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CFA Level II 2009

Equity & Fixed Income

CFA UK Surgery

April 2009

09 J2 Version 1.0

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April 2009

Dear Candidate

Welcome to this CFA surgery hosted by CFA UK and Quartic Training.

With this letter you will find some sample materials that may be referred toduring the surgery: afterwards, please use them to help with your studies overyour revision – you will find many worked examples, each with solutions providedat the end. Although these slides cover only a portion of your curriculum, theyshould help you on some of the tougher topics.

At Quartic we provide a wide range of CFA courses, including: 2 day Start-up for those relatively new to the markets 8 day Education, covering the entire curriculum 5 day Intensive Review, discussing each topic before getting lots of

question practice 6 hour mock.

As you can see, our materials provide you with an intuitive understanding offinance, not just a list of formulae for rote learning. It is this intuition that will getyou through the tougher exam questions.

Course structures are highly flexible, including evening and weekend options. Themodular structure enables you to fit your studies in with other commitments.

We also provide a high level of individual support, with study rooms, breakoutarea, snacks & internet, and of course tutors on hand every day before the exam.There is a tutor helpline until 9 p.m. in the final weeks before the exam to helpour candidates with their queries.

Please don’t hesitate to contact us if you would like to discuss your further CFAstudies.

Yours faithfully

Nicholas J Blain MA FCA FSI CFA

Chief ExecutiveQuartic Training Limited

Level II CFA Professional course timetableJune 2009 examination

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Quartic Training – Class Workbook, CFA Level II 2009

1 © Quartic Training Limited 2009

CFA UK: Equity & Fixed Income1

Quartic Tips for Equity & Fixed Income

CFA Level II 2009

CFA UK surgery

Quartic Training




Asset Valuation and Equity – outline

SS 10: Valuation Concepts

33: A Note on AssetValuation

35: Equity: Markets andInstruments

34: The Equity ValuationProcess

36: Return Concepts





SS 11: Global Context

37: Equity: Conceptsand Techniques

39: Industry Analysis

38: The FiveCompetitive Forces that

Shape Strategy

40: Valuation inEmerging Markets

41: Discounted DividendValuation





SS 12: Valuation Models

42: Free Cash FlowValuation

44: U.S. PortfolioStrategy: Seeking Value –

Anatomy of Valuation

43: Market-BasedValuation: Price

Multiples

45: Residual IncomeValuation




Reading 41

Discounted Dividend Valuation

A Note on Asset Valuation

Page references

CFA Program curriculum: Volume 4, page 281

Stalla Study Guides: SS 10-13, page 11-63

SchweserNotes: Book 4, page 149




Dividends, free cash flow and residual income

LOS 41.a/b: compare and contrast dividends, FCF, and R.I. as measures of cash flow in DCFvaluation, and identify the investment situations for which each measure is suitable.Determine whether DDM is appropriate for valuing a stock;

Subject to management discretion

Requires highly transparentaccounts

Suitable if no dividends or negativeFCF

Book value is starting point forvaluation

Residual incomebased

Can be volatile if the capitalstructure changes one year

Independent of dividend policy andcapital structure

Appropriate for controlling stake(but also for minority)

Free cash flowbased

Inappropriate for firms that do notpay dividends

Minority perspective: dividends donot necessarily reflect profitability

Theoretically justified

Less volatileDividend based

DisadvantagesAdvantagesModel category

When we value a business from future flows of funds, we use a variety of methods. We shallsee dividend, free cash flow and residual income based models.

Hence dividend models are the most suitable when (1) there is a dividend history; (2)dividends and profits are closely related; and (3) the minority perspective is appropriate.




DDM for one, two and multi-periods

One period:

LOS 41.c: Calculate the value of a common stock using theDDM for one-, two-, and multiple-period holding periods;

Two periods:

Multi-periods:

111

0)r1(

PDV

222

11

0)r1(

PD

)r1(

DV

nnn

22

11

0)r1(

PD

)r1(

D

)r1(

DV

Example: It is expected that Weakbow Inc will pay dividends of $3.00, $3.30 and $3.70 at theend of the next three years respectively. If an investor expects to hold Weakbow shares forthree years then sell them for $48, calculate the current value, using an 8% discount rate.

Solution: 65.46$08.1

4870.3

08.1

30.3

08.1

00.3V

320




Gordon growth model

If we adapt the DDM where the value is in a steady state and dividends grow at aconstant rate g, we have the Gordon growth model:

LOS 41.d: calculate the value of a common stock using the Gordongrowth model and explain the model’s underlying assumptions;

gk

D

gk

)g1(DP

e

1

e

00

Example

Wester International’s stock is priced at $118. The company recently paid adividend of $3.50 and has a growth rate of 8%. The risk free rate of interest is 6%and the market rate of return is 10%. Calculate the value of the company’s beta.

Quartic Quote: As well as basic stock valuation, be prepared to solve for any of the variables.

See Solution 5

Solution




DDM variations: two stage, H-model, three stage

LOS 41.j/p: Explain the assumptions and justifythe selection of the two-stage DDM, the H-model,the three stage DDM, or spreadsheet modelling…

We shall now look at some other models. In each case, consider the DDM as anextended NPV calculation. The algebra can look nasty, but if you focus on ourobjective (calculate dividends and discount them) then you can avoid most of it.

The models are as follows:

Initial high-growth phasefollowed by a steady statewith constant growth

Two stage model H-model Three stage model

Initial high growth declinessmoothly to steady statewith constant growth

Allows for 3 phases: growth, transitionaland mature, with high, medium andsteady state dividend growth

Q4 t

growth

gL

Q4 t

gL

gS

2HQ4 t

gL

Spreadsheeting approach: more detailed assumptions could be used with a spreadsheet,since growth rates and present values are straightforward to calculate in such a model.




Two stage DDM

Example

Piccante Inc has just paid a dividend of $2. This is expected to increase by 15% per year for thenext 5 years, after which a steady growth rate of 5% is expected. Using a rate of return of 10%,calculate the value of one Piccante share.

Solution

LOS 41.m: Calculate the value of acommon stock using the two-stage DDM,the H-model, and the three-stage DDM.

See Solution 6

Quartic Quote: Please ignore the algebra in the text – the formula is a variation on theannuity calculation and is unnecessary if you understand the solution shown here.




H-model


The H-model requires a formula:

L

LS0

L

L00

gr

ggHD

gr

g1DV

Q4 t

gL

gS

2H

H is the “half-life”, meaning that the reduction in growth rate takes 2H years.

Quartic Quote: If you break the formula down into the two components, the first part is thesteady state growth model; the second part is the area of the triangle – work it out!

Also please note that this formula in fact is a slight approximation to the manual approach.




H-model


Example

Piccante Inc has just paid a dividend of $2. The dividend growth rate is 20% and will reduceevenly to 4% over the next six years. Using a rate of return of 10%, calculate the value of onePiccante share.

Solution

See Solution 7




Three stage DDM


The three stage model is best applied using a timeline. Let’s demonstrate withtwo examples.

Example

Piccante Inc has just paid a dividend of $2. The dividend is expected to grow by 20% for 2 years,then 12% for the next 3 years, then the steady state growth rate of 4%. Using a rate of return of10%, calculate the value of one Piccante share.

Solution

Drawing the timeline:

T0 T1 T2 T4T3Q4

T5 T6

See Solution 8




Three stage DDM with the H-model


The three stage model can also incorporate the H-model.

See Solution 9

Example

Piccante Inc has just paid a dividend of $2. The dividend growth rate is 20% will continue for twoyears then will reduce evenly to 4% over the following six years. Using a rate of return of 10%,calculate the value of one Piccante share.

Solution




Reading 41: Discounted Dividend Valuation

Quartic Tips

• The DDM has a number of forms, all of which you must be very familiar with.

• You should know the different methods of estimating the required rate of return (CAPM, APT,bond yield plus) as well as the equity risk premium.

• Know how to calculate the HPR and hence the alpha.

• The one year, two year or multi-period holding period model is a basic DCF calculation.

• You must be familiar with the basic constant growth DDM as well as its link to justified P/Ecalculations.

• PVGO identifies how much of a share price relates to future growth.

• You must be able to apply the DDM to multi-stage scenarios: this includes the two stage, H-model and three stage approaches. The three stage model can incorporate the H-model as aterminal value. Some of these models can be reversed to calculate the implied rate of return.

• You should know how to compute sustainable growth, including with DuPont and the PRATmodel.




Reading 42

Free Cash Flow Valuation

Page references







FCFF and FCFE

LOS 42.a: define and interpret free cash flow to the firm(FCFF) and free cash flow to equity (FCFE).

• Free cash flows are cash flows that are “free” for distribution to equity (FCFE) or toall investors (FCFF).

