Upload
abdullaalothman
View
285
Download
0
Tags:
Embed Size (px)
DESCRIPTION
The thirs of a three part series on asset valuation. The slides are animated, the approach unusual. There are no formulas to puzzle through, everything is derived from scratch.
Citation preview
Abdulla Alothman 1
Asset Valuation
A Unified Approach
Part1 Valuation of Assets With Deterministic Payoffs
The Time Value of Money
The Theory
Abdulla Alothman 4
• Zero Coupon Bonds (The Building Blocks)
• The Term Structure of Interest Rates
• Bonds
Abdulla Alothman 5
VALUATION OF ZERO COUPON BONDS
Zero Coupon Certificates are IOU’s that make a single payment of $1.00 to the holder at maturity.
Abdulla Alothman 6
Objective: To find the “fair market value” of Z (t,r;T)
T = tnt
( ; )n
r t t
( , ; ) ????n
Z r t t =
Abdulla Alothman 7
Step 0: Collect necessary market dataThe Term Structure of Interest Rates at time t is observed and noted. We will denote this by:
1 2 3( ) { ( ; ), ( ; ), ( ; ) ( ; )}
nr t r t t r t t r t t r t tº ¼
( ; )r t T
T
ti r(t; ti)
t1 5%t2 6%t3 6.5%t4 6.8%
T = t5 7.3%
t
Abdulla Alothman 8
Step1: Simplify the task ,price a bond with only one period to maturity first..
T = t1t
1( ; )r t t
1( , ; )Z t r t =
Abdulla Alothman 9
Step1 cont…
T= t1 t
11
( , ; ) $1.00/ (1 )Z t r t r= +
1( ; )r t t
Abdulla Alothman 10
1(0,0.05;1) $1.00/ (1 0.05) 0.9524Z = + =
T=t1 t=0
Interest = 5% per period*
Example: Find the fair market price of Z(0, 0.05; 1)
Abdulla Alothman 11
Step 2: Pricing a Zero Coupon Bond with 2 periods to maturity
T = t2t
2( ; )r t t
2( , ; ) ????Z r t t =
Abdulla Alothman 12
The idea:
2( , ; ) ????Z r t t =
T = t2t t1
2$1.00/ (1 ( ; ))A r t t= +
A
A
Abdulla Alothman 13
The idea cont..…
22 2 2
( , ; ) / (1 ( ; )) $1.00/ (1 ( ; ))Z t r t A r t t r t t= + = +
t T = t1
2$1.00/ (1 ( ; ))A r t t= +
Abdulla Alothman 14
2(0,0.06;2) $1.00/ (1 0.06) 0.8900Z = + =
T=2 t=0
Interest = 6% per period*
Example: Find the fair market price of Z(0, 0.06; 2)
Abdulla Alothman 15
Final Step: Pricing a Zero Coupon Bond with n periods to maturity
T = tnt
( ; )n
r t t
( , ; ) ????n
Z r t t =
Abdulla Alothman 16
Key Idea: Pretend you’re at time tn and work backwards to tn-1, then tn-2, tn etc, to get…
( , ; ) $1.00/ (1 )nn
Z t r t r= +
T = tnt t2
We now have a general formula to price Z’s of any maturity.
tn-1t(n-2)t1 ……….
Abdulla Alothman 17
Example: Find the fair market price of Z(4, 0.08; 8)
T=8t=4
Interest = 8% per period*
4(4,0.08,8) $1.00/ (1 0.08) 0.7350Z = + =
Abdulla Alothman 18
Example: Let’s price the rest………..
