Interest rate risk teacher edition

Interest Rate Risk and an
introduction in fixed-income
management and two practical
cases included
By André Fail
Date: November 24, 2014
1

Interest Rate Risk
General notes:
• This presentation is for educational use so it is
not an investment advice!
• Please take your laptop or calculator with you
because it will help you solving the cases
2

Interest Rate Risk
A bit of my background as portfolio-manager
fixed-income/treasury
• Macro-economic strategy
• Behavioral finance
• Derivatives/Interest markets
• Portfolio management
• As treasurer responsible for funding, currency
trading & hedging
3

Interest Rate Risk
How does a workday look like for a portfolio-manager?
• He reads the newspaper & watch the news -> behavioral finance
• He checks the closings of equity and fixed income markets -> risk appetite!
• He checks the yield curve and especially the forward curve of money market and fixed income markets
• He checks the cash balances for cash withdrawals or cash inflows of the portfolios. A cash inflow or outflow can
result in mandate violations and even to deviations in the duration of the portfolio
• He checks new issues of bonds or coupon-payments->demand & supply
• He talks with his colleagues, brokers and strategists about the market developments and about new issues of
securities
• He trades bonds , derivatives and money market products
• He makes on request of client, account manager, risk and/or actuarial department portfolio proposals
• He answers and explains performances to clients, account-managers and reporting department
• He writes an outlook on interest rates
• He participates in investment policy councils as a representative of fixed income but should have view on equities
too
• He has to assist in pitching for new mandates
• He has to 4-eye presentations…
Conclusion: A portfolio manager is a real all-rounder with his eyes and ears tuned to the general public!
4

Interest Rate Risk
Today’s program:
1. What is interest Rate Risk?
2. What types of interest rate do we know?
3. How to calculate the value a bond?
4. What is a yield curve?
5. What is a zero curve?
6. What information can be derived from a zero curve?
7. What is duration and convexity of a bond?
8. How do we calculate the value of portfolio of bonds?
9. What about matching of liabilities?
5

Interest Rate Risk
Important assumptions:
• European markets, coupon payment once a year
• Accrued interest isn’t taken into account,
although markets do
• No difference between money market (act/360)
and bond market convention(30/360)
• Only bullet bonds are used in the analysis, thus
early redemption or optionality's aren’t taken
into account
6

Interest Rate Risk
Types of interest rates
• Central Bank Rate (Refi-rate)
• Money market rates (Euribor, EONIA)
• Government Bond Yields (German Bund)
• European Swap Rates
• Credit yields (a mortgage bond is just a special
credit)
7

Interest Rate Risk
A few graphs to illustrate interest rate risk
• 2yrs European Swaprate (10/20/14)
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Interest Rate Risk
9

Interest Rate Risk
10

Interest Rate Risk
11

Interest Rate Risk
The credit crisis in another perspective!
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Interest Rate Risk
What moves interest rates?
• Demand & Supply of cash flows
• Central Bank policy & policy rates
• Deviation from economic expectations (IFO, PMI,
productivity)
• Economic surprises, shocks, something huge and
unexpected e.g. terrorist attack, bankruptcy etc.
• Change in risk perception
• Rating agencies (S&P, Fitch, Moody’s)
• Inflation(CPI, CPI-core)
• Economic Growth(GDP)
13

Interest Rate Risk
DEFINITION OF 'INTEREST RATE RISK‘:
The risk that an investment's value will change due to a change in the
absolute level of interest rates, in the spread between two rates, in the shape
of the yield curve or in any other interest rate relationship. Such changes
usually affect securities inversely and can be reduced by diversifying
(investing in fixed-income securities with different durations) or hedging (e.g.
through an interest rate swap).
Source: http://www.investopedia.com/terms/i/interestraterisk.asp
14

Interest Rate Risk
• First a little example to illustrate interest rate
risk
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Interest Rate Risk
• Suppose Anna and Bello receive from their
Grandma both Euro 1000,=
• The grandchildren pursue each a different
investment strategy but both don’t plan to
consume the money or interest paid within
one year
• What do you think of their investment
strategy in the first year?
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Interest Rate Risk
This is Anna’s investment strategy:
• Every 6 months she invests in a term deposit
at the bank
• She reinvests accrued interest
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Interest Rate Risk
This is Bello’s investment strategy:
• Every 12 months he invests in a term deposit
at the bank
• He reinvests accrued interest
18

