微信客服：xiaoxionga100
微信客服：ITCS521
程序代写案例-PPHA 42510
时间：2022-04-25
PV for Uncertain Cash Flows
Lecture 7
Thomas S. Coleman
Harris PPHA 42510
Applied Financial Management
18 April 2022 Draft April 19, 2022
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 1 / 35
Outline
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 2 / 35
Uncertainty versus Risk
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 3 / 35
Uncertainty versus Risk
What Is Uncertainty? What is Risk?
Uncertainty: the spread of cash flows, the cause of risk
• Distribution of cash flows,
• Spread, often summarized by Standard Deviation
Risk: the price or value we assign to the Uncertainty
• Difference in PV between certain and uncertain CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 4 / 35
Uncertainty versus Risk
We are all Sloppy about Uncertainty vs Risk
Everyone uses “Uncertainty” and “Risk” interchangeably
• “Uncertainty” is about nature – spread of CFs
• “Risk” is about us – how much we love or hate uncertainty
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 5 / 35
Uncertainty versus Risk
Standard Deviation (Volatility) – Measures Uncertainty
Reminder about Standard Deviation
• Simply a convenient way to measure spread of distribution
We will often use Standard Deviation
• But nothing magical about it – sometimes use other measures
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 6 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 7 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
We Think We’re Smart, But Why Different Yields for UST
& FIS?
Turns out we don’t even how to PV our FIS bond
Let’s look at UST & FIS, and calculate yields
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• FIS: 30/360, semi 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=
4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 8 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
We Think We’re Smart, But Why Different Yields for UST
& FIS?
Turns out we don’t even how to PV our FIS bond
Let’s look at UST & FIS, and calculate yields
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• FIS: 30/360, semi 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 8 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
Our Discounting for FIS Totally Wrong
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
• FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
This cannot be right – such huge discounting differences
We should discount CFs at (roughly) same rate. Problem is promised CFs:
• The UST CFs will be paid; the FIS promise may be broken
• FIS maybe 90% chance $5, but 10% $0. Can’t even draw CF diagram!
Fact is, we don’t really have a clue how to PV the FIS bond!
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 9 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
CF Diagrams Misleading – Can’t Discount (so far)
We draw FIS and UST same way – but actually very different
• UST: actual known future CFs – we discount to get PV: CF(1+y)n
• FIS: promised CF
• Uncertain, distribution, say 90% $5, 10% $0
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Pictures make it look like we can discount, but we can’t (so far)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 10 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 11 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
Simple Example Shows Problems & Solutions
Real Estate project: invest today, get uncertain payout in 1 year
• Cash flow in one year expected $800
• Equal chance low ($711.41) or high ($888.59)
• (Known) value today is $714.29 and risk-free rate is
7%
$711.41 $888.59
Value of Office Building
1/2 1/2
$800
Avg highlow
• If future CF known, get PV by
discounting
• Uncertain CFs: how do we
“discount” multiple CFs???
• Don’t even know how to draw
CF diagram
$764.29k
PV=$714.29k
rf discount rate 7%
Yr 1
Discounting for Known
(Certain) CFs
$711k
p=1/2
PV=$714.29k
??
Yr 1
How do we Discount with
Multiple Uncertain CFs?
$889k
p=1/2
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 12 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
We Really and Truly Don’t Know What to Do
This is a fundamental conceptual problem
Right picture (uncertain CFs) – don’t know what to discount or how
• Bigger problem than people
usually let on
• Solution looks like simple
discounting
• Allows us to pretend problem
doesn’t exist
$764.29k
PV=$714.29k
rf discount rate 7%
Yr 1
Discounting for Known
(Certain) CFs
$711k
p=1/2
PV=$714.29k
??
Yr 1
How do we Discount with
Multiple Uncertain CFs?
$889k
p=1/2
But I want you to understand why risk-adjusted discounting works
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 13 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 14 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
Our Goal – Method for “Discounting” Uncertain CFs
Two fundamental issues we have to address:
1 How to get (certain) PV from uncertain CFs – this lecture
• Tool to convert uncertain CFs (distribution) to single PV today
• Analogue of simple discounting (PV = CF
(1+y)n ) applied to distribution
2 What is the right “price” of uncertainty – future lectures
This lecture is about first: the how
Preview of answer – looks just like discounting known CFs
• Convert distribution to promised or expected CF
• Add risk premium and discount at risk-adjusted rate: Promised CF(1+y+rp)n
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 15 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
Framework for Discounting Uncertain CFs
Theory issues we need to address
• What do we mean when we say “uncertain cash flows”? (Answer: Cash flow
distribution – graph of amount versus probability)
• Why can’t we just use discounting as we have for certain (known) cash
flows? (Answer: Multiple values, using average CFs simply does not work –
Expected utility and concavity.)