• “Free” means that the company has fulfilled all its obligations in respect of moresenior claimants.

Calculating free cash flow

FCFF is the available cash after all receipts and payments not involving any of theinvestors, either debt or equity. It can be calculated as:

FCFF = CFO + int (1 – t) + CFI

FCFE is the available cash after all receipts and payments not involving the equityinvestors, but after receipts and payments from/to debt investors. Calculation is:

FCFE = CFO + CFI + ΔDebt

Quartic Quote: You will see many other ways to calculate FCF, but they all boil down to thesame thing. You must understand each term here, as this will simplify the rest of the chapter.

CFI is net fixed assetinvestment, usually negative




Free cash flow models

LOS 42.b: Describe the FCFF and FCFE approaches to valuation,and contrast the appropriate discount rates for each model andexplain the strengths and limitations of the FCFE model.

Equity

Debt

Firm value =

Equity value =

• If leverage is stable, FCFE is more direct / easy to calculate than the FCFF

• If the firm varies its payout ratio, then FCFE may be difficult to forecast

1tt

e

t

k1

FCFE

1tt

t

WACC1

FCFF




FCFF and FCFE: the Quartic calculations

LOS 42.d: Discuss the appropriate adjustments to net income, earnings before interestand taxes (EBIT), earnings before interest, taxes, depreciation, and amortization(EBITDA), or cash flow from operations (CFO) to calculate FCFF and FCFE.

You need to be able to compute FCF from a variety of starting points. Ratherthan learning excessively, keep an eye on the target formulae:

FCFF = CFO + int (1 – t) + CFI FCFE = CFO + CFI + ΔDebtand

Notation: NCC = non-cash charges, WCInv = working cap investment, int = interest,t = tax rate, net int = int (1 – t)

n/a+ net int – ΔDebtFCFE

– net int + ΔDebtn/aFCFF

x (1 – t) + (NCC x t)– WCInv –net int to get CFO

x (1 – t) + (NCC x t) – WCInvto get [CFO + net int] then + CFI

EBITDA

– int then x (1 – t) to get NI,then + NCC – WCInv as above

x (1 – t) then + NCC – WCInvto get [CFO + net int] then + CFI

EBIT

+ NCC – WCInv to get CFO+ NCC – WCInv to get CFONet income

FCFEFCFFStarting point

Quartic Quote: Keep a clear head and realize that these calculations are all highly logical.Understanding this table will save you need to memorize so many formulae.




Free cash flow example: Conduit IncWe shall calculate free cash flow to both equity and the firm for Conduit Inc using thefollowing income statement and balance sheet.

Revenues $3,820

– Cost of goods sold (2,570)

Gross profit 1,250

– Depreciation (200)

+ Gain on sale of machine 240

– Other operating expenses (390)

Operating profit 900

– Interest (150)

Earnings before tax 750

– Income tax expense (30%) (225)

= Net income 525

Income statement of Conduit Inc for 20X1

Note: during the yearConduit sold a machine thathad cost $1200 and hadaccumulated depreciation of$900 at the time of disposal

LOS 42.e: calculate FCFF and FCFE given a company’sfinancial statements prepared according to U.S. GAAP orInternational Accounting Standards.




Example: Conduit Inc/2

Total liabilities & equity 3,560 2,990

Owners’ equity

Paid in common stock 500 450

Retained earnings 820 530

Total owners’ equity 1,320 980

Current liabilities 20X1 20X0

Accounts payable 480 440

Income tax liability 160 230

Dividends payable 60 50

Total current liabilities 700 720

Long-term debt 1,200 1,000

Deferred taxes 340 290

Total long-term liabs 1,540 1,290

20X1 20X0

Current assets

Cash 580 230

Accounts receivable 540 450

Inventories 350 370

Total current assets 1,470 1,050

Gross PP&E 3,450 4,000

Accum depreciation (1,700) (2,400)

Intangible assets 340 340

Total non-current assets 2,090 1,940

Total assets 3,560 2,990

Balance sheets (in $) of Conduit Inc for 31 December






Calculation for CFO indirect method

Net income 525

– Gain on sale of machine (240)

+ Depreciation 200

= Items from income statement 485

– Increase in receivables (90)

+ Decrease in inventories 20

+ Increase in payables 40

– Decrease in income tax liability (70)

+ Increase in deferred tax liability 50

Cash flow from operating activities 435


NCC =-240 + 200 + 50= 10

WCInv =90 – 20 – 40 + 70= 100





To calculate CFI, look at the PP&E information:

Opening cost x 4,000

+ Additions x 650

– Cost of disposals (x) (1,200)

= Closing PP&E x 3,450

From last year’s B/S

Balancing figure

From notes provided

From this year’s B/S

Conduit Inc

There was also a disposal: we are told that (1) it had book value of $1200 – $900 =$300, and (2) there was a profit on disposal of $240. Hence sales price was $540.

Overall, CFI = 540 – 650 = $(110).


ΔDebt can be calculated from the balance sheet as $1200 – $1000 = $200.





To conclude, we can now calculate FCF as follows:


FCFF

FCFE

Let’s check our understanding by starting from net income or EBIT:

EBIT = $900

Net income = $525


See Solution 10




Effect of capital transactions

LOS 42.h: Explain how dividends, sharerepurchases, share issues, and changes inleverage may affect FCFF and FCFE.

Certain transactions may impact free cash flow. In particular let us considerdividends, share repurchases, share issues and changes in leverage.

Firstly, understand that FCF represents the cash available for the benefit of investors,and certain transactions come out of this.

FCFE represents what is available for the benefit of stockholders. Dividends, shareissues and repurchases have no impact on FCFE, though all transactions to lenders(e.g. interest, new borrowings) will affect FCFE. For instance FCFE may besignificantly higher if a bond is issued in the year.

FCFF is calculated before any payments to/from debt or equity holders. Hencedividends, interest, share or bond issues will have no impact on FCFF.

Changes in leverage will likewise not affect FCFF. However FCFE will be affected by(1) bond issuance/redemption (issuance increases FCFE in that year), (2) interest(the new bond will incur interest that reduces FCFE).




Free cash flow valuation

LOS 42.j: Discuss the single-stage (stable growth), two-stage, and three-stage FCFF and FCFE models (including assumptions), and explain thecompany characteristics that would justify the use of each model.

Valuing a firm or its equity using free cash flow is a two-step problem.

Step 1 is to calculate FCFE or FCFF – we have now discussed this in detail.

Step 2 is to value each year’s cash flows, discounting appropriately. The modelshere are the same as with dividends, so you should be familiar with the calculations.

Key point: to value equity, calculate FCFE and discount at the cost of equity, ke. Tovalue the firm, calculate FCFF and discount at the WACC.

The constant growth model should look familiar (though note that the growth ratemay differ between equity and the firm):

ee

1

ee

e00

gk

FCFE

gk

)g1(FCFEEquity

f

1

f

f00

gWACC

FCFF

gWACC

)g1(FCFFFirm




Calculate the value of a company using FCF

LOS 42.k: Calculate the value of a companyusing the single-stage, two-stage, and three-stage FCFF and FCFE models.

The models used were seen in the Dividend section previously. For instance, youmay need to compute valuations given:

– two stage model, with high growth for a few years, then dropping to constant growth

– H-model, with high growth gradually declining to constant growth

– three stage model, with either high/medium/constant growth orhigh/declining/constant growth (use the H-model for the latter).

One possible calculation not seen previously is when the discount rate changes. Forinstance, suppose equity holders demand 15% for the first three years, then 10%.To discount future cash flows, you need a combination of the two rates:

T0 T1 T2 T4T3Q4

T5

$1000

1.15 1.15 1.15 1.10 1.10

Suppose FCFE in year 5 is $1000. The present value of this is:

$1000 1.102 1.153 = $543.40.




Reading 42: Free Cash Flow Valuation

Quartic Tips

• If you have a deep understanding of what cash flow is available to benefit (1) all investors or (2)equity holders, this section will be very logical.

• FCFF = CFO + net interest + CFI

• FCFE = CFO + CFI + change in debt

• Both of these can be rearranged so that FCF can be computed from NI, EBIT or EBITDA.

• When valuing, use the appropriate discount rate, i.e. cost of equity or WACC.

• You should know when each method is preferable and how different capital transactions affectthem (or not).

• Once you have calculated FCF, the valuation models are similar to those for dividends.




Reading 45

Residual Income Valuation


Page references







Residual income

• Residual income (RI) is the net income of a firm less a charge for equity capital

• It is a form of economic profit.

• The rationale for such a measure is that net income is after a charge for debt (i.e.interest, but no charge for equity.

LOS 45.a: Calculate and interpret residual income and relatedmeasures (e.g., economic value added, market value added).