ti Z(t; ti) r(t, ti)
t1 0.95248 5%
t2 0.88996 6%
t3 0.82785 6.5%
t4 0.7686 6.8%
T = t5 0.7031 7.3%
( ; )r t T
T
( ; )Z t T
T
Abdulla Alothman 19
Summary:1 2 3
( ) { ( ; ), ( ; ), ( ; ) ( ; )}n
r t r t t r t t r t t r t tº ¼
1 2 3( ) { ( ; ), ( ; ), ( ; ) ( ; )}
nZ t Z t t Z t t Z t t Z t tº ¼
Abdulla Alothman 20
Working Backwards:1 2 3
( ) { ( ; ), ( ; ), ( ; ) ( , )}n
r t r t t r t t r t t r t tº ¼
1 2 3( ) { ( ; ), ( ; ), ( ; ) ( , )}
nZ t Z t t Z t t Z t t Z t tº ¼
Abdulla Alothman 21
Extracting the term structure from market prices:
1 1 1 12 1/ 2
2 2 2 23 1/ 3
3 3 3 3
1/
1/ (1 ) (1/ ) 1
1/ (1 ) (1/ ) 1
1/ (1 ) (1/ ) 1
1/ (1 ) (1/ ) 1n nn n n n
Z r r Z
Z r r Z
Z r r Z
Z r r Z
= + Þ = -
= + Þ = -
= + Þ = -
= + Þ = -
Abdulla Alothman 22
1 1 12
2 2 23
3 3 3
0.8689 1/ (1 ) 15.09%
0.7890 1/ (1 ) 12.58%
0.7064 1/ (1 ) 12.28%
Z r r
Z r r
Z r r
= = + Þ =
= = + Þ =
= = + Þ =
Example:
Abdulla Alothman 23
VALUATION OF BONDS
Are IOU’s making a stream of payments to the holder over time.
Abdulla Alothman 24
Example:
……….. t1 t2 T = tn t3 t4t
( ; )t
B B t Tº
Abdulla Alothman 25
Note: Zero Coupons are Special Bonds:
………..
t1 t2 T= tn t3 t4t
Z(t ; T)
Abdulla Alothman 26
………..
t1 t2 T =tn t3 t4
C CCCC C
t
$1000.00
Objective: To find the “fair market value” of B(t,r,C;T)
( , ; ) ???B t C T
Abdulla Alothman 27
Step 0: Collect necessary market dataThe Term Structure of Interest Rates at time t, is observed and noted. We denote this by:
1 2 3( ) { ( ; ), ( ; ), ( ; ) ( ; )}
nr t r t t r t t r t t r t tº ¼
( ; )r t T
T
ti r(t; ti)
t1 5%t2 6%t3 6.5%t4 6.8%
T = t5 7.3%
Abdulla Alothman 28
Step 1: Value the Zero’s
ti r(t, ti) Z(t; ti)
t1 5% 0.95248
t2 6% 0.88996
t3 6.5% 0.82785
t4 6.8% 0.7686
T = t5 7.3% 0.7031
Abdulla Alothman 29
Step 2: Observe that a Bond is just a portfolio of Zero Coupon Bonds
Item Price Number of Items
Cost
Z1 C C*Z1Z2 C C*Z2
Z3 C C*Z3
………… ……….. ………….Zn 1000+C (C+1000)*Zn
Summing up the values in the third column therefore then gives…..
Abdulla Alothman 30
1
( , ; ) 1000i n
i ni
B t C n cZ Z=
=
= +å
The Bond’s fair market value
Abdulla Alothman 31
ti Z(t; ti) B(t,C; ti)
t1 0.9524 C*0.9524+952.48
t2 0.8900 C*(0.9524+0.8900)+889.00= 1.84C+889.96
t3 0.8279 C*(0.9524+0.8900+0.8279) + 827.90 = 2.67C+827.90
t4 0.7686 C*(0.9524+0.8900+0.8279+0.7686) + 768.6 = 3.44C+768.6
T = t5 0.7031 (0.95428+0.8900+0.8279+0.7686 +0.7031)*C+703.1 = 4.14C+703.1
Example:
Abdulla Alothman 32
3
4
( ,4; ) 4* 2.67 827.85 838.53
( ,5; ) 5* 3.4403 768.6 785.80
B t t
B t t
= + =
= + =
Abdulla Alothman 33
1
1
( , , ; ) 1000 )
F(t,y,t ) 1/ (1 ) 1000/ (1 )
k
k i ki
ki i
ki
P B t C t C Z Z
P C y y
=
=
= = +
= - + + + +
å
å
Given a family of Bonds:
r
Define a function:
F(t,y,t )k
* ( , ; )y y t cT=
*y
Yield to Maturity . Also known as Bond’s IRR
Yield to Maturity (IRR)
EXTRACTING THE PRICE OF ZERO’S FROM TRADED BONDS
The Bootstrapping Technique
Abdulla Alothman 35
1 1 1
2 2 1 2 2
1 2
1
1 2
( ; ) 1000
( ; ) ( 1000)
( ; ) ( 1000)
Solve for Z in equation 1.
