Transcript MBS Mathematics - Duration and convexity

Duration and convexity for Fixed-Income Securities
RES9850 Real Estate Capital Market
Professor Rui Yao
Duration and convexity: Outline
 I. Macaulay duration
 II. Modified duration
 III. Examples
 IV. The uses and limits of duration
 V. Duration intuition
 VI. Convexity
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A Quick Note
 Fixed income securities’ prices are sensitive to changes in interest rates
 This sensitivity tends to be greater for longer term bonds
CN
C1
C2
P0 

 ... 
1
2
(1  y ) (1  y )
(1  y ) N
 But duration is a better measure of term than maturity
 Duration for 30-year zero = 30
 Duration for 30-year coupon with coupon payment < 30
 A 30-year mortgage has duration less than a 30-year bond with similar yield
Amortization
Prepayment option
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I. (Macauly) duration
 Weighted average term to maturity
 Measure of average maturity of the bond’s promised cash flows
 Duration formula:
T
Dm   t  wt 
t 1
where:
wt 
P V(CFt )
T
P0   P V(CFt )
CFt /(1  y )t

P0
t 1
q
w
t 1
t
1
is the share of time t CF in the bond price
and t is measured in years
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Duration - The expanded equation
 For an annual coupon bond
 P V(Ct ) 
Dm   t  wt   t  

t 1
t 1
 P V(Bond)
 C1 
 C2 
 CN 
1

2

...

N
 (1  y ) 2 
 (1  y ) N 
(1  y )1 






 C1
CN 
C2


...

 (1  y )1 (1  y ) 2
N 
(
1

y
)


T
T
 Duration is shorter than maturity for all bonds except zero
coupon bonds
 Duration of a zero-coupon bond is equal to its maturity
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IV An Example – page 1
 Consider a 3-year 10% coupon bond selling at $107.87 to yield 7%.
Coupon payments are made annually.
10
 9.35
(1.07)
10
PV (CF2 ) 
 8.73
2
(1.07)
110
PV (CF3 ) 
 89.79
3
(1.07)
P riceof bond  9.35  8.73  89.79  107.87
PV (CF1 ) 
9.35  
8.73   89.79 

Duration (Dm )  1*

2
*
 3 *



107.87
107.87
107.87
 2.7458
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II. Modified duration (D*m)
Dm
D 
1 y
*
m
Direct measure of price sensitivity to interest rate changes
Can be used to estimate percentage price volatility of a bond
P
*
 Dm  y
P
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Derivation of modified duration
N
P
t 1
Ct
(1  y )t
Ct
P
1 N 


t


y 1  y t 1  (1  y ) t
  Dm
 
 P   Dm*  P
 1 y
1 P
  Dm*
P y
 So D*m measures the sensitivity of the % change in bond price
to changes in yield
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An Example – page 2
 Modified duration of this bond:
Dm* 
2.7458
 2.5661
1.07
 If yields increase to 7.10%, how does the bond price change?
 The percentage price change of this bond is given by:
P
  Dm*  y
P
= –2.5661  .1%
= –.2566%
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An Example – page 3
 What is the predicted change in dollar terms?
.2566
P
100
.2566

 $107.87
100
 $.2768
P  
New predicted price: $107.87 – .2768 = $107.5932
Actual dollar price (using PV equation): $107.5966
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Good
approximation!
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Summary: Steps for finding the predicted
price change
 Step 1: Find Macaulay duration of bond.
 Step 2: Find modified duration of bond.
 Step 3: Recall that when interest rates change, the change in a bond’s
price can be related to the change in yield according to the rule:
P
  Dm*  y
P
 Find percentage price change of bond
 Find predicted dollar price change in bond
 Add predicted dollar price change to original price of bond
 Predicted new price of bond
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V. Check your intuition
 How does each of these changes affect duration?
1. Decreasing the coupon rate.
verify this with a 10-year bond with coupon rate from 5% to 15%,
and ytm of 10%
2. Decreasing the yield-to-maturity.
verify this property with a 10-year bond with coupon rate of 10%,
and ytm from 5% to 15%
3. Increasing the time to maturity.
verify this property with a par bond with a coupon rate of 10%, and
term from 5 to 15 years
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V. Dollar Duration
 We have derived the following relationship between duration and price
changes (bond returns):
P
P   D*
m
y
 Hence

P
 Dm*  P
y
 Note the term on the RHS of the equation above measures the (absolute
value of) slope of the yield-price curve, which is also called dollar
duration
 We can then predict price changes using dollar duration:
P  (Dm*  P)  y  $dur y
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Duration with intra-year compounding
 In practice, lots of bonds do not pay annual coupon and we need to change
the formula a bit to account for it
 Some calculus (note: each step in summation is 1/m year so there are m*T
terms in total)
mT

Cn
P   
n
n 1  1  y / m 

 

mT

 1
Cn
Cn
P mT 
1
1



    n 



n


n 1
n 


y n1 
1  y / m m  (1  y / m) n1  1  y / m  m
 So dollar duration
mT
1
n  PV (Cn )
Dm  P 