Example

Given the following information, calculate residual income:

EBIT $200,000

Less interest expense ($30,000)

Pre-tax income $170,000

Less income tax expense ($50,000)

Net income $120,000

Capital is $1 million, with debt to capital of 40%. Cost of equity is 12%.

Solution

Equity charge is $600,000 x 12% = $72,000

Hence RI = $120,000 – $72,000 = $48,000.




Calculate future values of residual income

LOS 45.c: Calculate future values of residual income, …and calculate the intrinsic value of a share of commonstock using the residual income model.

1t

t1tt

033

22

11

00r1

rBEB...

r1

RI

r1

RI

r1

RIBV

We have described RI as being net earnings less an equity charge. This can be written as:

RIt = Et – (r x Bt-1) = (RoE – r) x Bt-1

expected EPS in year t book value of equityin year t – 1

required rate ofreturn on equity

We can now estimate the intrinsic value of a share as:

Quartic Quote: Get the logic: this model says that book value of equity is the cornerstone of ashare’s value – we then add the present value of future RI. This is a very different model to dividendand cash flow based calculations.




Residual income valuation: example

LOS 45.c: Calculate future values of residual income, …and calculate the intrinsic value of a share of commonstock using the residual income model.

An analyst wishes to value Resilience Inc using residual income. Today, book value of equity is $10per share. The following estimates are provided.

Return on equity will be 15% for three years, with payout ratio of 60%. After this, it is expectedthat the market price will converge to book value. Calculate residual income for each year andestimate the intrinsic value of a share. The cost of equity is 12%.

Residual income

Equity charge

Ending BV

Dividend

Earnings

Beginning BV

Year 3/$Year 2/$Year 1/$

Solution

See Solution 11




Drivers of residual income

• If RoE is equal to the required rate of return on equity, the justified market value of ashare of stock is equal to its book value; if RoE is higher than the required rate of returnon equity, the firm will have positive residual income.

• The fraction in this equation (i.e. the value minus B0) is the additional value generated bythe company’s ability to deliver returns in excess of the cost of equity.

LOS 45.d/e: Discuss the fundamental determinants or driversof RI. Explain the relationship between RI valuation and thejustified P/B ratio based on forecasted fundamentals.

If we assume constant growth of RI, constant RoE and dividend payout, we can show thevaluation of a share to be:

gr

BrRoEBV 0

00

Link to price-to-book ratio

• Residual income models are related to the P/B ratio since the justified P/B is directly linkedto expected future residual income.

• If RoE is greater than the required return on equity, market value will be greater than thebook value, and the justified P/B ratio will be greater than one:

gr

rRoE1

B

P

0

0




Single-stage residual income model

Example

A company has a book value of $30 per share. Return on equity is 15% and itsrequired return on equity is 12%. Dividend payout ratio is 70%. Calculate theintrinsic value per share using the residual income model.

Solution

LOS 45.f: Calculate and interpret theintrinsic value of a share of common stockusing a single-stage residual income model.


See Solution 12




Implied growth rate in residual income

Example

A company’s market price to book ratio is 1.6, with return on equity of 12%. Given anestimated required rate of return of 9% calculate the implied growth rate in residual income.

Solution

LOS 45.g: Calculate an implied growth rate in residualincome, given the market price-to-book ratio and anestimate of the required rate of return on equity.


gr

BrRoEBV 0

00

The RI valuation model we saw was:

This can be rearranged to:(which can be used to calculatethe implied growth rate).

00

0

BV

BrRoErg

gr

rRoE1

B

P

0

0

or

See Solution 13




Multi-stage RI model: continuing residual income

• Can residual income continue forever? A firm with a strong competitive strategy mayachieve RI for an extended period, but the infinite period of the GGM and FCF models maybe inappropriate.

• In modelling RI we need to make an assumption regarding what happens after the forecasthorizon. Such an assumption may be that:

– RI continues at the same level indefinitely

– RI declines to a mean reversion level over time, consistent with the industry

– RI declines to zero over time as RoE and r converge

– RI is zero (and stays there) immediately after the terminal year.

LOS 45.h: Explain continuing RI. List the commonassumptions and justify an estimate of continuing RI at theearnings forecast horizon, given co and industry prospects.

aggressive

conservative

• Residual income will continue after a specified earnings horizon depending on theindustry and the sustainability of a specific firm’s competitive prospects in the long term.

• A persistance factor, ω (omega), is defined as the ratio of one year’s RI to the next:if ω = 1 then RI continues as a perpetuity; if ω = 0 it drops to zero immediately.

• In practice, industry competition is likely to reduce economic profits over time, with RoEconverging to an industry norm.




Multistage residual income model

LOS 45.i: Calculate and interpret the intrinsic value of a share using amulti-stage RI model, given the required rate of return, forecasted EPSover a finite horizon, and forecasted continuing residual earnings.

The final calculation is a multistage RI model with continuing RI after the specified horizon.

The present value of the continuing RI can be calculated as:

r1

RIPV T

1T

Quartic Quote: Although not a proof, think of the extreme points. If ω = 0, PV is just RIT,discounted by one year (T to T-1). If ω = 1, this becomes a perpetuity with constant RI.

An analyst wishes to value Resilience Inc using residual income. Today, book value of equity is$10 per share. The following estimates are provided.

Return on equity will be 15% for three years, with payout ratio of 60%. After this, it is expectedthat the residual income will drop by 20% per year forever. Calculate the intrinsic value of ashare. The cost of equity is 12%.

Let’s revisit the Resilience example and add some extra RI:







E – eq chg 0.33710.3180.30Residual income


Solution

(See previous example for workings)

Value of RI from T3 onwards is:

See Solution 14

LOS 45.i: Calculate and interpret the intrinsic value of a share using amulti-stage RI model, given the required rate of return, forecasted EPSover a finite horizon, and forecasted continuing residual earnings.




Reading 45: Residual Income Valuation

Quartic Tips

• Residual income is a form of economic profit, subtracting an equity charge from accountingearnings.

• RI valuation equals book value of equity plus the PV of all future RI.

• Be able to project RI by calculating earnings and dividends from beginning book values, andhence compute RI from the equity charge.

• The main distinguishing feature of RI valuation is that book value is the first (and often largest)component.

• Be aware of accounting adjustments that may be needed – see FSA for details.

• Be able to solve the formula for implied growth rate.

• Understand the persistence factor, ω, in a multistage model and be able to use it in a valuationcomputation.




Study Sessions 10, 11 and 12

Solutions




Gordon growth model

If we adapt the DDM where the value is in a steady state and dividends grow at aconstant rate g, we have the Gordon growth model:

gk

D

gk

)g1(DP

e

1

e

00

08.0k

08.150.3

gk

)g1(D118

ee

0

ke = Rf + x (Rm – Rf), so 11.2% = 6% + x 4% = 1.3.

hence rearrange: %20.1108.0118

08.150.3ke

Quartic Quote: As well as basic stock valuation, be prepared to solve for any of the variables.

Solution 5

Example

Wester International’s stock is priced at $118. The company recently paid adividend of $3.50 and has a growth rate of 8%. The risk free rate of interest is 6%and the market rate of return is 10%. Calculate the value of the company’s beta.




Two stage DDM

Example

Piccante Inc has just paid a dividend of $2. This is expected to increase by 15% per year for thenext 5 years, after which a steady growth rate of 5% is expected. Using a rate of return of 10%,calculate the value of one Piccante share.

Solution

Calculating the first five dividends manually:

Solution 6

T0 T1 T2 T4

$2.30 $2.645

T3

$3.042 $3.498

Q4

T5

$4.023

Terminal value at T5: 48.8405.010.0

05.1023.4

gk

)g1(DP 5

5

Now use the CF function on the calculator:

CF0 = 0; C01 to C04 are the dividends as shown; C05 = 4.023 + 84.48 = 88.50

Setting I = 10% gives a solution of $63.90.

Quartic Quote: Please ignore the algebra in the text – the formula is a variation on theannuity calculation and is unnecessary if you understand the solution shown here.

x 1.15 x 1.15 x 1.15 x 1.15 x 1.15




H-model

Example

Piccante Inc has just paid a dividend of $2. The dividend growth rate is 20% and will reduceevenly to 4% over the next six years. Using a rate of return of 10%, calculate the value of onePiccante share.

Solution

Applying the formula, using H = 3:

67.50$

00.1667.34

04.010.0

04.020.032

04.010.0

04.012

gr

ggHD

gr

g1DV

L

LS0

L

L00

Solution 7




Three stage DDM

The three stage model is best applied using a timeline. Let’s demonstrate withtwo examples.

