An analytical approximation
of relative Cost-at-Risk
by Anders Holmlund
1. Introduction
tion compensation is realised on the portion of inflation-linked
debt that falls due.
The Debt Office has developed a simple model for arriving at an
On the basis of how much falls due during the coming year,
approximate figure for short-term relative Cost-at-Risk (RCaR)
the percentages of inflation-linked and foreign currency debt in
for Swedish central government debt. The method is not based
the total debt and the average coupon on the debt, we can
on simulation but on analytical calculations.
calculate how much the costs increase for one unit of increase
The analytical RCaR measure shows how much higher
in each risk factor. With the help of historical, option-implied or
than expected interest costs may be in a one-year perspective.
assumed relationships between factors, we can then calculate
In other words, it is a relative rather than an absolute measure.
confidence intervals for cost upturns. In other words, in this way
Expected interest costs are based on a situation where interest
we can arrive at an analytic approximation of RCaR.
rates and exchange rates are unchanged, and the inflation rate
is two per cent in keeping with the Riksbank’s target.
In “Central Government Debt Management – Proposed
Guidelines”, the Debt Office describes the new measure and how
On top of financial variables, we can add unexpected
increases in the primary borrowing requirement. Such increases are assumed to be financed according to how the debt is
structured from the beginning.
it is influenced by the characteristics of the debt. In the proposal,
All equations are found in section 4.
there is also a discussion about how the new measure should
be used in the management of government debt. This technical
3. Relative Cost-at-Risk as a function of
maturity profile and percentage of
foreign currency debt
memorandum describes exactly how RCaR is calculated and
what data are used in the analysis in the proposed guidelines.
2. The Debt Office model for
approximating relative Cost-at-Risk
The chart shows relative CaR as a function of the share of
yearly redemptions, based on different historical periods, as
well as data from the option market.
In principle, there are three risk factors for the cost of debt: the
interest rate (all types of debt), the exchange rate (foreign currency debt) and inflation (inflation-linked debt).
If the interest rate climbs during a year, average interest
rate on the debt rises by the increase in the interest rate multiplied by the portion of the debt whose interest rate is refixed.
95% RCaR in a one-year perspective
(Foreign currency a constant 29% of total, inflation-linked 14%.
No shock in primary borrowing requirement)
In this simple model, refixing of interest rates is assumed to be
equal to the portion of the debt that falls due (also called yearly
redemptions or maturities). Swedish and foreign interest rates
are assumed to be perfectly correlated, while real interest rates
are assumed to vary half as much as nominal ones.
If the krona weakens during a year, coupon payments on
the foreign currency debt rise, measured in Swedish kronor. In
addition, the Debt Office realises a larger (smaller) exchange
Cost increase (SEK bn)
50
40
30
20
10
Percentage of all debt
maturing in one year
loss (exchange gain) on the portion of the foreign currency debt
that falls due.1
If inflation is higher than expected during the year, coupon
payments on inflation-linked debt rise. In addition, more infla1
0
10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60%
1994-2002
1994-1997
1999-2002
Optionimplied
By cost, we are referring here to economic costs. One difference compared to cash-basis costs is that exchange losses are considered realised when the loans fall due, regardless of
whether the Debt Office refinances the loan or not. If the approach were instead cash-basis, we would look at the share that is repaid rather than the share that falls due.
RCaR increases linearly with the share of yearly redemptions.
Swedish nominal debt and foreign currency debt are assumed
This is consistent with the conclusions in the Debt Office’s report
to have the same share of yearly redemptions (25 per cent).
“Duration, Maturity Profile and the Risk of Increased Costs for
Inflation-linked debt is assumed to have a smaller share of
Central Government Debt”, which pointed out that the share of
yearly redemptions (5 per cent).
yearly redemptions is a better risk indicator than duration.
The calculation concerns the difference compared to
Depending on what historical period we use as the basis,
ex-pected cost
cost, not the absolute cost level. The basic assump-
RCaR rises by between SEK 2 and 4 billion for each five per
tion is unchanged interest rates, unchanged exchange rates,
cent climb in the maturity profile. The percentage of foreign
two per cent inflation and a primary balance that turns out as
currency debt in the total portfolio influences this figure – given
forecasted.
a lower percentage of foreign currency debt, relative CaR rises
more slowly if we increase the yearly maturity profile.