• What is uncertainty versus risk? (Answer: Uncertainty = distribution; risk =
cost or price of uncertainty – Certainty Equivalent.)
• How do we adjust for uncertainty? (Answer: Adjust cash flow distribution
until the expected value of the adjusted CFs equals the CE.)
Practical issue we need to address
• We want a method that works just like discounting, so we can use all the PV
and NPV ideas we have worked on
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 16 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
We Have Two Methods – Risk-Adjusted & Risk-Neutral
Both methods adjust the CF distribution
Risk-Adjusted Discounting – we use this most often
• Adjust CFs themselves
• But looks like discounting: CFadj = CForig1+rp then PV = CFadj1+y
• Used for most of our problems
Risk-Neutral Discounting – use this for options
• Adjust probabilities
• Used for derivatives, mainly options
• “Risk-neutral valuation” is bad name – really “Risk-adjusted probabilities”
You won’t read this lecture in most texts, but it is the way things work
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 17 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 18 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Value of Distribution – Look at Utility
Two things we need to do to PV uncertain CFs
1 Transform future known (certain) cash flows (CFs) into present value
• We know how to do this – Discounting
2 Transform distribution of uncertain CFs into a single known value.
• We do not yet have any idea how to do this.
• Expected Utility – Certainty Equivalent – single CF replaces distribution
• Want “trick” allowing us to continue using Average ($800)
Need to go back to basics: think about utility and value of the distribution
• How do we value the CF distribution?
• Simplify by ignoring discounting, stick to yr 1
• Want method to convert distribution to single,
certain, known value
• Call this Certainty Equivalent
• Also: trick to use $800
$711.41 $888.59
Value of Office Building
1/2 1/2
$800
Avg highlow
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 19 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Averaging Utility – Not CFs
We want utility or value of those two possibilities
We have to average the utility not CFs (expected utility not expected CF)
EU = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch) 6= u (p · Cl + (1− p) · Ch)
To make concrete, Constant Relative Risk Aversion: u(C ) = C
1−γ
1−γ , with γ = 8
Low Avg High
Probability 1/2 1/2
CF Dist’n $711.411 $888.59
Expect CF $800
Utility -15.490E-22 -3.266E-22
Expected Utility -9.378E-22
Certainty Eqv 764.28
But we can always ask “what single CF is equivalent to our utility?”
Solve for Certainty Equivalent:
U (CCE ) = EU (Cl ,Ch) = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 20 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Certainty Equivalent – CF that Gives Correct Utility
utility
utility of average
income, -6.8
income
800 888.59average of
income
u(CE)=.5*u(711.41)+.5*u(888.59)
711.41 CE
764.28
Certainty Equivalent (p=1/2)
average of
utility, -9.4
-3.3
-15.5
Solve for Certainty Equivalent:
U (CCE ) = EU (Cl ,Ch) = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 21 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Distribution → Utility → Certainty Equivalent
So far we have converted distribution to a single number, using expected utility
Next we see how to use Certainty Equivalent to get PV
1 Risk-Adjusted Discounting (risk premium): adjust CFs
2 Risk-Neutral Discounting: adjust probabilities
Finally, ask how we get the adjustments (to CFs or probabilities)
• From the market
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 22 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 23 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
“Trick” Allows Us to Use Original Average – Adjust CFs
Now perform a backwards trick: adjust CFs (in a neat way) so that
