Summary
Samenvatting - Financial Mathematics for Insurance (6414M0309Y)
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Extensive summary of the course Financial Mathematics for Insurance.
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Cash flow: the movement of money in our out of an investment or banknotation (xo
> xn) ,
x , . .
.,
Present value: determine the current wordt of a sum of money that is to received or paid in the future, accounting
(future value) FV
for the time value of money r)u >
PV (1 +
=
(present value) (interest rate)
: (1 + r)3
N
N
:
: ( + r)
X
X3
PV (xo ,
X, , . . .
.,
Xn) PV
~
XS
: (1 + r)2X2
V
: (1 + r)"
Compounding
>
interest depends on the period, we could have yearly interest, monthly interest etc.
when we want to have yearly interest rate, but we only get the interest rate per 6 months, we need
7
to tak compounding into account
ex. interest rate per 6 months = % 2
we have investment of E 1000
,
-
then after a year we have 1000 x 1 ,
02 = E1020 1020 + 1 , 02 =
E1040 ,
4
E1040
instead of 2 2 %: n 1000 + 1 , 04 =
&
O
m = number of payments a year r (1 )m note if , then (1 (m 1 + = + m d +
r
O
= interest paid per period
M
ex. : 2 %: 0 .
02
2
G r
S
= yearly interest rate r (1 02) 1 + =
+ 0 .
=
1 .
0404
r' =
0 . 040h
When we have cash flows, we have two options, put it on the bank or invest it
-
I 2
Ideal bank: swaps cashflow streams if Internal rate of return: the rate at which the initial investment
they have same PV equals the present value of future cash flows (makes the PV of
has interest rate
7
VIDEAL cash flows equal to zero), defines attractiveness of investment
0( *) + c
=
xo + +
...
+
Choosing between cash flow stream (Xo Xn) (yo y yn) ,
X ,
, . .
..
and ,
.
.
. .
..
I
2 *
Choose the one with largest PV, as bank Choose the one that has largest IRR rus ry
makes them equivalent with
(PVy ,
0
, ....
0) and (PV .
0 , ...,
0
Which one to choose or I 2
*
·
r
put money in investment
I VIDEAL
put money on bank
*
6
VIDEAL > r
,Problem: may lead to conflicting conclusions because the horizons is different (number or time points)
Ex. Cash flows IRR Ideal Bank uncertainty in time horizons D
"px
Xn)
M
(xo
M #
rx
M
,
X, .
. .
., . . .
N
"Pre
(yo ,
y .
. . .
., yn) ↑ ↑4949
*
ry . . . .
V V
As we see here, the horizons are different
To solve this the cashflow X can choose to again invest or put it in the bank
A type of investment
I
Bond market &
P = price of the bond C + F
"
F = face value, value of the bond O M
C = coupon, intermediate payments P
(kind of interest rate) V
What is a fair price of a bond? note yield = IRR of bond X =
k
r)
engin
-
P = F( , + +
used to compare bonds, tells us how interesting a bond is
7
=
F(1 + r( + C 1 -
( + r)" -
1
Bonds which are comparable should have comparable yields
·
There is a trade off between P and ↓
7
When we have two bonds (x and y), with r % =
2 and
*
ry = 3 %
, everyone wants to have bond y, so to adjust for
this we give bond x a higher price than bond y.
We need to be compensated when the probability that a cash flow is going to be paid is low
>
look at yield spreads which contains information about the financial health of a company
What investments risks are there when holding bonds?
Let's suppose, I am an insurer and I have a bond. At a certain time period, I get paid out a coupon, using this
coupon I can payout claims. However when the value of the coupon is larger than the amount that I need to
payout, I need to reinvest this difference again. Or the other way around, when I need to pay more than
the amount of coupon I received, I need to sell the bond.
However, yields change over time, which bring different risk:
>
Interest rate risk: bond value decreases, due to yield change
D
Reinvestment rate: coupons reinvested again against worse yield
&
, 8
Inflation risk: cash flows need to be higher to compensate
g
Liquidity risk: a bond is hard to sell or buy
We need to find a portfolio that minimizes these risks, we do this by analyzing how bonds react to market changes P
7
We analyse the effect of changes in yield on changes in bond prices AP
By
7
How sensitive is bond to yield changes, how sensitive priceto yield changes &
As pax e C 2 + F
We calculate P (1
It +... m
"
=
+ m
-P -
C
C+"... y
2x =
2pYax I
I
AP
>
· ... P p
--
DmAd
>
Macaulay duration
Modified duration Di
Ex. D =
90 X =
7 % Dm =
5
years
Now AJ = -
0 5 .
% from 7% to 6
.
5 % ), what is the new price?
P -Dm P Ax 90
Duration is a measure of the average time it takes for an investor to recover the
5 2 25
= =
005
.
-0
.
- . -
=
.
.
bond's price through its future cash flows, including both coupon payments and the
return of principal at maturity.
Hence new price is 92 25 .
Modified duration is a measure of the sensitivity of a bond's price to changes in
interest rates.
If I can control how a certain bond reacts to a change in yield. Then if I make a portfolio of bonds, I take many
bonds at the same time. They will all react to a change in yield, but then I can also calculate how my portfolio as a
total reacts to a change in yield. Then I can do some risk management, and try to ensure that the reaction is more
or less what I want it to be.
Create an bond portfolio which has
g Same PV as liabilities (enough value now in portfolio to pay liabilities)
Same (modified) duration as liabilities DBOND DLiabilities :
(PVBOND = PVAbilities)
>
I do not care about changes in cash flows if liabilities change in the same amount (as for example the
coupon prices go down, I do not care if I also have to pay less claims) immunization >
>
Invest in bonds with price and duration , such that
Di Pi
PVnAbilities =
a, P , + azPz +... + an PN = PUBOND
,
a P D, , + azPcDc +... + awPNDN
>
DLABILITIES =
,
a
P, D ,
P , + azP2 +... + an
PrDz
Po : DBOND
PrDN
Find di
, which determines the portfolio
= al PULIABILITIES
+
@2 PVLABILITIES +... + An PULABILITIES : DBOND
7
Results in portfolio which protects the risks in the liabilities against changes in yield
Risk is minimized because of immunization
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