Effect on cost of an interest rate increase
For this reason, it is interesting to view relative CaR as a
three-dimensional function of the maturity profile and the per-
(Eq. 1)
+
+
= cost increase in SEK billion
+
where Δcr
+
∆ cFX = Δr
∆ FX =wFX
S (K
+ FPnomrate
)
change
inFXinterest
+
Relative Cost-at-Risk as a three-dimensional function
+
∆ cr = ∆ r S (FPnom (1– wreal ) + FPreal wreal 0.5)
+
centage of foreign currency debt.
= central government debt in SEK billion
S
35
for nominal debt
(K+1=.96sub-index
∆ cPSBR =nom
σr )∆ PSBR
= sub-index for inflation-linked debt
real
30
+
+
maturity
profile
π – πe)= w
∆ cπ = (FP
real S (K real + FPreal )
w
= percentage (expressed as decimal)
+
Relative CaR
(1 yr 95%)
K + 1.96
σr Swedish and foreign interest rates are assumed
Please
note:
25
to move in tandem. Real interest rates are assumed to move
20
half
as much as nominal interest rates (which explains the 0.5
∆ PSBR
15
factor in the equation).
0.75
+
The chart is based on 12-month changes during the period 1998-2002
During this time period, CaR rises by SEK 2.3 billion for each
+
e
= sub-index for foreign currency debt
FX
+
debt represents 30 per cent of total debt. If the percentage of
+
= (K+1.96 σ )∆–PSBR
∆∆ccPSBR
r = ∆ r S (FPnom r(1 wreal ) + FPreal wreal 0.5)
σcr
5 per cent increase in the maturity profile, if foreign currency
+
0
+
+
Annual
redemptions
divided by total debt
0.25
+
+
0.05
+
FX debt divided by total debt
0.10
σ on cost if inflation is higher than expected
Effect
∆+crFX
= ∆σ FX wFX S (KFX + FPnom )
K
1.96
r
σC = σ T Ωσ + ∆ cPSBR
(Eq. 3)
e
–
π
π
) wreal S (K real + FPreal )
∆ cPSBR
π= (
∆
+
foreign currency debt is halved, CaR rises by SEK 1.2 billion for
the same interval on the maturity profile.
+
The same pattern applies if we keep the percentage of foreign currency debt constant and vary the maturity profile. In other
words, the percentage of foreign currency debt and the maturity
where π
+
+
0.15
+
0.20
+
0.5
0.25
+
0.30
+
+
1.0
0
0.35
+
+
5
+
σ cr =
+
∆ cr
σr
∆r
∆ cr = ∆on
r Scost
(FPnom
wreal ) + FP
Effect
of (a1–weaker
krona
real wreal 0.5)
c FX
FX
(Eq. 2)
σc =
σ
FX FX
∆ cFX = ∆ FX wFX S (KFX + FPnom )
∆ cr
π
σc =
σ (π – π )
e
where(π
Δ – π )= change in krona exchange rate (weakening)
ΔFX
π – πe)= w
∆ cπ = (K
real S (K real + FPreal )
coupon
10
= actual inflation
∆ cPSBR
∆ c=e(K+1.96 σr )∆ PSBR
σ cr = πr σr= expected inflation (in Sweden’s case 2 per cent)
∆r
+
profile have a mutually reinforcing effect on relative Cost-at-Risk.
K +FX1.96
σr
c FX
σc =
σFX
Borrowing
effects
FX requirement
+
4. Assumptions and equations for relative
Cost-at-Risk 2
The primary borrowing requirement may be larger than expect-
∆ cr
π
∆ PSBR
π
( –π)
+
σto
ed.σcIt=is assumed
π ) financed according to how the debt is
(π –be
e
Basic assumptions
e
structured. The interest rate on new loans is assumed to be the
∆ cr
∆r
The three risk factors – interest rate, exchange rate and infla-
σ c = coupon
σr on the debt plus 1.96 standard deviations on
average
tion – are assumed to be normally distributed.
the interest rate. 3
+
r
+
r
σFX
c FX
c
σ c =confidenceσinterval
FX
In the preparation of this memo, I encountered an error in the calculations. I have calculated a two-sided
instead of the one-sided interval I intended. This means that
σr FX
what we call 95 per cent RCaR is in fact 97.5 per cent RCaR. However, this error does not affect our qualitative
conclusions in the Proposed Guidelines.