CE = p · CF loadj + (1− p) · CF hiadj
utility
income
CE=.5*679.66 + .5*848.93
CE
764.28
Certainty Equivalent (p=1/2)
average of
utility, -9.4
-3.3
-15.5
679.66 848.93
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 24 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs by Risk Premium – To Use Original Average
We have multiple steps:
1 Get CE (certainty equivalent) from dist’n using utility
2 Adjust CFs so they average to CE:
CE = p · CF loadj + (1− p) · CF hiadj
3 Write adjustment as 1/1+rp:
CE = p ·
CF loorig
1+ rp
+ (1− p) ·
CF hiorig
1+ rp
CE =
1
1+ rp
[
p · CF loorig + (1− p) · CF hiorig
]
=
CF avgorig
1+ rp
4 Discount CE at risk-free, original avg CF at rf + rp:
PV =
CE
1+ rf
=
CF avgorig
(1+ rf ) (1+ rp)
=
CF avgorig(
1+ yrisky
)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 25 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs by Risk Premium – To Use Original Average
We have multiple steps:
1 Get CE (certainty equivalent) from dist’n using utility
2 Adjust CFs so they average to CE:
764.28 =
1
2
· 679.66+ 1
2
· 848.93
3 Write adjustment as 1/1+rp:
CE = p ·
CF loorig
1+ rp
+ (1− p) ·
CF hiorig
1+ rp
764.28 =
1
1+ .04673
[
1
2
· 711.41+ 1
2
· 888.59
]
=
800
1+ .04673
4 Discount CE at risk-free, original avg CF at rf + rp:
714.29 =
764.29
1+ .07
=
800
(1+ .07) (1+ .04673)
=
800
(1+ .12)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 25 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs Looks Like Discounting Average CF
Discounting Certainty Equivalent
at Risk-Free Rate
←→ Discounting Avg (Promised) CF
at Risk-Adjusted Rate
PV =
CE
1+ rf
=
CF avgorig
(1+ rf ) (1+ rp)
=
CF avgorig(
1+ yrisky
)
714.29 =
764.29
1+ .07
=
800
(1+ .07) (1+ .04673)
=
800
(1+ .12)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 26 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 27 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Return to UST & FIS
Apply our ideas of Risk Adjusted Discounting
• Ideas and theory can seem complicated
But using these ideas simple
• Everyone in financial markets uses them every day
• Calculate Yield, using “Promised CFs”
• Calculate as if the Promised CF is Known CF
• Yield we get is “Risk Adjusted”
• I did for many years, without knowing or understanding or caring about the
underlying theory
So let’s go back to UST & FIS and see what happens
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 28 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
FIS – Standard Yield Calculations – Promised CFs
Settle 31-dec-15 – Calculate FIS Yield using Promised CF
• Embedded in bond prices – market gives us risk premium
31-dec-2015 Coupon Maturity Price Yield
10-yr US Treasury 2.25% 15-nov-25 99.84375 2.268%
FIS 5.0% 15-oct-25 103.626
4.536%
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• Calculate FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=
4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For
4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 29 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
FIS – Standard Yield Calculations – Promised CFs
Market gives us risk premium – spread = 4.536% – 2.268% = 227bp
• Embedded in bond prices – market gives us risk premium
31-dec-2015 Coupon Maturity Price Yield
10-yr US Treasury 2.25% 15-nov-25 99.84375 2.268%
FIS 5.0% 15-oct-25 103.626 4.536%
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• Calculate FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 29 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Bond Market & FIS Example – Risk-Adjusted Yield
Market gives us risk premium – embedded in price (settle 31-dec-15)
• UST: 2.25% coupon, 15-nov-25 P=99.84375, Y=2.268%sab
• 1.02268 = (1+ rf )
• FIS: 5% coupon, 15-oct-25 P=103.626, Y=4.536%sab
• 1.04536 = (1+ rf ) · (1+ rp)⇒ 1+ rp = 1.02218 or rp = 2.218%
The UST CFs are known, the FIS only promised CFs, but CF diagram useful
We can just calculate yield, market gives us risk premium!
