Download Best Credit Data Bond Analytics Calculation Methodology Created by

Best Credit Data Bond Analytics Calculation Methodology
Created by:
Pierre Robert
CEO and Co-Founder
Best Credit Data, Inc.
50 Milk Street, 17th Floor
Boston, MA 02109
Contact Information:
[email protected]
1-978-502-2792
Last updated: March 1, 2016
YieldtoMaturity
Theyieldtomaturityformulaisusedtocalculatetheyieldonabondbasedonitscurrent
priceonthemarket.Theyieldtomaturityformulalooksattheeffectiveyieldofabond
basedoncompoundingasopposedtothesimpleyieldwhichisfoundusingthedividend
yieldformula.
Noticethattheformulashownisusedtocalculatetheapproximateyieldtomaturity.To
calculatetheactualyieldtomaturityrequirestrialanderrorbyputtingratesintothe
presentvalueofabondformulauntilP,orPrice,matchestheactualpriceofthebond.Some
financialcalculatorsandcomputerprogramscanbeusedtocalculatetheyieldtomaturity.
The yield to maturity is found in the present value of a bond formula:
For calculating yield to maturity, the price of the bond, or present value of the bond, is
already known. Calculating YTM is working backwards from the present value of a bond
formula and trying to determine what r is.
Example of Yield to Maturity Formula
The price of a bond is $920 with a face value of $1000 which is the face value of many
bonds. Assume that the annual coupons are $100, which is a 10% coupon rate, and that
there are 10 years remaining until maturity. This example using the approximate formula
would be
After solving this equation, the estimated yield to maturity is 11.25%.
YieldtoWorst(YTW)
Typicallyusedforcallablebonds.Ifabondiscallablebeforeitsmaturitydate,youwill
comparetheyieldtoeachcalldateandcallpriceaswellasmaturitydateandpriceandthe
lowestyieldingdateisthe(YTW).
Best Credit Data, Inc. 50 Milk Street, Boston MA, 02109
www.bestcreditanalysis.com
(YTW)Steps:
Step1
Notethepriceyoupaidforthebond,includingwhatpercentageofitsfacevalueyoupaid
andhowmanyyearsremainuntilthebond'smaturitydate.
Step2
Writedownandsetasideallcalldatesforacallablebond.
Step3
Assume,forthesakeofexplanation,thatinterestrateswillremainthesame.Foreveryyear
thatremainsbeforethematuritydate,dividetheanticipatedyearlyinterestpaymentbythe
amountofmoneyyoupaidforthebondmultipliedby100.Thiscalculationrepresentsthe
percentageofinterestyouwillearnforeachyearandcalldate.Ifyouplantoreinvestthe
interesteveryyear,calculateanewyearlyinterestpaymenteveryyear.Markwhichyears
arecalldatesandnotetheanticipatedinterestpercentages.
Step4
Subtracttheamountyoupaidforthebondfromitsfacevaluetodeterminetheanticipated
face-valueearnings,whichwillbeidenticalforeachcalldateorthematuritydate.Divide
theseearningsbythebond'sfacevalueandthenmultiplyby100.
Step5
Combinetheface-valuepercentagewiththeinterestpercentageforeachcalldate.Search
forthecalldatewiththelowesttotalpercentage,whichistheyield-to-worstdate.
Best Credit Data, Inc. 50 Milk Street, Boston MA, 02109
www.bestcreditanalysis.com
Modified Duration
Modified duration is a formula that expresses the measurable change in the value of a
security in response to a change in interest rates. Calculated as:
Where:
n = number of coupon periods per year
YTM = the bond's yield to maturity.
Effective Duration
Effective duration is a duration calculation for bonds with embedded options. Effective
duration takes into account that expected cash flows will fluctuate as interest rates
change. Effective duration can be estimated using modified duration if the bond with
embedded options behaves like an option-free bond. This behavior occurs when exercise
of the embedded option would offer the investor no benefit. As such, the security's cash
flows cannot be expected to change given a change in yield. For example, if existing
interest rates were 10% and a callable bond was paying a coupon of 6%, the callable
bond would behave like an option-free bond because it would not optimal for the
company to call the bonds and re-issue them at a higher interest rate.
Z-spread
TheZero-volatilityspread(Z-spread)istheconstantspreadthatwillmakethepriceofa
securityequaltothepresentvalueofitscashflowswhenaddedtotheyieldateachpointon
thespotrateTreasurycurvewhereacashflowisreceived.Inotherwords,eachcashflowis
discountedattheappropriateTreasuryspotrateplustheZ-spread.
