# Slides by jianghongl

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```									    Agenda

Some duration formulas (CT1, Unit 13, Sec. 5.3)

An aside on annuity bonds (Lando & Poulsen Sec. 3.3
– attached to hand-out)

Convexity (CT1, Unit 13, Sec. 5.4)

Immunisation (CT1, Unit 13, Sec. 5.5)

1                  October 26, 2010         MATH 2510: Fin. Math. 2
Duration

Measures the sensitivity of present
values/prices to changes in the interest rate.
It has ”dual” meaning:
 A derivative wrt. the interest rate
 A value-weighted discounted average of
payment times (so: its unit is ”years”)

2                 October 26, 2010     MATH 2510: Fin. Math. 2
Set-up:
 Cash-flows C at tktk

 Yield curve flat at i (or continuously
compounded/on force form:  )
 Present value of cash-flows:

A  k Ct k (1  i)tk  k Ct k P(tk )

3                   October 26, 2010         MATH 2510: Fin. Math. 2
Macauley Duration

The Macauley duration (or: discounted mean term) is
defined by
k tk Ctk (1  i)
tk

 :
A

Clearly a weighted average of payment dates.
But also: Sensitivity to changes in the force of interest.
Or put differently: To parallel shifts in the (continuously
compounded) yield curve.

4                    October 26, 2010                       MATH 2510: Fin. Math. 2
Duration of an Annuitiy

The duration of an n-year annuity making payments
D is (independent of D and) equal to
( Ia) n|
,
an|
(Note: On the Oct. 21 slides there was either a D too much or a D too little)

where as usual a  (1  v n ) / i with v  (1  i ) 1
n|

5                                         October 26, 2010                          MATH 2510: Fin. Math. 2
and (IA) is the value of an increasing annuity

( IA) n| : v  2v 2  3v 3  ... nv n
 v  v 2  v 3  ... v n  v(v  2v 2  3v 3  ... (n  1)v n 1 )
 an|  (1  i ) 1 ((IA) n|  nv n )

(1  i )an|  nv n
( IA) n| 
i

6                           October 26, 2010                  MATH 2510: Fin. Math. 2
Annuity Bonds

The remaining principal (outstanding notional) of an
annuity bond with (nominal) coupon rate r and
(annual) payment D satisfies
pt  (1  r ) pt 1  D

Repeated substitution gives

pn  (1  r ) p0  D j 0 (1  r ) j
n           n 1

7                   October 26, 2010              MATH 2510: Fin. Math. 2
Suppose (wlog) p0=100.

For the loan to be paid off after n periods we must
have pn=0, i.e.
100  D j 0 (1  r ) ( j n )  0
n 1

8                    October 26, 2010                 MATH 2510: Fin. Math. 2
This we can rewrite to solve for the yearly payment
r           100
D  100              n
      .
1  (1  r )       an|
In finance people will often refer to this as the annuity
formula.
The yearly payments (or: instalments) consist of
interest payments and repayment of principal.

9                    October 26, 2010         MATH 2510: Fin. Math. 2
Duration of a Bullet Bond

Using similar reasoning, the duration of a bullet bond w/
coupon payments D and notional R is

D ( Ia ) n|  Rnv n
Da n|  Rv       n

(Note: On the Oct. 21 slides D was missing in the denominator.)

10                                         October 26, 2010                MATH 2510: Fin. Math. 2
Convexity

The convexity of         Ct k   is defined as

k tk (tk  1)Ctk (1  i)
 (tk  2)
1  A
2
c(i ) :          
A i 2                   A

This is the ”effective” (or: ”volatility” version) – could
also do Macauley or Fisher-Weil style.

11                          October 26, 2010                        MATH 2510: Fin. Math. 2
Convexity and (effective) duration give a 2nd order
accurate (Taylor expansion) approximation to
changes in present value for (small) interest rate
changes

A(i   )  A(i)    1 A(i)          1 1  2 A(i) 2
                            
A(i)          A(i) i           2 A(i) i  2

1
 v(i)     c(i)   2  
2

12                    October 26, 2010            MATH 2510: Fin. Math. 2
A classical ”picture of” duration and
convexity

Approximating present value changes w/ duration and convexity

0.5

0.4
Relative change in present value

0.3

0.2

0.1                                                                       Exact
Dur. & conv. (2nd order)
0                                                                       Dur. (1st order)
0   0.01   0.02   0.03   0.04        0.05   0.06   0.07   0.08
-0.1

-0.2

-0.3

-0.4
Interest rate

13                                                             October 26, 2010                                     MATH 2510: Fin. Math. 2
Convexity is a measure of dispersion
around the duration

We can write Macauley duration as
  t twt
and Macauley-style convexity as
  t t 2 wt
A measure of dispersion around the duration is

 : t (t   ) 2 wt t (t 2   2  2t ) wt
 t t 2 wt  2 t twt   2 t wt     2

14                      October 26, 2010                 MATH 2510: Fin. Math. 2
Immunisation

Consider a pension fund that has assets At and    k

liabilities Lt .
k

We say that the fund is immunised against
  movements in the interest rate around i0 if
PVA (i0 )  PVL (i0 )
and

PV A (i0   )  PVL (i0   )

15                    October 26, 2010           MATH 2510: Fin. Math. 2
By Taylor expanding the difference between assets
and liabilities (known as the surplus) we get that the
”inequality condition” is fulfilled if
- the assets and the liabilities have the same
duration,
and
- convexity of the assets is higher than the
convexity of the liabilities

16                   October 26, 2010       MATH 2510: Fin. Math. 2
These are known as Redington’s conditions.

It doesn’t matter whether we use effective or Macaluey
duration.

Typical exercise approach: The ”equality conditions”
give two linear equations in two unknows; the
convexity condition follows from a dispersion
consideration.

17                   October 26, 2010       MATH 2510: Fin. Math. 2
Immunisation isn’t the be-all-and-end-
all of interest rate risk management

Note that an immunised portfolio is looks very
much like an arbitrage.
That tells us that considering only parallel shifts
to flat yield curves isn’t the perfect way to
model interest rate uncertainty.
And models with genuinely random behaviour
is the next topic.

18                 October 26, 2010      MATH 2510: Fin. Math. 2

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