∆ cr
π
σσthe
The number of standard deviations must naturally be adapted to what confidence interval we have in
measure.σ – π )
σ T–Ωσ
cC =RCaR
π
( πe)+ ∆c(πPSBR
2
+
3
2
σcr
e
+
+
+
+
∆ cr = ∆ r S (FPnom (1– wreal ) + FPreal wreal 0.5)
+
+
+
+
+
+
∆ cFX = ∆ FX wFX S (KFX + FPnom )
∆ cFX = ∆ FX wFX S (KFX + FPnom )
+
+ +
+
+
+
∆ cπ = (π – πe) wreal S (K real + FPreal )
∆ cπ = (π – πe) wreal S (K real + FPreal )
∆ cPSBR = (K+1.96 σr )∆ PSBR
∆ cPSBR = (K+1.96 σr )∆ PSBR
Examples of how relative Cost-at-Risk is calculated
(Eq. 4)
At the end of July 2003, Swedish central government debt
amounted to SEK 1,193 billion. The share of redemptions
where K + 1.96 σr = the interest rate at which the increased
borrowing requirement will be financed
K + 1.96 σr
within one year is assumed to be 25 per cent for nominal krona
debt and foreign currency debt, and 5 per cent for inflation-
= unexpected increase in the primary
Δ
ΔPSBR
linked debt. Foreign currency debt represented 28.9 per cent
+ + +
+ +
++ + +
++ + +
+
+
+
+
+
+ +
+
+
+
+ +
+
+
+
PSBR
+
+
∆ cPSBR = (0.05 +1.96 0.0161) 20
∆ cPSBR = 1.63
+
+
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+
+
+
+
∆ cPSBR = (0.05 +1.96 0.0161) 20
∆ cPSBR∆=c(K+=1.96
σr )∆ PSBR
1.63
3
+
+
+
+
+
+
+
+
+
++
++
+
+
++++
+
+
+
++
+
++ +
+
+
+
++
e
+
+
e
+
+
+
+
e
+
+
+
++
+
e
e
e
e
++
+
+
+ +
+
++
+
+
++
+
++ +++ + + ++ +
+
+
+
+
+
++
+
+
+
+
+ +
+++ +++
+
+
++ +
++
+ + + ++ ++ ++
+
+ +++
++
+
+
+
+
+
+
+
++
+
+
+
+
+
+
+++ +++ ++
+
+
+ +
+
+
++ + +
+
+
+
++ + ++ ++
+ +
+
+
++ + + +
+
+
+
+
+
+
+ +
+
+
+
+
++
++
++
+
++
+
+
+
+
+
+
++
+
++
+
+
+
++
+
++
+
++
e
e
+
+
+
+
+
+ ++
+
+
++ + + ++
+ +
++
++
+
+
+
+
+
+
+++ ++ +
+ + +
++ +
+
+ + +
+ +
+
∆ PSBR
borrowing requirement
of the total debt portfolio, and inflation-linked debt 13.7 per
∆ PSBR
cent. The average coupon is assumed to be five per cent.
∆ cr = ∆ r S (FPnom (1 – wreal ) + FPreal wreal 0.5)
∆ cr
σ cr = borrowing
σr requirement could be modelled in relaThe primary
r
∆
c
∆
∆ crr = ∆ r 1193
S (FP(nom
) + FP
0.25(1 –(1w–real
0.137
) +real0.05wreal0.120.50).5(Eq.
) 1)
r
σ cr =
σr variables, based on a historical correlation. But
tion
to the other
∆r
–
1193
0
.
25
1
0
.
137
+
0
.
05
0
.
12
0
.
5
(
)
(
c
=
r
261
.
0
∆
∆
– wreal
wreal 0w.5) )0.5)
(1nom
rrr
c FX requirement is influenced by many factors
real
∆ cr =S ∆ (rFPSnom(FP
the primaryFXborrowing
(1 )–+wFP
real ) + FPreal
real
σc =
σFX
=∆
261.0 (0.25 (1 – 0.137) + 0.05 0.12 0.5)
∆ ccr =
∆ rr 1193
∆
FXinclude, such as the GDP trend, divestment
that FX
this cdoes
not
–
r
FX
∆
∆
c
=
r
1193
0
.
25
1
0
.
137
+
0
.
05
0.12 0.5)
(
)
(
r
σc =
σ
FX FXcompanies
c
=
r
261
.