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www.bestcreditanalysis.com
Macaulay Duration
The Macaulay duration is the weighted average term to maturity of the cash flows from a
bond. The weight of each cash flow is determined by dividing the present value of the
cash flow by the price, and is a measure of bond price volatility with respect to interest
rates.
Macaulay duration can be calculated by:
Key Rate Duration
Key rate duration is holding all other maturities constant, this measures the sensitivity of
a security or the value of a portfolio to a 1% change in yield for a given maturity.
The calculation is as follows:
Where:
P- = Security's price after a 1% decrease in yield
P+ = Security's price after a 1% increase in yield
P0 = Security's original price
There are 11 maturities along the Treasury spot rate curve, and a key rate duration is
calculated for each. The sum of the key rate durations along a portfolio yield curve is
equal to the effective duration of the portfolio.
Best Credit Data, Inc. 50 Milk Street, Boston MA, 02109
www.bestcreditanalysis.com
Convexity
Convexity is a measure of the curvature in the relationship between bond prices and bond
yields that demonstrates how the duration of the bond changes as the interest rate
changes. Convexity is used as a risk-management tool, and helps to measure and manage
the amount of market risk to which a portfolio of bonds is exposed.
In the example above, Bond A has a higher convexity than Bond B, which means that all
else being equal, Bond A will always have a higher price than Bond B as interest rates
rise or fall.
As convexity increases, the systemic risk to which the portfolio is exposed increases. As
convexity decreases, the exposure to market interest rates decreases and the bond
portfolio can be considered hedged. In general, the higher the coupon rate, the lower the
convexity (or market risk) of a bond. This is because market rates would have to increase
greatly to surpass the coupon on the bond, meaning there is less risk to the investor.
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www.bestcreditanalysis.com
Option-Adjusted Spread (OAS) Definition Option-adjusted spread (OAS) is the spread
relative to a risk-free interest rate, usually measured in basis points (bp), which equates
the theoretical present value of a series of uncertain cash flows of an instrument to its
current market price. OAS can be viewed as the compensation an investor receives for
assuming a variety of risks (e.g. liquidity premium, default risk, model risk), net of the
cost of any embedded options.
General Applications
1. Evaluation of an interest-sensitive fixed-income security (in other words, a security
whose future redemption date and payment stream are influenced by interest rates
through the presence of an implicit embedded option). For example, is a 50 bp OAS
appropriate for an A-rated asset?
2. Facilitates comparison of assets. For example, is a AAA-rated asset at 90 bp over the
Treasury curve a better value than a B-rated asset with a 600 bp spread?
3. Profitability analysis. For example, keep OAS constant and see how price varies as
various factors are changed. Or, keep price constant and see how a change in various
factors impacts the OAS.
4. Product pricing. For example, when discounting interest-sensitive cash flows, what is
the appropriate spread over the risk-free interest rate that compensates the company for
the risks of the product?
5. Firm valuation or valuation of a book of business. For example, when discounting
interest-sensitive cash flows, what spread should be added to the risk-free interest rate in
order to compensate the buyer for the risks inherent in the firm or book?
Calculating OAS
1. Interest rate scenarios are generated stochastically, using a model consistent with the
current term structure and assumed level of volatility.
2. Free cash flows, reflecting interest-sensitive contractual features, are calculated at each
time step along each interest rate path.
3. The free cash flows are discounted at the risk-free interest rate plus a spread (the OAS)
to determine a value at time zero for each path.
4. An average of the path wise values is calculated.
5. Steps 3 and 4 are repeated, and the OAS is solved for which equates the average of the
path wise values to the current market price.
Strengths
1. Allows explicit measurement and representation of risks when pricing products.
2. Stochastic valuation provides a better representation and range of the modeled risk.
3. Stochastic analysis captures the effect of yield curve shape on interest-sensitive
variables and reflects the possibility interest rates will vary in the future.
Weaknesses
1. Measure is contingent on quality of model and assumptions.
2. Market price may not be available.
3. Only OAS’s of securities with similar “embedded options” can be compared.
4. Need for many scenarios may be time-consuming.
Best Credit Data, Inc. 50 Milk Street, Boston MA, 02109
www.bestcreditanalysis.com