Interest Rate Risk
They both save at the Savings Bank and this are
the quotes of the bank:
• 6 months 1%
• 12 months 2%
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Interest Rate Risk
Bello has chosen for the time deposit of 1 year,
so after one year his deposit has grown to:
1000 * (1 + 0,02) = 1020
So after exactly one year his account has grown
to Euro 1020,=
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Interest Rate Risk
Anna has chosen for the time deposit of a half year, so her
deposit has grown to:
1000 * (1 + 0,01 * 0,5)= 1005
So after exactly a half year her account has grown to Euro 1005,=
Note that in real-life calculations we would use:
(actual number of days)/365 instead of 0,5
21

Interest Rate Risk
• What can we say about the investment policy of
Anna?
• In a half year her money has grown
to Euro 1005.=
• For comparison the value of Bello’s at that same
moment has grown to Euro 1010.=
• Anna has to reinvest her money with the
accrued interest rate against an unknown
interest rate  this is interest rate risk!
22

Interest Rate Risk
• Could we say something about the expected
interest rate of the Anna’s deposit in the
second half of the year?
23

Interest Rate Risk
• Yes, we can tell something about the rate at
which Anna might reinvest her money for the
remainder of the year!
• In our real world we assume that there are
always market participants who take
advantage of arbitrage opportunities and
further that markets are efficient so there are
no trade barriers!
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Interest Rate Risk
• At the start of the two deposits we can derive
the expected interest rate for Anna’s deposit
in the second half of the year.
• At the start of the two deposits we assume
that all the current market expectations are
included in the prices.
• So we can calculate what interest rate market
participants expect to receive in the second
half of the year.
25

Interest Rate Risk
• In an efficient market both deposits are
expected to grow to the same future value
given the “market information” at the start of
the deposits.
26

Interest Rate Risk
We would expect after one year that Anna’s and
Bello’s will grow to the same future value, but
Anna makes a bet on future interest rates.
This bet is exactly interest rate risk!
27

Interest Rate Risk
• If Anna can reinvest after half a year against
higher rates as the market currently expects she
will end up with more money in her account after
one year than Bello. Hence if she reinvests
against lower rates than current expectations, she
will end up with less money than Bello!
• Bello doesn’t incur any interest rate risk during
the year (he knows exactly what will be in his
account after one year)
28

Interest Rate Risk
Time Anna’s account
[EUR]
Bello’s account
[EUR]
0 1000 1000
0,5 1005 1010 *)
1 ? 1020
29
*) Hence this is the expected value of his account at t=0,5, the actual value
could be influenced by market forces

Interest Rate Risk
In an efficient market we would expect that
Anna’s account will grow to Euro 1020.=
Thus we have to calculate:
1005 * ( 1 + r * 0,5) = 1020 <-> r = 3%
We call this 3% the expected 6 months forward
rate for a 6 month deposit
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Interest Rate Risk
Overview based on market expectations:
31
Time Interest
Rate
[%]
Anna’s
account
[EUR]
Bello’s
account
[EUR]
0 1% 1000 1000
0,5 3% 1005 *)
1 1020 1020
*) the value of his account is expected to be Euro 1010 after a half year but
this value could be influenced by market forces

Interest Rate Risk
It is clear Anna needs some investment advice
by you! Could you help her? What do you think
of her investment policy?
32

Interest Rate Risk
A way to illustrate the interest risk taken by
Anna is to assume future 6 months deposit rates
of 2% and 4%.
By comparing the results at different reinvest-ment
rates we get more insight in the risks taken
33

Interest Rate Risk
An overview of some possible results of Anna’s
investment policy
34
Time Anna’s
account
[EUR]
Anna’s
account
[EUR]
Anna’s
account
[EUR]
Bello’s
account
[EUR]
Reinvestment-rate
after 6
months
2% 3% 4%
0 1000 1000 1000 1000
0,5 1005 1005 1005
1 1015 1020 1025 1020
Average yearly
rate 1,50% 2,00% 2,50% 2,00%

Interest Rate Risk
• Now we turn to the bond market
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Interest Rate Risk
120
100
80
60
40
20
0
1 2 3 4 5 6 7 8 9 10
face_value
coupon
Pay off scheme of a bullet bond
36

Interest Rate Risk
• How to price a bond:
37

Interest Rate Risk
Note from the pricing formula:
• There is implicit an assumption that all income
is reinvested against the same yield, which is
seldom the case.
38