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Generally use spread: rp = 4.54% – 2.27% = 227bp
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 30 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Risk Premium From Market Yields
For FIS 31-dec-2015, decompose sensitivity into discounting and credit
• UST: 2.25% coupon, 15-nov-25 P=99.84375, Y=2.268%sab
• FIS: 5% coupon, 15-oct-25 P=103.626, Y=4.536%sab
Usually use approximation
• rp ≈ y − rf
• 227bp = 4.54% – 2.27%
Overall (risky) yield separates into two parts:
1 Risk-free discounting (rf ) – overall market
2 Risk premium (rp) – specific to FIS (or whatever
company)
2.27%
4.54%
0%
227bp Credit Spread
Risk Free
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 31 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Alternative (less useful): Risk Premium in Dollars
Risk Premium (1+rp) tells us value or price of uncertainty in rate or yield terms
Can also measure in up-front dollar terms
• Discount FIS at UST (risk-free) yield: what FIS would be if certain CFs
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=4.536%, P=103.626
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 4.536
Solve For 103.626
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=2.268%, P=
123.859
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 2.268
Solve For
123.859
Difference – $20.23 – is $ value of uncertainty
• But less useful than risk premium in yield (227bp)
• 227bp applies every year, to any similar CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 32 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Alternative (less useful): Risk Premium in Dollars
Risk Premium (1+rp) tells us value or price of uncertainty in rate or yield terms
Can also measure in up-front dollar terms
• Discount FIS at UST (risk-free) yield: what FIS would be if certain CFs
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=4.536%, P=103.626
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 4.536
Solve For 103.626
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=2.268%, P=123.859
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 2.268
Solve For 123.859
Difference – $20.23 – is $ value of uncertainty
• But less useful than risk premium in yield (227bp)
• 227bp applies every year, to any similar CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 32 / 35
Securitization
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 33 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
Examples
• Mutual Funds, hedge funds, Exchange Traded Funds
• Mortgage-backed and asset-back securities
• Common Stocks
• “Equity” (mutual fund) vs “bond” pools (MBS, “Securitization”)
Legal StructureR asons for Pooling
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
Examples
Legal Structure
• “Fund” and “Manager” separate legal entities
• Mutual Funds in US – Investment Co & Advisors Acts, 1940
• Hedge Funds – often off-shore (eg Caymans) – tax & regulatory
• Mortgage-backed and other asset-backed securities (bonds)
Reasons for Pooling
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
ExamplesLegal Structure
Reasons for Pooling
• Spreads & Diversifies risk (mortgage-backed bonds)
• Lowers cost (mutual funds); professional management (hedge funds)
• Investors: Access to assets unavailable otherwise
• Sponsors/Owners: Move assets off balance sheet (borrow)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Outline
• Pass-through: all investors get same CFs
• Tranched: sell senior & junior (equity) tranches
• Pre-specified rules (“waterfall”) for how cash is distributed
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Simple Trancheing Example
• “Senior”: gets 20% of CFs, first priority
• “Equity”: 80% of CFs, but lowest priority
• Equity loses money first
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Probability of 10% loss – pay back either $100 or $90
• Senior is first priority, always gets $20 (100%) back
• Equity suffers the loss, gets either $80 (100%) or $70 (87.5%)
• Senior riskless, Equity more risky (lose more than 10% of original)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Pricing (5yr, $5 coup, 5% risky, 4% risk-free yield)
• Underlying must be $100 (PV($5 @ 5%))
• Senior now risk-free, must be $104.45 (PV($5 @ 4%) – pay $20.89 for 20%
• Equity implied $98.89 (100=.2*104.45+.8*98.89) – pay $79.11 for 80%
Share of pool PV ($) $ Paid
Senior Tranche 20% $104.4518 −→ $20.8904
Equity Tranche 80% $98.8870 ←− $79.1096
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Lecture 7
Thomas S. Coleman
Harris PPHA 42510
Applied Financial Management
18 April 2022 Draft April 19, 2022
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 1 / 35
Outline
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 2 / 35
Uncertainty versus Risk
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 3 / 35
Uncertainty versus Risk
What Is Uncertainty? What is Risk?
Uncertainty: the spread of cash flows, the cause of risk
• Distribution of cash flows,
• Spread, often summarized by Standard Deviation
Risk: the price or value we assign to the Uncertainty
• Difference in PV between certain and uncertain CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 4 / 35
Uncertainty versus Risk
We are all Sloppy about Uncertainty vs Risk
Everyone uses “Uncertainty” and “Risk” interchangeably
• “Uncertainty” is about nature – spread of CFs
• “Risk” is about us – how much we love or hate uncertainty
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 5 / 35
Uncertainty versus Risk
Standard Deviation (Volatility) – Measures Uncertainty
Reminder about Standard Deviation
• Simply a convenient way to measure spread of distribution
We will often use Standard Deviation
• But nothing magical about it – sometimes use other measures
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 6 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 7 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
We Think We’re Smart, But Why Different Yields for UST
& FIS?
Turns out we don’t even how to PV our FIS bond
Let’s look at UST & FIS, and calculate yields
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• FIS: 30/360, semi 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=
4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 8 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
We Think We’re Smart, But Why Different Yields for UST
& FIS?