0
∆
∆
of state-owned
and
other
political
decisions.
We
∆ cr
r
π
∆ cr = ∆ r 261.0
σc =
σ
∆ cr = ∆ r S (FPnom (1 – wreal ) + FPreal wreal 0.5)
haveπ therefore
chosen
– πe) not(π –toπ ) do this, but have instead made
∆ cr (π
c
=
∆ FXw wFX0.5)S (KFX + FPnom–)
∆
σc =
σ
(Eq. 0
2).5
–
FX
∆
c
=
∆
r
S
(
FP
(
1
w
)
+
FP
∆real
cr = ∆
r 1193 (0.25 (1 0.137) + 0.05 0.12
r
nom
real
real
)
the primary
a static
πe) (∆π –cπr)requirement
= ∆ r S (FPnom
(1– wvariable.
(π – borrowing
real ) + FPreal wreal 0.5)
w
S
K
+
FP
(
)
c
=
FX
0
.
289
1193
0
.
05
+
0
.
25
∆
∆
)
FX
FX
FX ( nom
FX
–
∆∆
(1(–1interest
ww
FP
wreal
0.5
In∆
principle,
also
lead
to
–w
∆ cr = ∆ r 261
S (.0FPnom (1 – wreal ) + FPreal wreal 0.5)
S((FP
(FP
FP
FP
0.5
nom
real
real
ccrcr==
∆∆∆rrr SSincreased
)))+++costs
FP
)))a larger
r=
nom(1
real
real ww
real 0.5
nom
real
real
real
0w.289
0.05 +)0.25)
.4 Sw1193
∆ c+FX
FX = ∆ FX 103
(KFX +(FP
borrowing requirement,
which
must
be
financed.
It
is,
so
to
FX
FX
∆
∆
c
=
FX
w
S
(
K
FP
)
c
=
FX
∆
∆
∆
=
r
1193
1FX–+0FP
.137
(nom
FX
FX
nom FX
FX(0.S25 (K
nom))+ 0.05 0.12 0.5)
r
∆ cFX = ∆ FX wFX S (KFXFX
+ FPnom )
σcr
c
=
FX
103
.
4
∆
∆
FX =
c
=
FX
0
.
289
1193
0
.
05
+
0
.
25
∆
∆
–
speak,
a
“cost
on
top
of
the
cost
increase”.
But
this
is
a
second(
∆
∆
c
r
S
FP
1
w
+
FP
( 1193
FX
= ∆(FX nom.00.289
.05)+w0real
.25)0.5)
(0real
r ∆ c FX
real )
FX
wwFXFX SSS ((K(KKFXFX+++FP
FP
))
cFXFX===∆∆∆FX
FX w
FP
r = ∆ r 261
nom
σcr ∆∆∆ccFX
)
nom
FX
FX
nom
σ
ary effect
been
excluded
in our calculations.
=∆∆∆crFX1193
–
∆ cr =that
∆r r has
S (FP
.25 .4(1 – 0.137) + 0.05 0.12 0.(Eq.
5) 3)
= ∆103
FX.(40103
nom (1 wreal ) + FPreal w∆
real
r =+
(π) – πe) w S ∆(∆KccFXreal
c =0.5
FP
reale) ∆ FX
c–FX
w
KFP
(
∆ cπ = (π – πe) wreal S (Kπ real + FPreal ) real
σr
FXπ=
FX (KS
FX + FP
nom )
π
∆∆ccπ ==(∆∆
w
S
+
)
)
T e
real
real
r 261.0 real
r
cPSBRSS ((KK ++FP
))
==C((=π
πe))e)+ww∆wreal
(ππσ–––ππΩσ
cπ=Tπσ
FP
real
real
c––FXπ=ee)∆ w
FX 0S.289
1193
0.05 + 0.25)
∆π
)
Overall
∆∆∆ccπrisk
+ FP
real S (K real
real
real
π
∆
K
+
FP
c
=
(
(
real
real
π
real
real
real
π
0
.
137
1193
0
.
05
+()0.05)
∆
c
=
.
(
)
c
=
K
+
1
96
PSBR
∆
∆
(
)
(
σ
σC∆=cσ
Ωσ
∆ FX
wFX S= ((K
K+
FPnom
π
r
+ ∆ cPSBR
PSBR
FX =
FX1+
.96
PSBR
σr ))∆calculate
PSBR
From equations (1),∆ c(2)
and (3)
we
can
factor senc
=
FX
103
.