Interest Rate Risk
Other measures to quantify price risks
• Yield to maturity
• Macaulay Duration
• Modified Duration
• Convexity
39

Interest Rate Risk
Let’s have a closer look at our pricing formula of
bond:
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Interest Rate Risk
• Formula for Price:
P(P,C,y,t) = C/y * ( 1-(1+y)^-t) + 100 * (1+ y)^-t
P= price
C= coupon
t = time to maturity
y = yield to maturity
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Interest Rate Risk
• From the pricing formula it becomes clear that
most of the price is determined by the present
value of the nominal
42

Interest Rate Risk
• Formula for Mod Dur:
ModDur(P,C,y,t) =
((C/y^2)*(1-(1+y)^-t)+(t*(100-C/y)*((1+y)^-(t+1))))/P
P= price
C= coupon
t = time to maturity
y = yield to maturity
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Interest Rate Risk
Note the yields we use are yield to maturity (ytm)
Period Cash Flow P(1) P(0) P(1)
3,50% 4,50% 5,50%
104,5151 100,0000 95,7297
1 4,5 0,9662 0,9569 0,9479
2 4,5 0,9335 0,9157 0,8985
3 4,5 0,9019 0,8763 0,8516
4 4,5 0,8714 0,8386 0,8072
5 104,5 0,8420 0,8025 0,7651
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Interest Rate Risk
Note from the pricing table:
• Discount factors are calculated for every cash
flow and interest rate e.g. (1,035)^-5=0,8420
• All cash flow are multiplied with the discount
factors and added together in each column
• Have a look at these price swings! A 1%
decrease in yield to maturity results in a total
return of +4,51%. Similar an increase results in
a total return of -4,27%
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Interest Rate Risk
Some conclusions:
• Note that yield and price move in the opposite
direction! Thus:
• Yields move upwards, thus the price of the bond
decreases
• Yields move downwards, thus the price of the
bond increases
• If the price of the bond is 100 or trades at par the
yield to maturity equals the coupon
46

Interest Rate Risk
Another important risk measure is Macaulay
Duration which is the first derivative of the
pricing formula.
47

Interest Rate Risk
An example for the calculation of
Macaulay duration
Period Cash Flow DF PVCF t*PVCF
1 4,5 0,9569 4,3062 4,3062
2 4,5 0,9157 4,1208 8,2416
3 4,5 0,8763 3,9433 11,8300
4 4,5 0,8386 3,7735 15,0941
5 104,5 0,8025 83,8561 419,2807
100,0000 458,7526
i= 4,50% MacDur= 4,588
48

Interest Rate Risk
• Mostly used is modified duration which can be
derived easily from my Macaulay Duration:
MacDur/(1+i)
So modified duration of our example is: 4,39
49

Interest Rate Risk
Modified duration is a measure of price
sensitivity which will be studied in the next few
slides.
50

Interest Rate Risk
Modified duration is used as an approximation to calculate the
return of a bond when interest rates change.
51
yield change dP(ModDur) dP (exact) abs error
-1,00% 4,39% 4,52% 0,13%
-0,50% 2,19% 2,23% 0,03%
-0,10% 0,44% 0,44% 0,00%
-0,01% 0,04% 0,04% 0,00%
0,00% 0,00% 0,00% 0,00%
0,01% -0,04% -0,04% 0,00%
0,10% -0,44% -0,44% 0,00%
0,50% -2,19% -2,16% 0,03%
1,00% -4,39% -4,27% 0,12%

Interest Rate Risk
• Hence: if our goal is to calculate the total
return over a time span of 1 year, than we
have to take the coupon payment or return
into account.
• Thus if we assume a coupon payment of 4,5%
a year the total return on our bond over a
horizon of one year when interest rates rise
1% equals -4,39%+4,50%= + 0,11%
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Interest Rate Risk
Another measure of risk is convexity which is the second derivative of the
price formula. This measure enables to calculate price changes due to interest
rate changes.
Period Cash Flow (1+i)^-(t+2) t(t+1)CF t(t+1)CF/((1+i)^(t+2))
1 4,5 0,8763 9,00 7,89
2 4,5 0,8386 27,00 22,64
3 4,5 0,8025 54,00 43,33
4 4,5 0,7679 90,00 69,11
5 104,5 0,7348 3135,00 2303,69
3315,00 2446,66
i= 4,50% Convexity 24,47
given P=100
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Interest Rate Risk
With the help of the first and second derivative
of the price formula we can study the price
return of a bond without the need of continuous
recalculation:
Price change caused by convexity given yield
change of 1%:
= 0,5(24,47)(0,01)^2 = 0,12% (check the table!)
54