Turns out we don’t even how to PV our FIS bond
Let’s look at UST & FIS, and calculate yields
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• FIS: 30/360, semi 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 8 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
Our Discounting for FIS Totally Wrong
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
• FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
This cannot be right – such huge discounting differences
We should discount CFs at (roughly) same rate. Problem is promised CFs:
• The UST CFs will be paid; the FIS promise may be broken
• FIS maybe 90% chance $5, but 10% $0. Can’t even draw CF diagram!
Fact is, we don’t really have a clue how to PV the FIS bond!
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 9 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) We Think We’re Smart – But Can’t PV FIS
CF Diagrams Misleading – Can’t Discount (so far)
We draw FIS and UST same way – but actually very different
• UST: actual known future CFs – we discount to get PV: CF(1+y)n
• FIS: promised CF
• Uncertain, distribution, say 90% $5, 10% $0
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Pictures make it look like we can discount, but we can’t (so far)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 10 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 11 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
Simple Example Shows Problems & Solutions
Real Estate project: invest today, get uncertain payout in 1 year
• Cash flow in one year expected $800
• Equal chance low ($711.41) or high ($888.59)
• (Known) value today is $714.29 and risk-free rate is
7%
$711.41 $888.59
Value of Office Building
1/2 1/2
$800
Avg highlow
• If future CF known, get PV by
discounting
• Uncertain CFs: how do we
“discount” multiple CFs???
• Don’t even know how to draw
CF diagram
$764.29k
PV=$714.29k
rf discount rate 7%
Yr 1
Discounting for Known
(Certain) CFs
$711k
p=1/2
PV=$714.29k
??
Yr 1
How do we Discount with
Multiple Uncertain CFs?
$889k
p=1/2
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 12 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Simple Example for Solution
We Really and Truly Don’t Know What to Do
This is a fundamental conceptual problem
Right picture (uncertain CFs) – don’t know what to discount or how
• Bigger problem than people
usually let on
• Solution looks like simple
discounting
• Allows us to pretend problem
doesn’t exist
$764.29k
PV=$714.29k
rf discount rate 7%
Yr 1
Discounting for Known
(Certain) CFs
$711k
p=1/2
PV=$714.29k
??
Yr 1
How do we Discount with
Multiple Uncertain CFs?
$889k
p=1/2
But I want you to understand why risk-adjusted discounting works
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 13 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 14 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
Our Goal – Method for “Discounting” Uncertain CFs
Two fundamental issues we have to address:
1 How to get (certain) PV from uncertain CFs – this lecture
• Tool to convert uncertain CFs (distribution) to single PV today
• Analogue of simple discounting (PV = CF
(1+y)n ) applied to distribution
2 What is the right “price” of uncertainty – future lectures
This lecture is about first: the how
Preview of answer – looks just like discounting known CFs
• Convert distribution to promised or expected CF
• Add risk premium and discount at risk-adjusted rate: Promised CF(1+y+rp)n
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 15 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
Framework for Discounting Uncertain CFs
Theory issues we need to address
• What do we mean when we say “uncertain cash flows”? (Answer: Cash flow
distribution – graph of amount versus probability)
• Why can’t we just use discounting as we have for certain (known) cash
flows? (Answer: Multiple values, using average CFs simply does not work –
Expected utility and concavity.)
• What is uncertainty versus risk? (Answer: Uncertainty = distribution; risk =
cost or price of uncertainty – Certainty Equivalent.)
• How do we adjust for uncertainty? (Answer: Adjust cash flow distribution
until the expected value of the adjusted CFs equals the CE.)
Practical issue we need to address
• We want a method that works just like discounting, so we can use all the PV
and NPV ideas we have worked on
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 16 / 35
Problem – We Don’t Know How to PV Uncertain CFs (notes) Program for Valuing Uncertain CFs
We Have Two Methods – Risk-Adjusted & Risk-Neutral
Both methods adjust the CF distribution
Risk-Adjusted Discounting – we use this most often
• Adjust CFs themselves
• But looks like discounting: CFadj = CForig1+rp then PV = CFadj1+y
• Used for most of our problems
Risk-Neutral Discounting – use this for options
• Adjust probabilities
• Used for derivatives, mainly options
• “Risk-neutral valuation” is bad name – really “Risk-adjusted probabilities”
You won’t read this lecture in most texts, but it is the way things work
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 17 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 18 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Value of Distribution – Look at Utility
Two things we need to do to PV uncertain CFs
1 Transform future known (certain) cash flows (CFs) into present value
• We know how to do this – Discounting
2 Transform distribution of uncertain CFs into a single known value.
• We do not yet have any idea how to do this.