Substitute for Z in equation 2, and solve for Z
Subsitut
n n n n n
B t t C Z
B t t C Z C Z
B t t C Z C Z C Z
= +
= + +
= + + + +
Given Market Prices:
Step 1:
Step 2:
Step n:
L L
K
1 2 1e for Z ,Z ....Z in equation n,and solve for Z
n n-
Abdulla Alothman 36
( ) { ( ; ) : 0 }i
Z t Z t t t T= £ £( ) { ( ; ) : 0 }i
r t r t t t T= £ £
( ) { ( , ; ) : 0 , 0 }i i
B t B t C t t T C= £ £ £
Summary
( ) { ( , ; ) : 0 , 0 }i i
y t y t C t t T C= £ £ £
Term Structure Zero Prices
Yield Curves
Bond Prices
Applications
Abdulla Alothman 38
• Annuities
• Perpetuities
• Amortizing Loans
• Forwards
• Swaps
Abdulla Alothman 39
Flat Term Structure Assumption (Slides 39 – 55) :
( ; )r t T
T
r
Abdulla Alothman 40
An Annuity:
• Is just a Bond with a face value of Zero
Abdulla Alothman 41
………..
Example:
t1 t2 T =tn t3 t4
( , , ; )t
A A t C Tº r
t
Abdulla Alothman 42
……….. t1 t2 T =tn t3 t4
Pricing an Annuity:
( ; ) ( , , ; )A t T A t r C Tº
t
Abdulla Alothman 43
Step 0: Collect necessary market dataThe Term Structure of Interest Rates at time t, is observed and noted. We will denote this by:( ) { , , }r t r r r rº ¼
( ; )r t T
T
ti r(t, ti)
t1 r%t2 r%t3 r%t4 r%
T = t5 r%
Abdulla Alothman 44
Step 2: An Annuity is just a portfolio of Zero Bonds
Item Price Number of Items
Cost
Z1 C C*Z1Z2 C C*Z2
Z3 C C*Z3
………… ……….. ………….Zn C C*Zn
Summing up the values in the third column then gives…..
Abdulla Alothman 45
1
( , ; ) (1 (1 ) )i n
ni
i
cA t C t n c Z r
r
=-
=
+ = = - +åAn Annuity's fair market value
Abdulla Alothman 46
Example:
……….. t2 T =t5t3 t4
100 100100100 100
5100(0;5) (1 (1.05) ) 432.95
0.05A -= - =
t t1
Abdulla Alothman 47
A PERPETUITY:
• Is simply an ANNUITY with an infinite coupon stream
Abdulla Alothman 48
………..
t1 t2 T = t3 t4
Pricing an Perpetuity P(t,C,r;):
( ; ) ( , , ; )P t T P t r C Tº
Abdulla Alothman 49
Step 0: Collect necessary market data ……
• The Term Structure of Interest Rates at time t, is observed and noted. We will denote this by:
( ; )r t T
¥
Abdulla Alothman 50
Step 2: Item Price Number of
ItemsCost
Z1 C C*Z1
Z2 C C*Z2
Z3 C C*Z3
………… ……….. ………….
Zn C C*Zn
…………. ………… ……………
Summing up the values in the third column then gives…..
Abdulla Alothman 51
The Perpetuity’s fair market value!!