Interest Rate Risk
Nowadays we prefer to use DV01 instead of
Duration. This is the measure of change in value
of 1bp (=0,01%)
DV01= -MarketValue * ModDur * 0,01*0,01
DV01 = -100 * 4,39 * 0,01 * 0,01 = -0,0439cent
Assume P=Euro 100.=
55

Interest Rate Risk
Modified Duration or DV01 and Convexity are
used to construct investment portfolios given
risk constraints.
56

Interest Rate Risk
Calculation of the modified duration of a
portfolio is a market weighted average of the
duration of the underlying bonds.
Euro 200 in bond A with ModDur=5
Euro 500 in bond B with ModDur=8
Euro 300 in bond C with ModDur=15
Total portfolio is Euro 1000 and ModDur=9,5
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Interest Rate Risk
We can calculate our DV01 of our portfolio:
DV01 = -1000 * 9,5 * 0,01 *0,01 = -0,95
So 1bp movement in yields will cost/benefit you
approximately Euro 1.=, or you could chose for a
hedge which offsets this price-movement…
58

Different types of interest rates:
• The central bank rate (REFI)
• Money market rates (EURIBOR, EONIA)
• Treasury or Government bond yields
• Swaps-rates (European)
• Credit-rates and credit spread
• Other interest rates which are usually derived from
either government or swap rates which give an
indication of credit quality e.g. your mortgage rate,
your lending rate when you have an overdraft on your
account..
59

Interest rate risk
An other way of talking of interest rate for a
given maturity(t):
i(t) = T(t) + SS(t) + CS(t)
i(t) = interest rate
T(t) = treasury rate (usually German government bond yield)
SS(t) = swaps spread
CS(t) = credit spread
60

Interest Rate Risk
• The yieldcurve (20/10/14)
Perhaps hardly
visible, around
10yr maturity
there is a “gap”
in the curve
which is caused
by derivative
instruments
(bundfuture).
61

Interest Rate Risk
• What information can be derived from a yield
curve?
• Zero curve or zerobond-curve which can be
used to calculate the present value of a bond
or cash flows
• Future interest-rates which are currently
expected by the markets can be derived from
a zerobond-curve
62

Interest Rate Risk
Assume a hypothetical par yield (bond price=100 ) curve:
Note that the rates of 0,5 and 1 year are in fact zero rates,
because in Europe interest coupon is paid annually.
Maturity
[yrs]
Yield[%] Zero Curve
Yield[%]
0,5 2,500 2,500
1 3,000 3,000
2 3,250 3,254
3 3,500 3,511
4 4,000 ?
5 4,500 ?
63

Interest Rate Risk
• How to derive the zero curve for the 2nd year
Z(2)?
100 = 3.25/(1.03) + 103.25/(1+Z(2))^2)
Z(2) = 3.254%
For every maturity point repeat the calculation but
remind that every cash flow is discounted against
its own zero yield.
64

Interest Rate Risk
• How to derive the interest rate of 6 months in 6
months time? Effectively what is the forward rate?
R6x12 =( ((1+R12)^(1)/ (1+R6)^(0,5)) )^2 - 1
R6x12 = 3,50%
• Note this ()^2 because we need to calculate an annual
rate
• Another way to calculate interest rates within a year is
by linear interpolation, when the rates are small the
results are nearly the same
65

Interest Rate Risk
Another application of this calculation, that it
can be used to forecast the yield curve when
expectations about central bank and/or money
markets change.
66

Interest Rate Risk
• How to derive the interest rate of one year in
one years time? Effectively what is the
forward-price?
Fz(1,1) = (Z(2))^2 / Z(1) - 1
Fz(1,1)= (1,03254)^2/(1,03) - 1
Fz(1,1)=3.508%
67

Interest Rate Risk
• From this hypothetical curve we derive the
zero curve
• And we can derive market expectations, the
market expects a one year rate in one years
time to be 3,508%
68

Interest Rate Risk
• From this hypothetical curve we derive the
zero curve
• Also market expectations can be derived from
this curve, for example the market expects a
one year rate in one years time to be 3,508%
69