4
∆
∆
e
wFX S ((0K.05
FX +
πc––FX=π
πe) –w
0.real
137
+FP
0.nom
05))
∆ ccππ =
= (∆π
(K real + FP
∆
16
.e3 S1193
96
PSBR
∆∆PSBR
real )
.96
(Kthree
cPSBR
K
PSBRTogether with historical or
r ))
π (π π ) wreal S (K real + FPreal )
===((K
+++11.1.96
∆∆∆ccfor
PSBR
σσσ
sitivities
the
risk
factors.
r )r∆
PSBR
c
=
FX
0
.
289
1193
0
.
05
+ 0.25)
∆
∆
e
(
∆ cππ = (π –FXπe)π –16
.e3 01193
0π
.137
.05 +(00..05
(0FP
πe) wrealdeviations,
∆ cπ = (π –standard
S (K realwe
+ FP
.137
1193
05)+ 0.05)
∆ c∆πFX
=( w
)
option-implied
can
calculate
K + 1.96 σstandard
real )
c
=
S
K
+
∆
(
)
r
FX ∆ c FX = ∆ FX
FX 103.4 FX
nom
K + 1.96 σr
–=π(e)π
∆∆ccπ = (=∆π∆
16
.ee3)period
deviations
for
cost
increases by risk factor.
Historical
data
for–
the
1994
–.05
2002 25
provide
π
16
.
3
c
FX
0
.
289
1193
0
(
) the following
π
K
+
1
96
σ
.
–
π
π
K
+
1
96
σ
.
FX
∆
w
S
K
c
=
1π.61(%
) real
( real++0.FP
σr
Kc+ 1.96
σKrr +
r 1.96
real )
(
)
=
PSBR
∆
∆
σ
r
PSBR
Factor
sensitivities
are
obtained
by
dividing
equations
(1),
standard
deviations
(for
twelve-month
changes):
∆ cr = ∆ r S (FPnom (1– wreal ) + FPreal wreal
0.5)
cr FX==∆ ∆
∆
∆ PSBR
61
42(%π 103
– πe.)4 0.137 1193 (0.05 + 0.05)
σFX
c61π.FX
=
∆ PSBR
(2) and (3) by the change
in each respective risk factor and mul42%
%
=∆ c161π..12
σFX
PSBR
π – πee) w
∆with
=
π
PSBR
∆PSBR
16real
.3 S (K real + FPreal )
σ
tiplying∆
the standard deviation of each respective risk factor.
r
σrπ61(% 1.61%
c
∆
Kc+FX1.=96∆σFX
∆
w
S
(
K
+
FP
)
r
r
r
σ
e
1
.
12
%
FX ∆ c
FX
nom
π
σc =
σr
π – π ) 0.137 1193 (0.05 + 0.05)
42(%
σFX =∆σc6π.=
σ cr = r σr
FX =e 6.42 %
∆ r (Eq.
5)
–12π%) wreal
∆∆cccr r
π
∆
S (K real + FPreal )
c
=
r
∆
(
rr ∆
π
σ
1
.
πe%
∆σcππ = (π1–.12
.3
πr
= r σσσr r
) .016
σσσcrc =
c =∆ r
e
r
1
.
61
%
261
σcc = (σπr – πe) 1.61
% 11934.202
(π∆r r– π ) wreal c S (K real + FPreal ) FX cFX
∆∆cPSBR
0.137
∆
(0.05 + 0.05)
π= ∆
π
r
FX
σc =
σFX
FX
FX
4
202
1
.
61
%
261
.
0
=
σ
=
6
.
638
6
42
103
4
c
FX
6
.
42
%
=
=
σ
σ
e
σ
(Eq. 6)
c
∆ccFXπ = (πFX– π ) 16.3
ccFX
FX FX
FX c
π =
FXFX
=c=r (FXKFX+1σσ.σ
638
42
%1.the
103
)
96
PSBR
σ
∆σcrσσ
∆
%
16%.vector
3.4 = 460..196
σ
c c∆
FX
σ116rπr ..12
r
This
that
standard
PSBR
=
61
%
261
FX
σccr means
61
FX
c
∆
π
FX
4.202 deviations for cost
1.12
61
%.0 261with
.0 202
σ
r
σ c =c
σr FX π
FX
c
c
∆
π
σ
σ
=
–
r
π
π
c
(
)
FX
∆r
σ
e
0
.