Example

Piccante Inc has just paid a dividend of $2. The dividend is expected to grow by 20% for 2 years,then 12% for the next 3 years, then the steady state growth rate of 4%. Using a rate of return of10%, calculate the value of one Piccante share.

Solution

Drawing the timeline:

T0 T1 T2 T4

$2.40 $2.88

T3

$3.2256 $3.6127

Q4

T5

$4.0462

T6

x 1.20 x 1.20 x 1.12 x 1.12 x 1.12


CF0 = 0; C01 to C04 are the dividends as shown; C05 = 4.0462 + 70.13 = 74.18


Terminal value at T5: 13.7004.010.0

04.10462.4

gk

)g1(DP 5

5

Solution 8




Three stage DDM with the H-model

The three stage model can also incorporate the H-model.

Solution 9

Example

Piccante Inc has just paid a dividend of $2. The dividend growth rate is 20% will continue for twoyears then will reduce evenly to 4% over the following six years. Using a rate of return of 10%,calculate the value of one Piccante share.

Solution

The first two dividends are: D1 = $2.40, D2 = $2.88. Then apply the H-model:

96.72$

04.2392.49

04.010.0

04.020.0388.2

04.010.0

04.0188.2

gr

ggHD

gr

g1DV

L

LS2

L

L22


CF0 = 0; C01 = 2.40, C02 = 2.88 + 72.96 = 75.84






To conclude, we can now calculate FCF as follows:

FCFF = CFO + int (1 – t) + CFI = 435 + 150 x 0.7 + (110) = $430

FCFE = CFO + CFI + ΔDebt = 435 + (110) + 200 = $525

Let’s check our understanding by starting from net income or EBIT:

– int then x (1 – t) to get NI, then+ NCC – WCInv as above

i.e. (900 – 150) x 0.7 = 525 (NI)then + 10 – 100 CFO = 435

Now subtract 110 and add 200 to getFCFE = $525

x (1 – t) then + NCC – WCInv toget [CFO + net int] then + CFI

i.e. (900 x 0.7) + 10 – 100 + (110) FCFF = 430

EBIT = $900

+ NCC – WCInv to get CFO

i.e. 525 + 10 – 100 CFO = 435

Now subtract 110 and add 200 to getFCFE = $525

+ NCC – WCInv to get CFO

i.e. 525 + 10 – 100 CFO = 435

Now add 150 x 0.7 – 110 to get FCFF= $430

Net income = $525


Solution 10




Residual income valuation: example

An analyst wishes to value Resilience Inc using residual income. Today, book value of equity is $10per share. The following estimates are provided.

Return on equity will be 15% for three years, with payout ratio of 60%. After this, it is expectedthat the market price will converge to book value. Calculate residual income for each year andestimate the intrinsic value of a share. The cost of equity is 12%.

E – eq chg

BVbeg x 12%

BVbeg + E – D

E x 0.6

BVbeg x 15%

0.33710.3180.30Residual income

1.34831.2721.20Equity charge

11.910211.23610.60Ending BV

1.01120.9540.90Dividend

1.68541.591.50Earnings

11.23610.6010.00Beginning BV


Solution

76.10$12.1

3371.0

12.1

318.0

12.1

30.000.10V

3210

Solution 11




Single-stage residual income model

Example

A company has a book value of $30 per share. Return on equity is 15% and itsrequired return on equity is 12%. Dividend payout ratio is 70%. Calculate theintrinsic value per share using the residual income model.

Solution

g = retention ratio x RoE = (1 – 0.70) x 0.15 = 4.5%


42$

045.012.0

3012.015.030

gr

BrRoEBV 0

00

Solution 12




Implied growth rate in residual income

Example

A company’s market price to book ratio is 1.6, with return on equity of 12%. Given anestimated required rate of return of 9% calculate the implied growth rate in residual income.

Solution

Given that we are not given book values, use the second form of the equation:

P/B = 1 + (RoE – r) / (r – g)

1.6 = 1 + (0.12 – 0.09)/(0.09 – g)

g = 4.00%


gr

BrRoEBV 0

00

The RI valuation model we saw was:

This can be rearranged to:(which can be used to calculatethe implied growth rate).

00

0

BV

BrRoErg

gr

rRoE1

B

P

0

0

or

Solution 13





0534.18.012.1

3371.0

r1

RIPV T

1T



E – eq chg 0.33710.3180.30Residual income


Solution

36.11$

12.1

0534.1318.0

12.1

30.000.10V

210

(See previous example for workings)

Value of RI from T3 onwards is:

Solution 14




Quartic Tips for Equity & Fixed Income

Fixed Income Investments

CFA Level II 2009




Fixed Income Investments – outline

SS14: Valuation Concepts

52: GeneralPrinciples of

CreditAnalysis

54: TermStructure andVolatility of

Interest Rates

55: ValuingBonds withEmbedded

Options

SS15: Analysis and Valuation

56:Mortgage-

BackedSector of theBond Market

57: Europe’s WholeLoan Sales Market

Burgeoning asMortgage Credit

Market Comes of Age

58: Asset-Backed

Sector of theBond Market

59: ValuingMortgage-Backed and

Asset-BackedSecurities

53: TheLiquidity

Conundrum




Reading 54

Term Structure and Volatility of Interest Rates

Term Structure of Interest Rates

Page references







Review of bootstrapping and the spot rate curve

Bootstrapping is the process of calculating theoretical spot rates from on-the-run Treasury bonds.

The process is iterative, starting from the shortest period, and the starting point is the par yieldcurve, the curve of Treasury securities trading at par.

Each step solves the basic valuation equation (noting that the price is always $1000):

Example: You are given that the yields of 1, 2 and 3 year par bonds are 4%, 6% and 7%respectively. Calculate the 1, 2 and 3 year spot rates.

Solutions

nn

2

21 z1

pc

z1

c

z1

cpricebond

%06.6z1000

z1

1060

04.1

6022

2

1 year rate: %4z1000z1

10401

1

2 year rate:

3 year rate: %12.7z1000

z1

1070

0606.1

70

04.1

7033

3

2





Key rate durations

LOS 54.f: Compute and interpretthe yield curve risk of a security ora portfolio, using key rate duration.


Key rate

Yield

MaturityQ4

Original curve

Interest rate risk, as measured by duration, has a significant weakness: it assumes thatmovements in the yield curve are parallel. What if we have a twist? Bonds with different couponstructures (but the same duration) may change price at different rates.

Key rate durations (Thomas Ho, Journal of Fixed Income, Sept 1992) try to describe interestrate risk using different points on the yield curve. They are defined as the change in value of aportfolio for a change in a single spot rate. In percentage terms, they represent the percentagedecrease of a security or portfolio given a 1% increase in one rate on the yield curve, as shown:

Ho described 11 keymaturities, rangingfrom 3 months to 30years. Other rates canbe deducted withlinear interpolation.




Key rate durations: example

LOS 54.f: Compute and interpretthe yield curve risk of a security ora portfolio, using key rate duration.

20 yr10 yr5 yr

0%100%0%Portfolio B

33.3%0%66.7%Portfolio A

Portfolio constitution Rate duration of zeros

20105Duration

20 yr10 yr5 yrMaturity


See Solution 1

Quartic Quote: We have approximated effective duration of a zero to be equal to its maturity.

Example

Suppose we have two portfolios consisting entirely of zero coupon bonds. Portfolio Aconsists of two-thirds in 5 year zeros, one-third in 20 year zeros. Portfolio B is entirelymade up of 10-year zeros. Calculate the effective durations and key rate durations of thetwo portfolios.

20 yr10 yr5 yr

Portfolio B

Portfolio A

Key rate duration

Portfolio B

Portfolio A

Effective duration




Key rate durations: example, cont’d

Although the effective durations of the two portfolios are the same, the key rate durations are quitedifferent.

LOS 54.f: Compute and interpret theyield curve risk of a security or aportfolio, using key rate duration.


Calculate the impact on portfolio values for Portfolios A and B given the following two scenarios:

1. Parallel upward shift in yield curve by 50 basis points

2. Steepening twist, with 5 year rate falling 50 b.p., 10 year rate unchanged, 20 year raterising 80 b.p.

Solution

Quartic Quote: Note that the effective duration is the same as the sum of the key ratedurations.

See Solution 2




Portfolio profiles

Plotting key rate durations enable us to interpret the spread of cash flows in a portfolio:

LOS 54.f: Compute and interpret theyield curve risk of a security or aportfolio, using key rate duration.


Q4

3m 1y 2y … … 25y 30y

Q4

3m 1y 2y … … 25y 30y

Q4

3m 1y 2y … … 25y 30y

Ladder portfolio Barbell portfolio Bullet portfolio

Cash flows are evenlyspread across thedifferent maturities

More cash flows at theshort and long end ofthe maturity spectrumthan in the middle

High concentration ofcash flows at anintermediate maturity

Quartic Quote: It is the distribution of cash flows that provides the key rate durations. Itmay be that the effective durations of these three portfolios is the same.