Interest Rate Risk
• As we saw in our little example the market
expects a one year rate in one years time of
3.508%
• If you believe this rate is too high you better
buy this bond today! You could calculate your
profit if your expectation become true and the
rate remains at 3%.
• If you believe this forward rate is too low…You
better don’t invest or even better sell!
70

Interest Rate Risk – Case 1
Not long ago many Dutch banks offered a yearly
increasing savings account with a total maturity
of 5 years
The goal of this exercise is to derive the par-bond
yieldcurve
71

The bank offered these rates, with annual
increasing interest rates:
72
year interest
1 1,00%
2 1,50%
3 2,00%
4 2,25%
5 2,50%

• How would you describe these rates?
• Which risk do you incur if you would deposit
your money with this bank?
• Derive the zero yieldcurve of this bank?
• Derive the par bond curve of this bank?
73

How would you describe these rates?
• These rates are actually 1 year forward rates
• You could argue that the yields represent the
banks yield or credit curve
74

Which risks do you incur if you would deposit
your money with this bank?
• Credit risk the bank might go bankrupt
• Inflation risk, the purchasing power of your
deposit might decrease
• Interest rate risk, because rates could increase
75

Derive the zero curve
76
year interest zero curve
1 1,00% 1,00%
2 1,50% 1,25%
3 2,00% 1,50%
4 2,25% 1,69%
5 2,50% 1,85%
=((PRODUCT($D$6:D8+1))^(1/year))-1

Or otherwise with help of a calculator, the zero
rate in year 3 is:
(((1+0,01)*(1+0,015)*(1+0,02))^(1/3))-1 = 0,015
Or 1,50% is the average yield of 3 year zero bond
77

Calculate the par yield curve
Year 1 = 1% (or equals the zero rate)
78

• Year 2:
x *( (1,01)^-1 ) + (100+x )* ((1,0125)^-2) = 100
x *(( (1,01)^-1 ) + ((1,0125)^-2)) = 100 * (1 - ((1,0125)^-2) )
1,96556 * x = 2,4538
x = 0,01248 or 1.25%
79

• How to calculate for year t:
Σ((1+y(1))^-1) + ((1+y(2))^-2) + .. + ((1+y(t))^-t)
80

81
The resulting par yield curve in the last column
year interest zero curve discountfactors sum discountfactor coupon par bondrate
1 1,00% 1,00% 0,9901 0,9901 1,00 1,00%
2 1,50% 1,25% 0,9755 1,9656 1,25 1,25%
3 2,00% 1,50% 0,9563 2,9219 1,49 1,49%
4 2,25% 1,69% 0,9353 3,8572 1,68 1,68%
5 2,50% 1,85% 0,9125 4,7697 1,83 1,83%
=((1+0,01) * (1+0,015) * (1 +0,02))^(1/3))-1
add the
discountfactors
(1+zerorate(3))^-3
100*(1-((1+0,015)^-3))/2,9219

Interest Rate Risk
As a starting point we return to our par bond-yieldcurve
82

Interest Rate Risk
• Another application of bonds is to hedge
liabilities by matching the investment in cash
flow/duration
83

Interest Rate Risk
• A simple cash flow matching example
• We have to pay a cash flow of Euro 1000 at
the end of year 4
• How could we hedge this cash flow?
84

Interest Rate Risk
Assume our hypothetical par yield curve:
(par curve = all bond prices are 100)
Note that the rates of 0,5 and 1 years are in fact zero rates
Maturity
[yrs]
Yield[%] Zero Curve
Yield[%]
0,5 2,500 2,500
1 3,000 3,000
2 3,250 3,254
3 3,500 3,511
4 4,000 4,041
5 4,500 4,586
85

Interest Rate Risk
• Calculate the present value of the liabilities
• PV(L) = 1000 * (1+0,04041)^-4 = 835,45
• Calculate the Mod duration of the liability
• ModDur = 4/(1+0,04041) = 3,84
86

Interest Rate Risk
Here is our bond universum which we can use to select the
correct bonds for the hedge
Bond
Maturity
[yrs]
Yield[%] Mod Dur
A 0,5 2,5 0,49
B 1 3 0,97
C 2 3,25 1,91
D 3 3,5 2,80
E 4 4 3,63
F 5 4,5 4,39
87

Interest Rate Risk
• We calculated that our liability has a modified
duration of 3,84
• Now we have to select a combination of bonds
that match best, intuitively you choose for a
combination of bond E and F or just E
• But all other combinations are also good, they
depend on your view on the future development
of the yield-curve, your risk appetite and risk
constraints (convexity, investment-mandate, etc.)
88