196
1
.
12
%
16
.
3
increases
in
the
three
factors
is
c
σc =
FX
σ
–
σc = σFX
= .46.638
6.42
103
(π π(Eq.
) 7)
%
(π – π )
e
=%66..42
= 6.638
42
%.4 103
π
c
∆∆∆ccrcr r σσπ(– ππ – π )
π π=
π
c c==
(π(–ππ– π) )
1
.
61
%
σσσ
e
π
σ
σ
σ
0
.
196
1
.
12
%
16
.
3
e
σ
1
.
12
%
cr
π
0.196
1.12% 16.3
π––ππe ) ( )
K +FXc1.96(c(π
σ
FX
cπr
σ c = ( r– πσ)FX)
4.202
1.61% 0.55
261.0
σFX =1σc6.420%,.11
FX
FX
=%
6.42%–00.55
0,.111
r
11
.103
02 .4 = 6.638
Ωσ=π 0.1σ
c1.12
σ
c
∆
r
π
c
r
π
∆σPSBR
r
where,
for
example,
means
the
standard
deviation
in
cost
σ
=
σ
c
(cπ – π )
11
.16
02.3.0
Ω = 0,..1σ
40.196
202
1.02
.12
61%
%–00.155
261
e
cc
55
–00,.111
r
r r(π – π )
1
0,.11
0.55
σσbased
c
increaseσ
on interest rate change and σr is a standard
FX
c c
σr
,..σ
55
=– 016.02
.42%– 01.103
.4 = 6.638
c
11
Ω =r 00Ω
=
0.11
1 02 – 0.02
∆ cthe
deviation
in
r interest rate risk factor.
T
π1.61%
40.,.202
261
.
0
rr
σ
σ cr σ
σ
=σσ
1
11 0.0196
.55 4.202
σ
σ
σ
Ωσ
c
1
.
12
%
16
.
3
c
=
∆
+
r
T
c
r r
C
0,.55
0matrix
.02 – 0for
1 twelve-month
σC = σbetween
Ωσ + ∆the
cPSBR
0–,.55
.02
1
Since∆the
correlation
risk factors is not one,PSBR The FX
correlation
changes in the same
=
σ
=
6
.
638
6
.
42
%
103
.
4
T
1
0
,.
11
0
202 = 68.7
4.202
6.638 10.1960,.11
0.11 0.55
1 – 0.55
.02 64.638
T
c
r = σT Ωσ
c
∆
+
σ
σ
Ωσ
c
=
∆
+
C
PSBR
σσσ
the total
risk
will
be
less
than
the
sum
of
the
individual
risks.
period
is
σ
Ωσ
c
=
C
∆
PSBR
+
cC
π
PSBR
c FX
.120%
16.130.111
4σ.202
0.196
1.02 – 0155
.02 460..196
638
68.7
.11
02
Ω6.=1638
FX
55– –00.0196
c
202 4=.202
=
σ crisk
σ
Overall
1,.11 00,..11
0.55
σr isFX FX
00..11
55 0–.155
01.02 – 01.02 60..196
.02
4.202 46..202
6380,.155
196–000,.11
638 6=.638
68.7 = 68.7
60..638
.196
0.11
1 – 0.02
(Eq. 8)
T cr
∆
π
10.55– –00.0.02
Ω = 0.11
.02 – 01.02 01.196 0.196
σσc C== σ Ωσe + ∆
σcPSBR
55
(π – π ) (π – π )
1
0,.11 0.55 4.202
1
0
,.
11
0
.
55
0
,.
55
–
0
.