1
( , ; ) (1 (1 ) )i
ii
C CA t C t n c Z r
r r
=¥- ¥
=
+ = = - + =å
Abdulla Alothman 52
………..
t+1 t+2 T = t+3 t+4
1
50( ,0.06,50) 50 1/ (1.06) 833.33
0.06
ii
i
P t=¥
=
= = =å
t
Example: Find the fair market price of P(t, 0.06,50)
Abdulla Alothman 53
t1 t2 T =tn t3 t4
Example: Calculating Loan Payments:
1
1 1
So:
/ 1/ (1 ) * / (1 (1 ) )
i n
ii
i n i ni n
ii i
LOAN PMT Z
PMT LOAN Z LOAN r LOAN r r
=
=
= =-
= =
=
= = + = - +
å
å å
……………….
Interest = r% per period*
Abdulla Alothman 54
t1 t2 T =t5 t3 t4
Example :
51000* 0.05/ (1 (1 0.05) ) 230.97PMT -= - + =
……………….
Interest = 5% per period*
$1000.00
t
Abdulla Alothman 55
LOAN AMORTIZATION (Optional):
This is just the “Depreciation” of a Loan and has nothing to do with valuation, but rather, with how bonds are accounted for by companies.
Abdulla Alothman 56
LOAN PMT INTEREST 5% PRINCIPLEPAY DOWN
BALANCE
$1000.00 -$230.9 -$50.00 -$180.90 $819.10
$819.00 -$230.9 -$40.95 -$189.95 $629.05
629.05 -$230.9 -$31.45 -$199.44 $429.71
429.71 -$230.9 -$21.455 -$209.45 $220.26
220.26 -$230.9 -$11.13 -$219.89 $0.00
LOAN AMORTIZATION TABLE
Abdulla Alothman 57
FORWARDS:
Are contracts which allow the (Buyer / Seller ) to lock in today (time t) the Future (Buying / Selling ) price of an asset at some future date t+i.
Abdulla Alothman 58
Assumption: We take the following Market Data as an given…. The Term Structure of Interest Rates at time t:
1 2 3( ) { ( ; ), ( ; ), ( ; ) ( ; )}
nr t r t t r t t r t t r t tº ¼
( ; )r t T
T
ti r(t, ti)
t1 5%t2 6%t3 6.5%t4 6.8%
T = t5 7.3%
Abdulla Alothman 59
Motivating question one:How much do you need to invest at time ti to receive $1.00 at time ti+1?
t T ti ti+1
$1.00
Abdulla Alothman 60
Analysis…At time t construct the following portfolio:
1. Buy one Zero maturing at time ti+1 . Cost: Zi+1 .
2. Finance your purchase by selling (going short) n =Zi+1/Zi Zero’s maturing at time ti . Revenue: n*Zi= Zi+1 .
,
t = 0 ... ti ti+1
-Zi+1 ... 0.00 $1.00
+n*Zi=Zi+1 ... -n*$1.00(Zero’s mature)
0.00
Net = $0.00 -n=-Zi+1/Zi $1.00
Transaction Cash Flows
Abdulla Alothman 61
t T
Solution:
ti ti+1
How much do you need to invest at time ti to get $1.00 at time ti+1
Answer =Zi+1/Zi
$1.00
Abdulla Alothman 62
Motivating question two:What is the effective one period future interest rate f(t; ti; ti+1)* you’ve locked in for your time ti investment?
