E = X + c0 B/E = X – p 0 will fluctuate as market condition changes.
Note: Value of forward at expiration VT (T ) = ST – F 0(T ), At initiation, V0 (T ) = 0
Fixed and float rates are paid on the notional principal (NP does not => Forward value + Present value of forward price = Spot underlying asset
change hands) price
At inception, cashflows = 0
Portfolio method:
Strategy 1: You hold the stock, valued now at 𝑆G .
Strategy 2: You buy the stock at 𝐹' 𝑇 , which is now valued at 𝑆G
U G
Note: 𝑆' = # + . , 𝐹' 𝑇 = 𝑆' 1 + 𝑟 G
)"&
Call: Put:
ITM: S T > X ITM: S T < X 2 nd approach:
ATM: S T = X ATM: S T = X 1. Buy the forward contract.
OTM: S T < X OTM: S T > X 2. Short the underlying asset and lend/deposit 𝑆0 at rate 𝑟.

//Contingent Claims// Credit Derivatives

Options A derivative contract between a credit protection buyer and seller.
Call option – right to buy the underlying by a specified date (expiration) at This can be thought of buying and selling of insurance against a specific
a certain price (strike)
credit loss.
Put option – right to sell the underlying by a specified date (expiration) at
a certain price (strike) Credit Default Swap (CDS)
Note: Options can be Exchange traded or OTC Derivative contract between a credit protection buyer and a credit To prevent arbitrage, we must have:
protection seller, in which the buyer makes a series of payments to the 𝐹' 𝑇 = 𝑆' 1 + 𝑟 G
Basic Option terminologies: seller and receives a promise of compensation for credit losses resulting
- Underlying assets
from the default (or other pre-defined credit event) of a third party. Pricing of forwards with dividends and costs
- Strike (exercise) price: Price at which underlying will be bought/sold
𝐹' 𝑇 = (𝑆' −𝛾 + 𝜃) 1 + 𝑟 G , where 𝛾 = cash dividends, 𝜃 = storage cost
- Expiration date: Date at which the option contract expires Note: Buyer forgoes cash dividends/benefits, avoid storage costs
- Option premium/price: Amount buyers pays to the seller for the contract * # G#*
Note: European options can only be exercised at maturity, whereas Value of forward: 𝑉* 𝑇 = 𝑆* − 𝛾 − 𝜃 1 + 𝑟 − 𝐹' 𝑇 1+𝑟
American options can be exercised from anytime till maturity. Note: For Forward Rate Agreements(FRAs), underlying is the interest
rates, hence forward price = forward rate
Pricing of futures Different approach to pricing a SWAP -
=> Similar to that of the forward. - Swaps are priced to be consistent with the term structure of interest
Daily settlement (MTM): Futures price is reset to the settlement price and rates for LIBOR based liabilities.
the value is reset to zero. - For the floating rate payer, the cash flows on a swap are initially
Note: equivalent to a long position in a fixed rate bond priced at par, and an
- When interest rates are constant or independent of futures price, equal value short position in a floating rate bond
forwards and futures are priced the same. - For the fixed rate payer, the equivalent cash position is the exact
- When interest rates are positively correlated with futures price, LONG opposite
futures contract are more desirable than forwards (since higher futures +Pfloat -F(r0) -F(r1) -F(r2) -F(r3) … -F(1+rN-1)
profits can be reinvested at higher future interest rates, though negligible)

-Pfixed +cF +cF +cF +cF … +F(1+c)

Pricing of Interest Rate Swaps
Recall: At initiation, value of a swap = 0 (no cashflows are exchanged)
Fixed rate is set so that : Hence, The current market value of the swap is the difference between
PV of the future fixed-rate payments = PV of the future floating-rate the present value of the fixed and floating payments.
payments.
Note: The Fixed interest rate = Swap rate => Pricing an IRS is to determine Valuing a Floating Rate Bond
Fact: Floating rate bonds always are priced at par at reset dates.
Calculation of Swap Rate Proof:
The payments of the fixed-rate payer are known, the floating-rate At time N-1 there is one remaining payment of principal and interest,
payments are unknown – they depend on the value of the reference rate equal to F×(1 + 𝑟V#) ). => “FV” Its value at time N-1, PN-1, is F× (1 +
(say, future LIBOR) at the reset dates. 𝑟V#) )/ (1 + 𝑟V#) ) = F.
!"#$%& '( )*+, -! * .%&-')
𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔– 𝑟𝑎𝑡𝑒 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 × 3𝑚𝑜𝑛𝑡ℎ 𝐿𝐼𝐵𝑂𝑅 × /01

Stepping back to time N-2, PN-2 = ( F× (𝑟V#$ ) + PN-1)/ (1 + 𝑟V#$ ) = F×(1 +

One can use the forward (futures) contracts to lock in the forward (or 𝑟V#$ )/ (1 + 𝑟V#$ ) = F.
future) rate:
- Future interest payments based on these locked-in rates are used to Continuing in this way, it is clear that the price equals the face value on all
calculate the PV of the floating-rate payments. reset dates, including at time 0.
- The forward rate implied spot rates are used as the discount factors in
calculating the PV of the floating-rate payments. Swap pricing: implications of arbitrage
𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 At initiation, PV of Fixed = PV of Float payments
) Since PV of float payments = Face value of float rate bonds
= , = 1/ (1 + 𝐿𝐼𝐵𝑂𝑅) 1 + 𝑓) 1 + 𝑓$ … 1 + 𝑓V#)
)"&,
Then PV of fixed rate payments = PV of fixed rate bonds

Recall: Bootstrapping
Therefore, fixed rate on the swap is determined by setting the present
value of the future fixed rate payments equal to par.
Present Value (PV) of fixed-rate payments can also be calculated as:
PV of 𝑓𝑖𝑥𝑒𝑑– 𝑟𝑎𝑡𝑒 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠
-./0 12 34516- 7
= 𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒× ∑ 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 × 89:
×𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟

Since PV of fixed = PV of float, Therefore;

𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒 The PV of a swap can be
PV of 𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔– 𝑟𝑎𝑡𝑒 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 thought of as two bonds -
= fixed and float
𝑑𝑎𝑦𝑠 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
∑ 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 × ×𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟
360