Yield volatility

Yield volatility is computed using standard deviation (or variance) of interest rate yields.

LOS 54.g: Compute andinterpret yield volatility.


1t

t

y

y

1.Calculate the “daily yield ratio”,comparing each day’s yield to the

previous day’s:

2. Take the natural log of the dailyyield ratio (to make it continuously

compounded):

1t

tt

y

yln100X

3. Calculate variance and standarddeviation for the sample data:

1T

XX

s

2T

1tt

2

4. Annualize the variance:

yearindaysss dailyann

yearindaysss daily2

ann2

Quartic Quote: The understanding is more important than the number-crunching. Yieldvolatility is measuring how much yields are likely to change over a period. However, youshould understand that the volatility is relative to yields: if volatility is 25% and yields are 4%,then the standard deviation of yields is 25% of 4%, i.e. 1% movement in yield.




Yield volatility: example

Example

Calculate the annual yield volatility for the 2 year US Treasury note given the following yieldobservations over a week. Assume 250 working days in a year.

LOS 54.g: Compute andinterpret yield volatility.

5.01%4.89%4.99%5.06%5.13%5.00%Yield

FriThuWedTueMonFriDay


Solution

100 x ln(y.r.)

Yield ratio

FriThuWedTueMonDay

Sample deviation:

Sample variance:

Hence the annualized sample deviation:

See Solution 3




Reading 54: Term Structure and Volatility of Interest Rates

Quartic Tips

• Much of fixed income analysis derives from the yield curve. Know the changes (parallelshift, twist, butterfly shift), which bonds may be used to construct it, and understand thethree theories of its shape (pure expectations, liquidity preference, preferredhabitat).

• Key rate durations describe interest rate risk when the yield curve moves in a non-parallel manner. Make sure you can interpret key rate durations and calculate bondportfolio values in any given scenario.

• Yield volatility measures how much movement there is in yields (as opposed to theactual yield values) and is needed in the next chapter. Understand its calculation andinterpretation, as well as the distinction between historic and implied volatility.





Reading 55

Valuing Bonds with Embedded Options

Valuing Bonds with Options

Page references







Relative value analysis

Popular benchmarks

• the Treasury market

• a specific bond sector with required credit rating, and

• a specific issuer.

Types of yield spreads

• Nominal yield spread

• Zero volatility spread, and

• Option adjusted spread (OAS).

LOS 55.a: Evaluate, using relativevalue analysis, whether a security isundervalued or overvalued.


Yield spreads are used to analyse a bond’s relative value. The spread is theamount of yield produced by a bond above the appropriate benchmark.

Typically, if a security’s spread is higher than the required spread, it is consideredcheap, assuming the same benchmark. But if the security has embedded options,then only the option adjusted spread (OAS) is relevant.

We’ll come back to this later once we have defined OAS fully.




Review of yield spread measures (1 of 3):nominal spread

The nominal spread is the difference between the yield to maturity (YTM) of a particular bondand the benchmark bond.

Example: A 10-year, 5% Treasury bond is trading at par, while a 10-year, 8% corporate bondhas a YTM of 7.40% and is priced at 104.19. Calculate the nominal spread.

Solution: YTM of Treasury bond = 5%, since it is trading at par.

Hence nominal spread = YTMcorporate – YTMTreasury = 7.40% – 5%= 2.40%.

Limitations

The nominal spread (like the YTM) has some limitations:

• This measure fails to take into consideration the term structure of spot rates (it represents asingle discount rate for valuing all the cash flows)

• Nominal spread ignores the possibility of cash flows changing as a result of embedded options.

Question: Assuming that we are using an equivalent option-free Treasury bond as benchmark,what risks does the nominal spread compensate investors for? (Answer in a bit.)

LOS 55.b: Evaluate the importanceof the benchmark interest rates ininterpreting spread measures.





Zero-volatility spread (or Z-spread or static spread): This is a measure of thespread that is added to each spot rate on the benchmark yield curve for the discountedcash flows to equal the bond’s trading price.

Q4

Treasury yield curve

spotrates

maturity

Z-spread

Where the nominal spread is the spread above a singlepoint on the Treasury par yield curve, the Z-spreadrepresents a parallel shift of the entire spot rate curve.

The z-spread and nominal spread will differ more for(1) steep yield curves and (2) amortizing securities.

Review of yield spread measures (2 of 3): Z-spread

Question: Assuming that we are using an equivalent option-free Treasury bond as benchmark,what risks does the Z-spread compensate investors for?






Q4


spotrates

maturity

Z-spread

OAS

option cost

An issuer has two bonds with identical features, except for an embedded option in one of them:

Bond A (option-free): Z-spread = 140 b.p. OAS = 140 b.p.

Bond B (callable): Z-spread = 185 b.p. OAS = 135 b.p.

Which of these two bonds is (1) the cheaper, and (2) the better value?

Review of yield spread measures (3 of 3): OAS



Answer:Wemustcomparelikewithlike.Bischeaper,butAisproviding5b.p.moreyieldforthesamerisks(i.e.credit/liquidity).




Importance of benchmarks



OAS

Zero volatility spread

Nominal spread

Priced risksBenchmarkSpread type

Issuer yield curve

Sector yield curve


Option & liquidity risk

Credit, option & liquidity risk


Issuer spot rate curve

Sector spot rate curve

Treasury spot rate curve

Option & liquidity risk



Issuer spot rate curve

Sector spot rate curve

Treasury spot rate curve

Liquidity risk

Credit & liquidity risk

Credit & liquidity risk

The following table shows different spread measures and the risks priced into eachmeasure, depending on the benchmark:

Quartic Quote: This table is important, but understand its content: using the issuer’s curveremoves credit risk, and measuring the OAS removes option risk.




Binomial model: preamble

We saw at Level I that many bonds have embedded options, such as call/put features,refundability, conversion, prepayment, etc. Depending on who benefits from the option, thevalue of the bond will be different from the option-free equivalent:

In both cases, a higher yield volatility increases the option’s value (increasing the putablebond’s value, decreasing a callable bond).

How do we value such a bond, when we do not know the future cash flows?

In particular, the value of some securities (see MBSs later) is “path dependent”: if rates risethen fall a prepayment option may be less likely to be exercised than if rates fall then rise, evenif the start and end points are the same.

Putable bond

Bond

Put

Callable bond

Bond

Call

LOS 55.e/f: Illustrate the relationship among the values of a callable (putable)bond, the corresponding option-free bond, and the embedded option. Explain theeffect of volatility on the arbitrage-free value of an option.

Value of a callable bond= Value of option-free bond

– value of call option

Value of a putable bond= Value of option-free bond

+ value of put option




Binomial model: structure

We shall construct a “binomial tree” of spot/forward rates, each rate being for a single period.At each point in time, the rate may rise or fall, with 50:50 probability:

LOS 55.c: Illustrate the backwardinduction valuation methodology withinthe binomial interest rate tree framework.


Q4

up

downR0

up

downR1,H

up

downR1,L

R2,HH

R2,HL

R2,LL

The actual rates depend on interest rate volatility: the greater the volatility, the wider apart the“H” and “L” rates will be.

Although rates are likely to be provided, to move vertically up a node you must multiply by e2,i.e. the exponential of 2 x volatility. For example, if R2,LL = 4% and = 20%, then:

• R2,HL = 4% x e2x0.2 = 5.97%

• R2,HH = 5.97% x e2x0.2 = 8.90%.




Arbitrage-free valuation

The concept of arbitrage-free valuation is essential to the binomial model. Essentially thismeans that the model must provide valuations that match the market price, i.e. to remove anyarbitrage possibility.

How does this work?

The rates are constructed one period at a time, given a set of risk-free Treasury bonds exactlymatching the periodicity and dates of the tree. Assume that a period = one year.

R0: this is the one-year spot rate, calculated by finding the yield on a one-year annual-payTreasury bond (i.e. a single cash flow discounted to give today’s market price).

R1: given that we can calculate R1,H from R1,L (as shown above) we can identify (by trial anderror) what value of R1,L produces a theoretical value of a two-year Treasury bond equal to itsmarket price.

And so on. This creates a model that should be able to calculate accurately any Treasurysecurity.

The model has now been “tweaked” to today’s market conditions and from this we can extendthe use of the tree to other bonds that are not risk-free.






Valuation using backward induction

The procedure for using the binomial tree for valuing a Treasury bond is known asbackward induction. This involves starting at the right-hand nodes and steppingback one step at a time until we reach time zero.