1*Definition ( ; ; ) :
i if t t t
+The forward interest rate locked in at time t for an investment starting time ti and maturing at time ti+1
ti+1ti
1( ; ; )
i if t t t
+
1/
i iZ Z
+
$1.00
Abdulla Alothman 63
Analysis:1 1 1
1
11 1
11
1
1 ( ; ; ) $1.00/ ( / ) /
But:
$1.00/ (1 ) , 1
Substituting for , gives:
1+ ( ; , ) (1 ) / (1 )
So:
(1 )( ; , ) 1
(1 )
i i i i i i
ll l
i i
i ii i i i
ii
i i ii
f t t t Z Z Z Z
Z r l i i
Z Z
f t t t r r
rf t t t
r
+ + +
+
++ +
++
+
+ = =
= + = +
= + +
+= -
+
Abdulla Alothman 64
Solution:
t(i+1)ti
1/
i iZ Z
+
$1.001
11
(1 )( ; ; ) 1
(1 )
ii
i i ii
rf t t t
r
++
+
+= -
+
Abdulla Alothman 65
The picture:
……….. t1 t2 T = tn t+i ti+1t
$1.00
11
(1 ( ; ))ii
r t t ++
+
(1 ( ; ))i
ir t t+ 1(1 ( ; ; ))
i if t t t
++´
Abdulla Alothman 66
Generalizations: Using the same arguments as in the previous slides, will, mutatis mutandis, yield the following….
11
11
22 2
22
33 3
33
(1 ( ; ))(1 ( ; ; ))
(1 ( ; ))(1 ( ; ))
(1 ( ; ; ))(1 ( ; ))
(1 ( ; ))(1 ( ; ; ))
(1 ( ; ))
(1 ( ; ))(1 ( ; ; )) ( )
iii
i i ii i
iii
i i ii i
iii
i i ii i
ik i ki
i i ki k
r t tZf t t t
Z r t tr t tZ
f t t tZ r t t
r t tZf t t t
Z r t t
r t tZf t t t
Z
++
++
++
++
++
++
++
+
++ = =
++
+ = =+
++ = =
+
++ = =
K
(1 ( ; ))
k
ii
r t t
+
+
Abdulla Alothman 67
Summary:The forward rate f(t ; ti; ti+1) is the one period rate that would have to prevail at time ti – unknown at time t – for you to be indifferent between the following two investments:
• Investing at time t, for i+1 periods at the rate of r(t ; ti+1) per period.
• Investing at time t for i periods at the rate of r(t,ti) per period , then, when your deposit matures, rolling the investment over for one more period at the , currently unknown, rate of r(ti ; ti+1).
Abdulla Alothman 68
Example:1 2 3
( ) { ( ; ), ( ; ), ( ; ) ( , )}n
r t r t t r t t r t t r t tº ¼
( ; )r t T
T
ti r(t, ti)
t1 5%t2 6%t3 6.5%t4 6.8%
T = t5 7.3%
Abdulla Alothman 69
11/ 1 1/ 11
11
21/ 2 1/ 22
22
31/ 3 1/ 33
33
4
(1 ( ; ))1 ( ; ; ) ( ) { }
(1 ( ; ))(1 ( ; ))
1 ( ; ; ) ( ) { }(1 ( ; ))
(1 ( ; ))1 ( ; ; ) ( ) { }
(1 ( ; ))
1 ( ; ; ) (
iii
i i ii i
iii
i i ii i
iii
i i ii i
ii i
i
r t tZf t t t
Z r t tr t tZ
f t t tZ r t t
r t tZf t t t
Z r t tZ
f t t tZ
++
++
++
++
++
++
+
++ = =
++
+ = =+
++ = =
+
+ =4
1/ 4 1/ 44
45
1/ 5 1/ 555
5
(1 ( ; ))) { }
(1 ( ; ))(1 ( ; ))
1 ( ; ; ) ( ) { }(1 ( ; ))
ii
ii
iii
i i ii i
r t t
r t tr t tZ
f t t tZ r t t
++
++
++
+
+=
++
+ = =+
Example Cont...:
Abdulla Alothman 70
Example cont: The time t +1 forward term structure
1 2 1 3 1 4 1 5( ; 1) { ( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )}f t t f t t t f t t t f t t t f t t t+ º
( ; 1; )f t t T+
5t
t1 f(t, t+1;t+2))
t2 7.025%%t3 7.2634%t4 7.411%t5 7.884%t6 na
1t
Abdulla Alothman 71
Example cont.. The time t +2 forward term structure
2 3 2 4 2 5( ; 2) { ( ; ; ), ( ; ; ), ( ; ; )}f t t f t t t f t t t f t t t+ º
( ; 1; )f t t T+
5t
t2 f(t, t+1;t+2))
t3 7.50%%t4 7.61%t5 8.172%t6 nat7 na
2t
Abdulla Alothman 72
Exercises:1. Verify the time t+1 forward term structure…2. Verify the time t+2 forward term structure3. A client wishes to buy 10 ounces of gold from you
at time t2. Market data at time t are given below:
t t2
Price of Gold (ounce)
$850.00 ?