(1  y / m) n1
*
 Duration
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*
m
D
mT
1
 PV (Cn ) 

n


1  y / m n1 
P 
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V. Effective Duration– Numerical Approximation
 Instead of calculating modified duration based on weighing the time of
cash flow with the present value share of the CF, and then modify by
dividing by (1+y), we can numerically approximate the modified
duration from the slope of price-yield chart:
Dm*  
P 1

y P
 The slope of the yield-price curve is the dollar duration

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P
 Dm*  P
y
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V. Effective Duration – Numerical Approximation
 We can approximate the slope of the graph / dollar duration by averaging
the forward and backward slope (“central difference method”)
P 1  P  Po Po  P  P  P
 
 

y 2  y  yo yo  y  2  y
 The duration is then
P P  P

y
2  y
 The modified duration then can be estimated as
Dm*  
P  P
2  y

1
Po
 This approach directly uses the idea that the duration measures price
sensitivity to interest rate
 Can duration be negative?
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VI. Duration and Convexity – Numerical
Approximation
price
P-
Po
P+
y-
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y
°
y+
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yield
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VI. Convexity
 Duration is the first order approximation for percentage change in bond
prices for a one percent change in yield to maturity
 For a fixed rate non-callable bond, duration underestimates change when
yield falls and overestimates when yield rises
 The difference is captured by convexity
 Convexity is typically positive for bond
 Is this good or bad?
 Mortgage is a difficult product to evaluate due to embedded call options
 Duration tends to become shorter when interest becomes lower as borrower
prepays mortgage
 Duration becomes longer when interest rate becomes higher as borrower
holds on to his mortgage
 Negative convexity
 Opposite to the case of a typical bond
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V. Convexity
 The forecast of price response using dollar duration is


dP
 y
dy
 Essentially it is a linear projection using the slope measured at yo
P   Dm*  y  P   Dm*  P  y 
 However as soon as you move away from yo the slope will change
 The rate of slope change
d 2P
dy2
is captured by dollar convexity
y  yo
 A little calculus yields (take first order derivative of dollar duration with
respect to yield)
T
Ct
d 2P
1
2

(
t

t
)


dy2 (1  y) 2 t 1
(1  y)t
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V. Convexity – Numerical Approximation
 What is the predicted dollar duration at y+ using dollar duration from yo
and convexity measure at yo?
$dur y  y   $dur y  y
o
d 2P
 2  y
dy
 So the average of slopes at y+ and yo , which gives a better
approximation of changes in prices when yield changes, is


1
1 d 2P
$dur y  y   $dur y  y  $dur y  y 
o
o
2
2 dy2
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 y
y  yo
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V. Convexity

The forecast of price response using duration and convexity
2

1
d
P
P   $dur y  y 
o

2 dy2


2

1
d
P
*
 y   y   Dm  P  y 
2

2
dy
y  yo


 y 2
y  yo
In percentage term
P
1 1 d 2P
*
  Dm  y 
P
2 P dy2


 y 2
y  yo
The term
1 d 2P
P dy2
y  yo
1
1

P (1  y ) 2
T
Ct
1
(t  t ) 


(1  y )t (1  y ) 2
t 1
2
T
 (t  t
t 1
2
)
PV (Ct )
P
is referred to as convexity
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V. Convexity with intra-year coupons
 Dollar convexity with intra-year coupons
d 2P
1

dy2 (1  y / m) 2
mT
Cn
1
(
n

n
)



(1  y / m) n m2
n 1
2
 Convexity with intra-year coupons

 1
Cn
2
(
n

n
)

P

 (1  y / m) n
  m2
n 1


mT
PV (Ct ) 1
1
2

(
n

n
)

 2

(1  y / m) 2 n 1
P
m
1 d 2P
1

P dy2 (1  y / m) 2
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V. Effective Dollar Convexity – Numerical
Approximation
 Instead of analytical formula, in practice $ convexity is frequently
approximated using numerical methods based on price-yield relations
 P  Po Po  P


2

y
y
d P 

2
dy
y


  P  P  2  Po
(y) 2
 Numerical approximation is very useful when cash flow size and timing
are uncertain due to built-in options in the bond payments
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V. Homework


30 year T-bond has a yield to maturity of 3.0% and price at par, and coupon is
paid annually.
1. Find out analytically the following measure at 3.0%:
 A. duration
 B. modified duration
 C. dollar duration
 D. dollar convexity
 E. convexity
2. Also calculate B, C, D, and E using numerical approximations using a step (delta y) of 1
basis point. How accurate is the approximation compared with analytical solutions
from part 1?
3. Use dollar duration measure to predict price when yields change from 1% to 5% at 0.5%
interval.
4. Use both duration and convexity to predict bond price when yields change from 1% to
5% at 0.5% interval.
note: you can use either the analytical duration/convexity or their numerical
approximation for question 3 and 4.
5. What do you conclude when comparing results from 3 and 4? Which one is more
accurate and why?
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