• Expected Utility – Certainty Equivalent – single CF replaces distribution
• Want “trick” allowing us to continue using Average ($800)
Need to go back to basics: think about utility and value of the distribution
• How do we value the CF distribution?
• Simplify by ignoring discounting, stick to yr 1
• Want method to convert distribution to single,
certain, known value
• Call this Certainty Equivalent
• Also: trick to use $800
$711.41 $888.59
Value of Office Building
1/2 1/2
$800
Avg highlow
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 19 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Averaging Utility – Not CFs
We want utility or value of those two possibilities
We have to average the utility not CFs (expected utility not expected CF)
EU = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch) 6= u (p · Cl + (1− p) · Ch)
To make concrete, Constant Relative Risk Aversion: u(C ) = C
1−γ
1−γ , with γ = 8
Low Avg High
Probability 1/2 1/2
CF Dist’n $711.411 $888.59
Expect CF $800
Utility -15.490E-22 -3.266E-22
Expected Utility -9.378E-22
Certainty Eqv 764.28
But we can always ask “what single CF is equivalent to our utility?”
Solve for Certainty Equivalent:
U (CCE ) = EU (Cl ,Ch) = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 20 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Certainty Equivalent – CF that Gives Correct Utility
utility
utility of average
income, -6.8
income
800 888.59average of
income
u(CE)=.5*u(711.41)+.5*u(888.59)
711.41 CE
764.28
Certainty Equivalent (p=1/2)
average of
utility, -9.4
-3.3
-15.5
Solve for Certainty Equivalent:
U (CCE ) = EU (Cl ,Ch) = U (Cl ,Ch) = p · u(Cl) + (1− p) · u(Ch)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 21 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Defining Certainty Equivalent – CF Equivalent to Utility
Distribution → Utility → Certainty Equivalent
So far we have converted distribution to a single number, using expected utility
Next we see how to use Certainty Equivalent to get PV
1 Risk-Adjusted Discounting (risk premium): adjust CFs
2 Risk-Neutral Discounting: adjust probabilities
Finally, ask how we get the adjustments (to CFs or probabilities)
• From the market
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 22 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 23 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
“Trick” Allows Us to Use Original Average – Adjust CFs
Now perform a backwards trick: adjust CFs (in a neat way) so that
CE = p · CF loadj + (1− p) · CF hiadj
utility
income
CE=.5*679.66 + .5*848.93
CE
764.28
Certainty Equivalent (p=1/2)
average of
utility, -9.4
-3.3
-15.5
679.66 848.93
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 24 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs by Risk Premium – To Use Original Average
We have multiple steps:
1 Get CE (certainty equivalent) from dist’n using utility
2 Adjust CFs so they average to CE:
CE = p · CF loadj + (1− p) · CF hiadj
3 Write adjustment as 1/1+rp:
CE = p ·
CF loorig
1+ rp
+ (1− p) ·
CF hiorig
1+ rp
CE =
1
1+ rp
[
p · CF loorig + (1− p) · CF hiorig
]
=
CF avgorig
1+ rp
4 Discount CE at risk-free, original avg CF at rf + rp:
PV =
CE
1+ rf
=
CF avgorig
(1+ rf ) (1+ rp)
=
CF avgorig(
1+ yrisky
)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 25 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs by Risk Premium – To Use Original Average
We have multiple steps:
1 Get CE (certainty equivalent) from dist’n using utility