Cost of Insurance* 5% of Value 5% of Value
(Z1 ,Z2,Z3) (0.95,0.90,0.85) Na
Abdulla Alothman 73
Exercices cont…..
The client is concerned that the price at time t= 2 might be higher than today and wishes to lock in a rate today.
What rate would you quote him and why ?
Abdulla Alothman 74
AnswersQ.3
Action t t2
Buy 10 Gold: -8500.00
Insurance Premium 1:
- 850.00
Loan 9350.00
Payback2 Loan+Interest
-10,388.88
Client Payment 10,388.88
Cash Flow 0.00 0.00
2
1)Insurance costs 2* 0.05* 8500
2)Total to reapay:
8500*(1+2*0.05)/ Z
Abdulla Alothman 75
Exercices cont…..
4. The Current Euro / USD exchange rate is 1.00 /2.00 .
A USD Z1 is selling for 0.95.
A Euro Z1 bond is selling for 0.90.
The client wants to buy from you 100 Euros at time t+1. He would like however, to lock in the rate today. What rate should you quote him?
Abdulla Alothman 76
SWAPS:• Are agreements between two parties, party A and
Party B to exchange a series of future cash flows….
PARTY A PARTY Bt
PARTY A
PARTY A
PARTY B
PARTY B
t+1
t+n
……
. ……
.
*In this course we are only going to looking at a basic interest rate swaps
Abdulla Alothman 77
The Basic Interest Rate Swap
• Are agreements between two parties, party A and Party B to exchange a series of future cash flows….
PARTY A PARTY B
PARTY A
PARTY A
PARTY B
PARTY B
……
.
……
.
1t
2t
nt
1( ; )r t t
1 2( ; )r t t
C
C
C1
( ; )n n
r t t-
1( ; ) :Is the one period rate at time
C : Is some fixed constant to be determinedi i i
r t t t-
Abdulla Alothman 78
Exercise: Given the Market data below, find the value C which makes the both cash streams have identical t0 values.
ti r(t, ti)
t1 5%t2 6%t3 6.5%t4 6.8% t5 7.3%
( ; )r t T
T
Abdulla Alothman 79
Step 1: Price the Zeros ti r(t, ti) Z(t; ti)
t1 5% 0.95248
t2 6% 0.88996
t3 6.5% 0.82785
t4 6.8% 0.7686
T = t5 7.3% 0.7031
Abdulla Alothman 80
Solution: Think about how to generate the random stream....
100 100 100 100100
1100 ( ; )r t t+
1 2100 ( ; )r t t+
2 3100 ( ; )r t t+
3 4100 ( ; )r t t+
4 5100 ( ; )r t t+
100
5100Z
1 1 2 2 3 3 4 4 5{ ( ; ), ( ; ), ( ; ), ( ; ), ( ; )}r t t r t t r t t r t t r t t
5
Solution:
1 0100 0t
V Z= -
Abdulla Alothman 81
Think about how to generate the fixed stream....
100
C
5
1
Solution:
t ii
B C Z=
= å
C C C C
Abdulla Alothman 82
Set both streams equal and solve for C...
5
5
51
5
5
4
Solving for C give
1
s
00
:
C
100
=100(1 ) / 7.
,
1
; )
8
(
%
i
ti
i
i
C Z B t C
Z
Zt V
Z
=
=
= = = -
- =å
å
Abdulla Alothman 83
END OF PART 1