Steps:

1. enter all spot rates

2. enter terminal cash flows

3. backwards induct, discounting year at a time.

At each node, do the following:

• discount the two values to the right (i.e. one year ahead) at the current one-year spot rate, taking the average result

• add on any cash flow (i.e. the coupon).

Let’s demonstrate the process with an example.






Valuation using backward induction



Example

You are given a two-year annual-pay 5% option-free Treasury bond, with rate R0 =4% and R1,L = 4.5% and volatility of 20%. Calculate the value of the bond.

Q4

R1,H = 6.7132%V1,H = 983.95 + 50

= $1033.95

R1,L = 4.5%V1,L = 1004.78 + 50

= $1054.78

V2,HH = $1050

V2,HL = $1050

V2,LL = $1050

R0 = 4.0%V0 = 1004.20*

* This value is [(1033.95 + 1054.78) 2] 1.04 = 1004.20

Hopefully, if you understood the idea behind arbitrage-free valuation, you willappreciate that in fact the actual 2-year 5% Treasury bond is trading in the marketfor $1004.20. After all, this was how the R1,L rate was identified.




Backward induction: another Treasury bond



Example

You are given a three-year annual-pay 7% option-free Treasury bond, with rateR0 = 4%, R1,L = 4.5%, R2,LL = 5% and volatility of 20%. Calculate the value ofthe bond.

Q4

R1,H =V1,H ==

R1,L = 4.5%V1,L ==

R0 = 4.0%V0 =

V3,HHH =

V3,HHL =

V3,HLL =

V3,LLL =

R2,LL = 5.0%V2,LL ==

R2,HL =V2,HL ==

R2,HH =V2,HH ==

See Solution 4




Callable and putable bond valuation

If the bond is a callable, we need to apply the call rule at each node. Firstly wemust know when the call option may be exercised (e.g. just at T2 (if European) orat any time after T0 (if American)).

The call rule is that the call value represents an upper cap to the bond price at thatnode. For instance if a calculated value is $1030 and the bond is currently callableat 102, then the price is set to $1020.

Otherwise, the backward induction is identical in process.

For a putable bond, the rule is similar, except that the put represents a lower capon the bond price.

For example, if a bond is currently putable at 97 and a calculated value is $950,then the node price is set to $970.

LOS 55.d/i: Compute the value of a callable bondfrom an interest rate tree. Calculate the value of aputable bond, using an interest rate tree.

Quartic Quote: This new rule allows significant flexibility – for example options may havedifferent strike prices at different dates; or one may be European and the other American.Whatever the options, just identify the specific rule at each date before applying to nodes oneat a time.




Backward induction: a callable bond


Example

You are given a three-year annual-pay 7% bond, callable at T2 only at 101. Ratesand volatility are as before. Calculate the value of the bond.

Q4

R1,H = 6.7132%V1,H = 983.29 + 70

= $1053.29

R1,L = 4.5%V1,L = 1026.67 + 70

= $1096.67

R0 = 4.0%V0 = $1033.63

V3,HHH = $1070

V3,HHL = $1070

V3,HLL = $1070

V3,LLL = $1070

R2,LL = 5.0%V2,LL = 1010.00 + 70

= $1080.00

R2,HL = 7.4591%V2,HL = 995.73 + 70

= $1065.73

R2,HH = 11.1277%V2,HH = 962.86 + 70

= $1032.86

From this you can see that the callable bond is valued at $1033.63, while the equivalentoption-free bond was worth $1035.72. Hence the call option has value $2.09. (This is ratherlow, but it was only exercised at one of the nodes.)

LOS 55.d: Compute the value of acallable bond from an interest rate tree.




Option adjusted spread (OAS)

We saw before how OAS is shown on a diagram, as well as the absence of the option riskfrom its pricing. How exactly is the option risk adjusted out of the yield spread?

LOS 55.g: Interpret an option-adjustedspread with respect to a nominal spreadand to benchmark interest rates.


Working definition: the option adjusted spread is the amount that, whenadded to the rates at each node of a binomial tree, result in a calculatedarbitrage-free value equal to the market price.

How this works in practice is as follows.

Suppose we have a callable corporate bond. Relative to Treasuries this contains credit,liquidity and option risks.

If we were to use the same tree with the benchmark rates, the calculated price is simply toohigh. Let’s say we adjust every rate up (using trial & error) by the same amount until thecalculated price equals the market price. We now have a spread that compensates for allthree risks (in effect, the Z-spread).

Now incorporate the call rule at each node. The model is incorporating the impact of theoption, and the calculated price is lower. Now add a constant spread again until we have themarket price: this is the OAS and compensates for credit and liquidity risks only, since theoption has already been adjusted.




Spreads and relative values

You are given the following information regarding some corporate bonds. For each one, identifywhether it is relatively cheap or expensive and explain why. Spreads are in basis points.

LOS 55.a/g: Evaluate, using relative value analysis, whether asecurity is undervalued or overvalued. Interpret an OAS withrespect to a nominal spread and to benchmark interest rates.

All bonds from thisissuer have same

rating and liquidity.-2010Same issuerB4

Other BBB+ option-free bonds have Z-spreads of 140 b.p.

120150AA corporateBBB+3

Other AA bonds haveOAS of 95 b.p.

9090TreasuriesAA2

50

OAS Conclusion

Other option-freeAA+ bonds have Z-spreads of 40 b.p.

-20TreasuriesAA+1

Market infoZ-sprBenchmarkRatingBond

See Solution 5




Reading 55: Valuing Bonds with Embedded Options

Quartic Tips

• You should be thoroughly familiar with the three spread measures (nominal, zero volatility andOAS) and how they compensate investors given different benchmarks.

• You should understand the construction of a binomial interest rate tree and its dependenceon volatility.

• Be able to use backward induction, given a binomial tree and the description of a bond.

• You should be able to describe arbitrage-free valuation and how it applies to benchmark bondsgiven a binomial tree.

• Understand how call and put options are applied to a binomial tree and calculate values.

• The OAS is the parallel shift in rates that should be applied to a tree that is calculating a bondwith an embedded option. Identify when bonds are cheap or expensive, given spread data.

• Describe how effective duration and convexity can be calculated from a binomial model.

• There are many descriptions of convertible bonds: you should know the definitions and beable to calculate them, given a bond and its straight and equity equivalents.





Reading 56

Mortgage-Backed Sector of the Bond Market

MBS

Page references







Mortgage loan

LOS 56.a: Describe a mortgage loan and illustratethe cash flow characteristics of a fixed-rate, levelpayment, fully amortized mortgage loan.

MBS

The mortgage rate refers to the interest rate for a mortgage loan.

Types of mortgages

• Fixed rate level-payment fully amortized mortgages (FRM)

• Adjustable rate mortgages (ARM)

• Balloon mortgages

• Growing equity mortgages

• Reverse mortgages

• Tiered payment mortgages

• Many others …

Apart from the FRM description, you need no details of the mortgage types.

A mortgage loan is a loan acquired against real estate using it as collateral. Theborrower agrees to pay a predetermined set of payments to pay off the loan overan agreed period.




Mortgage loan, cont’d

MBS

Fixed-rate level-payment fully amortized mortgages (FRM)

• Mortgage rate is fixed for the life of the loan, hence fixed-rate

• Payment amount is fixed over life of the loan, hence level-payment

• Last monthly payment pays-off all remaining principal, hence fully amortized

Fixed-rate, level-pay, fully amortized mortgage

0

500

1,000

1,500

2,000

2,500

21

42

63

84

105

126

147

168

189

210

231

252

273

294

315

336

357

Period

pa

ym

en

t

Loan Amount: 300000

Annual Interest Rate: 7.25%

Term (years): 30

Payment

Period

Payment

Amount Interest Principal

1 2,047 1,813 234

2 2,047 1,811 235

3 2,047 1,810 237

4 2,047 1,808 238

… … … …

357 2,047 49 1,998

358 2,047 37 2,010

359 2,047 25 2,022

360 2,047 12 2,034

interest

principal

payment

LOS 56.a: Describe a mortgage loan and illustratethe cash flow characteristics of a fixed-rate, levelpayment, fully amortized mortgage loan.




Mortgage passthrough

LOS 56.b: Illustrate the investmentcharacteristics, payment characteristics, andrisks of mortgage passthrough securities.

MBS

A mortgage passthrough security (MPS) is a security created from a pool of residentialmortgages. A passthrough may contain only a few loans to thousands of loans with similarcharacteristics. A mortgage is called securitized when it is included in a mortgage pool as acollateral for mortgage passthrough security.

A weighted average coupon (WAC) and weighted average maturity (WAM) for apassthrough security are calculated from the underlying mortgage loans.

The coupon rate on the passthrough security is called the passthrough rate. This is less thanthe WAC as a result of the servicing and guarantee fees.