2 Adjust CFs so they average to CE:
764.28 =
1
2
· 679.66+ 1
2
· 848.93
3 Write adjustment as 1/1+rp:
CE = p ·
CF loorig
1+ rp
+ (1− p) ·
CF hiorig
1+ rp
764.28 =
1
1+ .04673
[
1
2
· 711.41+ 1
2
· 888.59
]
=
800
1+ .04673
4 Discount CE at risk-free, original avg CF at rf + rp:
714.29 =
764.29
1+ .07
=
800
(1+ .07) (1+ .04673)
=
800
(1+ .12)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 25 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Risk-Adjusted Discounting: Adjust CFs
Adjusting CFs Looks Like Discounting Average CF
Discounting Certainty Equivalent
at Risk-Free Rate
←→ Discounting Avg (Promised) CF
at Risk-Adjusted Rate
PV =
CE
1+ rf
=
CF avgorig
(1+ rf ) (1+ rp)
=
CF avgorig(
1+ yrisky
)
714.29 =
764.29
1+ .07
=
800
(1+ .07) (1+ .04673)
=
800
(1+ .12)
Looks like regular yield-to-maturity, just a higher “risk-adjusted” rate
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 26 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 27 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Return to UST & FIS
Apply our ideas of Risk Adjusted Discounting
• Ideas and theory can seem complicated
But using these ideas simple
• Everyone in financial markets uses them every day
• Calculate Yield, using “Promised CFs”
• Calculate as if the Promised CF is Known CF
• Yield we get is “Risk Adjusted”
• I did for many years, without knowing or understanding or caring about the
underlying theory
So let’s go back to UST & FIS and see what happens
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 28 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
FIS – Standard Yield Calculations – Promised CFs
Settle 31-dec-15 – Calculate FIS Yield using Promised CF
• Embedded in bond prices – market gives us risk premium
31-dec-2015 Coupon Maturity Price Yield
10-yr US Treasury 2.25% 15-nov-25 99.84375 2.268%
FIS 5.0% 15-oct-25 103.626
4.536%
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• Calculate FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=
4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For
4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 29 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
FIS – Standard Yield Calculations – Promised CFs
Market gives us risk premium – spread = 4.536% – 2.268% = 227bp
• Embedded in bond prices – market gives us risk premium
31-dec-2015 Coupon Maturity Price Yield
10-yr US Treasury 2.25% 15-nov-25 99.84375 2.268%
FIS 5.0% 15-oct-25 103.626 4.536%
• UST: 2.25% coupon, 15-nov-25 (31-dec-15) P=99.84375, Y=2.268%
Type Settle Mat CPN% YLD% Price
Given A/A Semi 12.312015 11.152025 2.25 99.84375
Solve For 2.268
• Calculate FIS: 5% coupon, 15-oct-25 (31-dec-15) P=103.626, Y=4.536%
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 103.626
Solve For 4.536
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 29 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Bond Market & FIS Example – Risk-Adjusted Yield
Market gives us risk premium – embedded in price (settle 31-dec-15)
• UST: 2.25% coupon, 15-nov-25 P=99.84375, Y=2.268%sab
• 1.02268 = (1+ rf )
• FIS: 5% coupon, 15-oct-25 P=103.626, Y=4.536%sab
• 1.04536 = (1+ rf ) · (1+ rp)⇒ 1+ rp = 1.02218 or rp = 2.218%
The UST CFs are known, the FIS only promised CFs, but CF diagram useful
We can just calculate yield, market gives us risk premium!
fixed 5% coupon
100
PV=103.6
discount at 4.54%
FIS Bond
. . .
Yr 2Yr 1 Yr 10. . .
fixed 2.25% coupon
100
PV=99.84
discount at 2.27%
US Treasury
. . .
Yr 2Yr 1 Yr 10. . .