02
1
∆ cPSBR = (K+1.96 σr )∆ PSBR
6.1638
.196
1 – 0.02 6.638 = 68.7
= =
04.11
0.02 0.11
where Ω denotes the correlation matrix between the risk factors
(.202
K+1.96
∆ΩccPSBR
σr )∆0–PSBR
∆
PSBR = (0.05 +1.96 0.0161) 20
– 0increases
.02 1 will 0then
196be
0,.55variance
– 0.02for unexpected
1
and σ is a column vector with standard deviations for each risk
The overall
cost
0.05
+(1K.σ
96
0PSBR
.0161
)0.155
20 0,.11 0.55 4.202
= c(1.63
∆ cPSBR
K
+1=.96
PSBR ∆
r
+r1).∆
96
σ
r )∆ PSBR
PSBR
factor. σ
If cwe add the (static) unexpected increase in the primary
.202 6.638 0.196 0.11
1 – 0.02 6.638
= 68.7
(Eq. 8)
=4(1.63
∆ccPSBR =
0.05 +1.96 0.0161
20
∆
PSBR ∆ c
borrowing requirement, the equation becomes:
1 0).00161
0.55
,.11 )– 0020
55 14.2020.196
PSBR = (0.05 +1.96
..02
σr
= c1.63
∆ 4c.PSBR
202∆
.638= 1.63
0.196 0.11
1 – 0.02 6.638 = 68.7
(Eq. 9)
∆ c6PSBR
T
PSBR = (K+1.96 σr )∆ PSBR
σC = σ Ωσ + ∆ cPSBR
0.55 – 0.02 1
0.196
∆ cPSBR = (0.05 +1.96 0.0161) 20
K+1.96 σr )∆ PSBR
∆ cPSBR = (1.63
+
+
+
4.202
1.61% 261.0
σcr
σcFX = 6.42% 103.4 = 6.638
σcπ
0.196
1.12% 16.3
and the standard deviation is thus the square root of 68.7,
foreign currency debt. The dollar percentage is approximately
which equals SEK 8.3 billion. This means that with 67 per cent
equivalent to the sum of the USD, GBP and JPY percentages.
certainty, the actual
cost
1
0,.11of the0.debt
55 during next year will deviate by a maximum of SEK 8.3 billion from the forecast.
1
– 0.02
Ω = 0.11
0,.55 – 0.02
1
In Cost-at-Risk calculations, we are only interested in the
risk that the cost will be larger than expected. We can then say
that with 95 per cent probability, the cost will be no more than
1
0.11
Inflation is measured as the twelve-month change in the
Consumer Price Index (CPI).
Monthly data have been used, and all historical series have
been obtained from the EcoWin database programme.
We use year-on-year changes in the monthly data. For exam-
0,.11 0.55 4.202
ple, this means that the period 1998 to 2002 contains underlying
1 – 0.02 6.638 = 68.7 monthly data between January 1997 and December 2002.
SEK 8.3 billion × 1.96 = SEK 16.3 billion higher than forecasted.
4.202 6.638 0.196
0.55 – 0.02meanwhile
1
0.deterio196
If the primary borrowing requirement
Option-implied calculations
rates by SEK 20 billion, the cost rises further:
(Eq. 4)
∆ cPSBR = (K+1.96 σr )∆ PSBR
+
+
∆ cPSBR = (0.05 +1.96 0.0161) 20
∆ cPSBR = 1.63
Implied volatility for one-year swaptions starting in one year has
been used to approximate the risk of interest rate increase in a
one-year perspective.
Implied volatility for foreign currency options on USD and
EUR against SEK has been used to approximate the risk of a
weakening of the krona in a one-year perspective. The two
A surprisingly large borrowing requirement – at the same time
implied volatility measures have been linearly weighted, i.e.
as interest rates, exchange rates and inflation show adverse
without taking into account that the correlation between USD/
changes – can thus result in an unexpected cost increase of
SEK and EUR/SEK may be less than one.
around SEK 18 billion.
Inflation uncertainty and correlation are also approximated
with this method, using historical data. Year-on-year changes
5. Data for the calculations
Historical calculations
Twelve-month Treasury bills are used as a measure of inter-
during the period December 2001 to December 2002 are used
for both parameters.
6. References
est rate increase. This is a reasonable approximation, since a
large proportion of borrowing each year is in Treasury bills and
“Central Government Debt Management – Proposed Guidelines”,
since a large proportion of the movements in the yield curve is
document number 2003/1665, Swedish National Debt Office.
explained by parallel shifts.
“Duration, Maturity Profile and the Risk of Increased Costs
for Central Government Debt, document number 2003/1590,
EUR. This is an approximation of the actual structure of the
Swedish National Debt Office.
Wildeco
The change in exchange rate is measured as a change in a
currency basket made up of 35 per cent USD and 65 per cent
Visiting address: Norrlandsgatan 15 • Postal address: SE-103 74 Stockholm, Sweden
Telephone: +46-8-613 45 00 • Fax: +46-8-21 21 63 • E-mail [email protected] • Internet: www.rgk.se