Payment characteristics are similar to a mortgage loan, except a prepayment assumption isincluded in the cash flow projections. This leads to prepayment risk, which is not present inmost other fixed income securities.

One factor that exacerbates prepayment risk is the feature of FRM mortgages that permitshomeowners to prepay their mortgages without penalty. [In fact, penalties were introduced on somemortgages in the 1990s, though this is ignored for much of this chapter’s analysis.]

The majority of MPSs are called “agency passthrough securities” and are issued by:

• Ginnie Mae: Government National Mortgage Association

• Fannie Mae: Federal National Mortgage Association

• Freddie Mac: Federal Home Loan Mortgage Corporation




These are securities collateralized by mortgages loans. Payments by borrowers consist of(1) interest (including a servicing fee), (2) principal repayments and (3) curtailmentsor principal prepayments.

Mortgage passthrough securities

Although cash flows are monthly, payment amounts are uncertain. The level and timing ofprepayments is unknown and is the source of much of the risk for investors. Prepayment riskincreases as yields fall.

Mortgage

Mortgagepool

Mortgage

Mortgage

MPS 1

MPS 2

MPS 3

INVESTO

RS

Q4

LOS 56.b: Illustrate the investmentcharacteristics, payment characteristics, andrisks of mortgage passthrough securities.

MBS




Prepayment conventions

LOS 56.c: Calculate the prepaymentamount for a month, given the singlemonthly mortality rate.

MBS

Valuing MPSs can be challenging, since cash flows are not known.

A prepayment assumption needs to be made. This has several components.

Firstly there is a monthly and an annual assumption on what proportion ofoutstanding principal will be prepaid in each period.

The monthly rate is the single monthly mortality rate (SMM), and the annual rateis the conditional prepayment rate (CPR).

Note that the SMM in month N is month N’s prepayment as a percentage of {principaloutstanding at start of month N less scheduled repayments for the month}.

Given the compounding nature of prepayments, we can link the two:

12SMM1CPR1 12/1

CPR11SMM or (a bit less intuitively):

The other convention is how we predict what SMM/CPR actually are each month: forthis we use the Public Securities Association (PSA) prepayment benchmark.

Quartic Quote: When you do calculations, you should notice that CPR is just less than 12 xSMM; or SMM is just above one twelfth of CPR. Useful shortcut if you’re short of time!




Single monthly mortality (SMM)

LOS 56.c: Calculate the prepaymentamount for a month, given the singlemonthly mortality rate.

MBS

Example 1

Calculate the SMM for the 15th month of a mortgage, if the outstanding principal of themortgage at the end of the 14th month is $280,000. Month 15 has scheduled principal paymentsof $6,000 and expected prepayments of $2,000.

Solution

Example 2

Calculate the expected prepayment for month 321 of a mortgage, if the forecasted SMM is0.48%, outstanding balance at the beginning of the month is $70,719 and scheduled principalpayment is $1,608.

Solution

Example 3

If SMM is given as 0.48%, calculate the conditional prepayment rate, CPR.

Solution

See Solution 7




PSA prepayment benchmark (also referred as prepayment speeds, PSA speeds)is a convention to represent prepayment pattern of a mortgage pool.

“100% PSA” or “100 PSA” is defined as a CPR of 0.2% in the first month, rising by0.2% per month through to 6% in month 30, then remaining at this rate:

if t 30, CPR = 6% x t 30

if t 30, CPR = 6%

where t is the number of months since the mortgage was originated (loan age).

Alternative assumptions are a linear multiple of 100 PSA, rising evenly to 30 months.

PSA benchmark

LOS 56.d: Compare and contrast the CPR tothe Public Securities Association (PSA)prepayment benchmark.

MBS

Q4

CPR 6%

30 Age in months

100% PSA

50% PSA

150% PSA

3%

9%




Example

Calculate the CPR, for different points on the chart.

Solution

CPR and PSA example

MBS

200 PSA100 PSA50 PSAMonth

10

251

138

20

See Solution 8

Q4

CPR

10

100 PSA

50 PSA

200 PSA

20 138 251

LOS 56.d: Compare and contrast the CPR tothe Public Securities Association (PSA)prepayment benchmark.




Average life

LOS 56.e: Explain why the average life of amortgage-backed security is a more relevantmeasure than the security’s maturity.

MBS

A typical treasury or corporate bond pays off principal at the end of the maturity period(i.e., bullet maturity). But for a typical mortgage pool principal is repaid over the life ofthe pool (i.e., an amortizing instrument). The term “maturity” for a mortgage poolrefers to the last scheduled principal payment due, which may of little use to aninvestor who will have received virtually all of their principal prior to this date.

A measure similar to treasury bond maturity for a mortgage pool is “weightedaverage life (WAL)” or “average life”. The longer the WAL, the higher the interestrate risk.

Since a mortgage loan can be prepaid, the higher the prepayment speed, the shorterthe WAL. Shorter WAL results in lower interest rate risk but higher reinvestmentrisk.

The following table shows examples of how the WAL changes for a 30 year bond with aprepayment speed assumption:

4.88 yr8.87 yr14.68 yr20.40 yr30 yr

400 PSA200 PSA100 PSA50 PSA0 PSA




Prepayment risk

LOS 56.f: Explain the factors thataffect prepayments and the types ofprepayment risks.

MBS

Factors affecting prepayment risk

• Level of mortgage rates (lower rates precipitate refinancing of loans)

• Path of mortgage rates (prepayments are path-dependent – see later)

• Housing turnover (good economic conditions or lower rates encouragehomeowners to upgrade and hence prepay their existing mortgages)

• Seasoning (young mortgages will prepay less)

Types of prepayment risk

Contraction risk – A reduction in the mortgage rates increases prepayments,thus shortening average life and increasing reinvestment risk.

Extension risk – An increase in the mortgage rates decreases prepaymentswhich lengthens average life thus increasing interest rate risk.

Quartic Quote: Prepayment risk produces similar concerns to investors as callable bonds,with price compression and negative convexity at low rates.




Collateralised mortgage obligation (CMO)

LOS 56.g: Illustrate how a CMO is createdand how it provides a better matching ofassets and liabilities for institutional investors.

MBS

Tranche types:

• Sequential-pay tranche

• Accrual tranche

• Planned amortization class (PAC)

• Supporting class

• Interest-only (IO) and principal-only (PO) tranches

• Floaters and inverse floaters

A collateralised mortgage obligation (CMO) is a structured security which usesmortgage backed securities as collateral.

A CMO has multiple groups with many tranches in each group. Each tranche hasdifferent prepayment risk characteristics such as extension and contraction risks. ACMO cannot eliminate prepayment risk but it can distribute the risk among differentclasses of bonds by redistributing the collateral cash flows.




CMO sequential pay tranches

LOS 56.h: Distinguish among the sequential paytranche, the accrual tranche, the planned amortizationclass tranche, and the support tranche in a CMO.

MBS

Sequential-pay tranche

All principal payments received from underlying collateral will be paid out to the senior class (here,Tranche A) until it is paid off and then entire principal received will be paid out to next most seniorclass (Tranche B) until it is paid off and so on. Interest paid to each class is based on the unpaidprincipal balance on that class at the beginning of the month.

Unpaidprincipalbalance

Age of the mortgage security

Average tranche life (months)

82725521500

93897143200

1241098261100

157127967850

180150110900

ZCBAPSA

Accrual tranche (Z-tranche)

The accrual tranche, also known as the Z-tranche, accumulates all payments, including principaland interest, until principal for the rest of the classes in the group are paid off and then trancheZ receives payments. Typically all sequential tranche CMOs contain at least one accrual tranche.

Tranche A

Tranche C

Tranche ZTranche B

no interest paid on Z:




CMO planned amortization class tranches

MBS

Planned amortization class

The prepayment risk in a mortgage security can be managed by making cash flows more predictable,if actual prepayment speed is with in a specified band. This is known as planned amortized class(PAC). PAC tranches have a higher priority (seniority) than the other classes in a CMO structure.

Initial PAC collar

The original minimum and maximum PSA speeds for the PAC tranche are known as the initial PACcollar or initial PAC band. The effective PAC collar depends upon the size of the supporting classesand previous prepayments.

Age of the mortgage security

Unpaidprincipalbalance

PACTranche A

SupportTranche S

7221300

8962200

10962100

1276250

150900

SupPAC – A*PSA

Sample average life (months)

*PAC-Ahas collar:50-200PSA





MBS

Support tranches

Support tranches, as the name implies, protect PAC tranches from prepaymentrisk, providing two-sided prepayment protection.

If the actual prepayments are within the specified PAC band then the supporttranche does not contribute to the PAC – however as soon as actual prepaymentsleave the band (either way) then the support kicks in.