Generally use spread: rp = 4.54% – 2.27% = 227bp
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 30 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Risk Premium From Market Yields
For FIS 31-dec-2015, decompose sensitivity into discounting and credit
• UST: 2.25% coupon, 15-nov-25 P=99.84375, Y=2.268%sab
• FIS: 5% coupon, 15-oct-25 P=103.626, Y=4.536%sab
Usually use approximation
• rp ≈ y − rf
• 227bp = 4.54% – 2.27%
Overall (risky) yield separates into two parts:
1 Risk-free discounting (rf ) – overall market
2 Risk premium (rp) – specific to FIS (or whatever
company)
2.27%
4.54%
0%
227bp Credit Spread
Risk Free
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 31 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Alternative (less useful): Risk Premium in Dollars
Risk Premium (1+rp) tells us value or price of uncertainty in rate or yield terms
Can also measure in up-front dollar terms
• Discount FIS at UST (risk-free) yield: what FIS would be if certain CFs
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=4.536%, P=103.626
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 4.536
Solve For 103.626
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=2.268%, P=
123.859
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 2.268
Solve For
123.859
Difference – $20.23 – is $ value of uncertainty
• But less useful than risk premium in yield (227bp)
• 227bp applies every year, to any similar CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 32 / 35
PV for Uncertain CFs: Utility and Certainty Equivalent Return to UST & FIS
Alternative (less useful): Risk Premium in Dollars
Risk Premium (1+rp) tells us value or price of uncertainty in rate or yield terms
Can also measure in up-front dollar terms
• Discount FIS at UST (risk-free) yield: what FIS would be if certain CFs
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=4.536%, P=103.626
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 4.536
Solve For 103.626
• FIS: 5% coupon, 15-oct-25 (31-dec-15) Y=2.268%, P=123.859
Type Settle Mat CPN% YLD% Price
Given 360 Semi 12.312015 10.152025 5 2.268
Solve For 123.859
Difference – $20.23 – is $ value of uncertainty
• But less useful than risk premium in yield (227bp)
• 227bp applies every year, to any similar CFs
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 32 / 35
Securitization
1 A Digression on Shorting & Leverage – moved to Lecture 5
2 Uncertainty versus Risk
Section II of “Practical Guide to Yield Curves, Discounting, and Derivatives” (Canvas);
ch 2 of “Practical Guide to Risk Mgmt” (Amazon or CFA)
3 Problem – We Don’t Know How to PV Uncertain CFs (notes)
Lecture Notes and Section II of “Practical Guide to Yield Curves” (Canvas)
We Think We’re Smart – But Can’t PV FIS
Simple Example for Solution
Program for Valuing Uncertain CFs
4 PV for Uncertain CFs: Utility and Certainty Equivalent
Defining Certainty Equivalent – CF Equivalent to Utility
Risk-Adjusted Discounting: Adjust CFs
Return to UST & FIS
5 Securitization
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 33 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
Examples
• Mutual Funds, hedge funds, Exchange Traded Funds
• Mortgage-backed and asset-back securities
• Common Stocks
• “Equity” (mutual fund) vs “bond” pools (MBS, “Securitization”)
Legal StructureR asons for Pooling
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
Examples
Legal Structure
• “Fund” and “Manager” separate legal entities
• Mutual Funds in US – Investment Co & Advisors Acts, 1940
• Hedge Funds – often off-shore (eg Caymans) – tax & regulatory
• Mortgage-backed and other asset-backed securities (bonds)
Reasons for Pooling
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Securitization – Pooling Assets & Cash Flows
Pooling of assets and repackaging of the underlying cashflows
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
Public
Public
.
.
.
shares to public
ExamplesLegal Structure
Reasons for Pooling
• Spreads & Diversifies risk (mortgage-backed bonds)
• Lowers cost (mutual funds); professional management (hedge funds)
• Investors: Access to assets unavailable otherwise
• Sponsors/Owners: Move assets off balance sheet (borrow)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 34 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Outline
• Pass-through: all investors get same CFs
• Tranched: sell senior & junior (equity) tranches
• Pre-specified rules (“waterfall”) for how cash is distributed
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Simple Trancheing Example
• “Senior”: gets 20% of CFs, first priority
• “Equity”: 80% of CFs, but lowest priority
• Equity loses money first
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Probability of 10% loss – pay back either $100 or $90
• Senior is first priority, always gets $20 (100%) back
• Equity suffers the loss, gets either $80 (100%) or $70 (87.5%)
• Senior riskless, Equity more risky (lose more than 10% of original)
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
Securitization
Pass-Through vs Trancheing (Waterfall)
Pooling Entity
Company or SPV
Asset 1
Asset N
.
.
.
Asset 1
Asset N
.
.
.
various share
tranches to public
Public
Public
senior tranch
top 20%
equity tranch
bottom 80%
CF Tranching
or Watefall
Pricing (5yr, $5 coup, 5% risky, 4% risk-free yield)
• Underlying must be $100 (PV($5 @ 5%))
• Senior now risk-free, must be $104.45 (PV($5 @ 4%) – pay $20.89 for 20%
• Equity implied $98.89 (100=.2*104.45+.8*98.89) – pay $79.11 for 80%
Share of pool PV ($) $ Paid
Senior Tranche 20% $104.4518 −→ $20.8904
Equity Tranche 80% $98.8870 ←− $79.1096
Coleman (Harris 42510) PV & Uncertain CFs 18-apr-22 35 / 35