Support tranches are typically structured into multiple levels, with descendingpriorities. Each support tranche can again be structured into PAC tranches, knownas PAC II and PAC III etc. The PSA band of the PAC II tranche is narrower than thePAC I tranche, PAC III tranche’s PSA band (collar) is narrower than the PAC IItranche and so on.

In general, support tranches have the greatest level of prepayment risks, bothextension risk and contraction risk, since they absorb shocks created byprepayments, while protecting the PAC senior tranches.

CMO planned amortization class tranches





Stripped MBS

LOS 56.j: Explain the investmentcharacteristics of strippedmortgage-backed securities.

MBS

Principal-only (PO) strip

PO strips receive only the principal component of the payments from theunderlying collateral, purchased at a deep discount.

Interest-only (IO) strip

IO strips receive only the interest part of the payments from the underlyingcollateral, and receive no principal (thus no par value).

PO strip

MPS

Q4

Price

Rates

IO strip

Quartic Quote: Interest only strips have the curiousfeature of negative duration, since they move in thesame direction as interest rates (at low rates). The IOstrip receives no principal, so when rates are low andprepayments high, the result is that the IO strip’s cashflows stop much earlier, with no compensation. The POstrip, by contrast, has extremely high interest rate risk,since higher rates mean not only a higher discount rate,but also a longer period to wait for the cash flow.




CMO tranche analysis

LOS 56.i: Evaluate the risk characteristics andrelative performance of each type of CMO tranche,given changes in the interest rate environment.

MBS

Behaves as sequential-pay tranche

Behaves as sequential-pay tranche

PAC tranche (actual PSAoutside the collar)

Value decreasesValue increasesIO tranche

Contraction riskExtension riskSupporting tranche

Value increases due tolower interest rates

Value decreases due tohigher interest rates

PAC tranche (actual PSAwith in the collar)

Value increasesValue decreasesPO tranche

Contraction riskExtension riskAccrual tranche

Senior tranches facecontraction risk

Junior tranches faceextension risk

Sequential-pay tranche

Decreasing interestrates

Increasing interestrates

Tranche Type




Reading 56: Mortgage-Backed Sector of the Bond Market

Quartic Tips

• You should understand the basic features of the American mortgage market, in particular fixed-rate level payment fully amortized mortgages.

• You need to know the characteristics of passthrough securities (MPSs), including the nature ofthe cash flows, WAC, WAM and WAL.

• Know how to calculate SMM and CPR from one another and be able to apply the PSAbenchmark. You should appreciate that this is a cash flow convention rather than a directvaluation model.

• Understand prepayment risk, including contraction and extension risks.

• There are various types of CMO. For sequential pay and PAC tranches, be familiar with howcash flows are distributed amongst the tranches, and how the risks are affected by thestructure.

• MPSs can be stripped into IO and PO strips: know the pricing and interest rate risk of both.

• Know the differences between non-agency and agency MBSs in terms of quality, risk andguarantee.

• Understand the features of CMBSs and how they differ from residential MBSs.

MBS




Study Sessions 14 and 15

Solutions




Key rate durations: example

Example

Suppose we have two portfolios consisting entirely of zero coupon bonds. Portfolio Aconsists of two-thirds in 5 year zeros, one-third in 20 year zeros. Portfolio B is entirelymade up of 10-year zeros. Calculate the effective durations and key rate durations of thetwo portfolios.

Portfolio constitution Rate duration of zeros

20 yr10 yr5 yr

0100Portfolio B

6.6703.33Portfolio A

Key rate duration

100% x 10 = 10Portfolio B

(66.7% x 5) + (33.3% x 20) = 10

Quicker: 3.33 + 6.67 = 10

Portfolio A

Effective duration

20105Duration

20 yr10 yr5 yrMaturity


Solution 1

Quartic Quote: We have approximated effective duration of a zero to be equal to its maturity.

20 yr10 yr5 yr

0%100%0%Portfolio B

33.3%0%66.7%Portfolio A




Key rate durations: example, cont’d

Although the effective durations of the two portfolios are the same, the key rate durations are quitedifferent.


Calculate the impact on portfolio values for Portfolios A and B given the following two scenarios:

1. Parallel upward shift in yield curve by 50 basis points

2. Steepening twist, with 5 year rate falling 50 b.p., 10 year rate unchanged, 20 year raterising 80 b.p.

Solution

1. Using effective duration, both portfolios will fall in value by 0.5% x 10 = 5%.

2. Using key rate durations, the movements in the two portfolios will be:

Portfolio A: (3.33 x 0.5) + (0 x 0) + (6.67 x -0.8) = -3.67%

Portfolio B: (0 x 0.5) + (10 x 0) + (0 x -0.8) = 0%.

Quartic Quote: Note that the effective duration is the same as the sum of the key ratedurations.

Solution 2




Yield volatility: example

Example

Calculate the annual yield volatility for the 2 year US Treasury note given the following yieldobservations over a week. Assume 250 working days in a year.

5.01%4.89%4.99%5.06%5.13%5.00%Yield

FriThuWedTueMonFriDay


Solution

2.4244-2.0244-1.3931-1.37392.5668100 x ln(y.r.)

1.02450.98000.98620.98641.026Yield ratio

FriThuWedTueMonDay

Sample deviation: s = 2.2574 (daily volatility)

Sample variance: s2 = 5.0960

Hence the annualized sample deviation = 2.2574 x √250 = 35.69%.

Solution 3




Backward induction: another Treasury bond


Example

You are given a three-year annual-pay 7% option-free Treasury bond, with rateR0 = 4%, R1,L = 4.5%, R2,LL = 5% and volatility of 20%. Calculate the value ofthe bond.

Q4

R1,H = 6.7132%V1,H = 983.29 + 70

= $1053.29

R1,L = 4.5%V1,L = 1031.00 + 70

= $1101.00

R0 = 4.0%V0 = $1035.72

V3,HHH = $1070

V3,HHL = $1070

V3,HLL = $1070

V3,LLL = $1070

R2,LL = 5.0%V2,LL = 1019.05 + 70

= $1089.05

R2,HL = 7.4591%V2,HL = 995.73 + 70

= $1065.73

R2,HH = 11.1277%V2,HH = 962.86 + 70

= $1032.86

Solution 4




Spreads and relative values

You are given the following information regarding some corporate bonds. For each one, identifywhether it is relatively cheap or expensive and explain why. Spreads are in basis points.

Bond is callable. It isexpensive as OAS isnegative compared to

similar risk.

All bonds from thisissuer have same

rating and liquidity.-2010Same issuerB4

Bond is callable. OAS of120 < 140 hence bond

is expensive.

Other BBB+ option-free bonds have Z-spreads of 140 b.p.

120150AA corporateBBB+3

Bond is option-free. 90< 95 hence expensive.

Other AA bonds haveOAS of 95 b.p.

9090TreasuriesAA2

50

OAS

The bond is putable, asZ < OAS. 50 > 40

hence bond is cheap.

Conclusion

Other option-freeAA+ bonds have Z-spreads of 40 b.p.

-20TreasuriesAA+1

Market infoZ-sprBenchmarkRatingBond

Solution 5




Single monthly mortality (SMM)

MBS

Example 1

Calculate the SMM for the 15th month of a mortgage, if the outstanding principal of themortgage at the end of the 14th month is $280,000. Month 15 has scheduled principal paymentsof $6,000 and expected prepayments of $2,000.

Solution

SMM is the prepayments as a percentage of “available” principal:

SMM = $2,000 ($280,000 – $6,000) = 0.00730 or 0.73%.

Example 2

Calculate the expected prepayment for month 321 of a mortgage, if the forecasted SMM is0.48%, outstanding balance at the beginning of the month is $70,719 and scheduled principalpayment is $1,608.

Solution

Prepayment = 0.0048 x ($70,719 – $1,608) = $331.73.

Example 3

If SMM is given as 0.48%, calculate the conditional prepayment rate, CPR.

Solution

CPR = 1 – [1 – 0.0048]12 = 0.0561 or 5.61%.

Solution 7




Example

Calculate the CPR, for different points on the chart.

Solution

CPR and PSA example

MBS

200 PSA100 PSA50 PSAMonth

4%2%1%10

12%6%3%251

12%6%3%138

8%4%2%20

Calculation examples:

CPR(Month 10, 200 PSA) = (2 * 6) * (10/30) = 4%, since (t < 30)

CPR(Month 138, 50 PSA) = (0.5 * 6) = 3% , since (t > 30)

CPR(Month 251, 100 PSA) = (1 * 6) = 6%, since (t > 30)

Solution 8

Q4

CPR

10

100 PSA

50 PSA

200 PSA

20 138 251

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