Interest Rate Risk
• For example if your view is to create a barbell
which is a combination of short and long
maturity paper, so you could suggest to use a
combination of bond A and F as a solution.
89

Interest Rate Risk
Your portfolio which consists of a combination
of A and F would consist of:
0,49*(1-x) + 4,39*x = 3,84
x= 0,86
So our portfolio of total Euro 835,45 would
consist of 14% A (Euro 116,96) and of
86% F (Euro 718,49)
90

Interest Rate Risk
• Our little example does have more than one
solution
• An optimal could be constructed by also checking
convexity
• A solution is dependent on your own views, e.g. a
barbell leads to a higher reinvestment risk, every
quarter you will have to roll into another bond, if
the interest rates are higher you could make a
profit.. But you could also lose in market-to-market-
value in the long bond.
91

Interest Rate Risk
• If you would choose to invest all your money in
bond E then you’re nearly matched but you have
still a reinvestment risk of the cash flow from the
coupon payments
• There is a small deviation from the duration of
the liabilities does your mandate allow this?
• Remind that movements of the curve can lead to
the situation that the hedge has to be checked
regularly which can lead to more portfolio
adjustments  transaction costs!
92

• The goal of this case is to make you more
familiar with cash-flow matching by studying
an early retirement-scheme
93

• Assume, your best friend who lives in Belgium
and whose age is currently 46 and he/she wants
to invest in an early retirement scheme which will
start in exactly 16 year and will run for 5 years
• It is your goal to receive Euro 100,000 at the start
of each year in today’s purchasing terms
• You expect an annual inflation of 2,5% par annum
• He asks you, for assisting him with his investment
strategy.
94

Here are some zerobond-rates:
(derived swap-rates as of Nov. 10, 2014)
95
time to maturity zero-rate yields
16 1,75%
17 1,80%
18 1,90%
19 2,00%
20 2,10%

• On the next slide you will find all the Belgian
government bonds which are currently available
• Assume no accrued interest
(pricing-date of the bondyields is Nov 10, 2014)
96

ticker coupon coupon date time to maturity yield to maturity Clean Price modified duration
BGB 2,75 mrt/16 1,3 -0,02% 103,62 1,35
BGB 3,25 sep/16 1,8 -0,01% 105,90 1,89
BGB 4 mrt/17 2,3 0,00% 109,23 2,36
BGB 3,5 jun/17 2,6 0,00% 108,95 2,69
BGB 5,5 sep/17 2,8 0,01% 115,41 3,10
BGB 4 mrt/18 3,3 0,05% 113,05 3,59
BGB 1,25 jun/18 3,6 0,09% 104,12 3,64
BGB 4 mrt/19 4,3 0,16% 116,47 4,73
BGB 3 sep/19 4,8 0,24% 113,18 5,16
BGB 3,75 sep/20 5,8 0,40% 119,20 6,38
BGB 4,25 sep/21 6,8 0,57% 124,51 7,61
BGB 4 mrt/22 7,3 0,68% 123,59 8,07
BGB 4,25 sep/22 7,8 0,79% 126,10 8,67
BGB 2,25 jun/23 8,6 0,94% 110,72 8,69
BGB 2,6 jun/24 9,6 1,13% 113,25 9,69
BGB 4,5 mrt/26 11,3 1,37% 132,58 12,34
BGB 5,5 mrt/28 13,3 1,57% 146,87 15,12
BGB 4 mrt/32 17,3 1,88% 131,08 17,31
BGB 3 jun/34 19,6 2,02% 115,70 17,51
BGB 5 mrt/35 20,3 2,02% 149,24 21,34
BGB 4,25 mrt/41 26,3 2,20% 140,62 24,74
BGB 3,75 jun/45 30,6 2,31% 131,32 26,04
97

• How would you advise your friend about how
to invest and on risks taken?
• As he read something about yields which are
currently very low could you advise him
another investment strategy? What are the
risks of this alternative?
98

Interest Rate Risk – Case 2- Answer
How to tackle this challenge?
• Draw a graph
• Determine the pay-offs with help of the inflation-assumption
• Determine the present value of the pay-offs with help of the zerobond-curve
• Hence that the (Macaulay) duration of a zerobond equals the time to maturity of that
zerobond
• The weights of the present value of the cashflows determine the weighted average of
yield and duration which lead to the modified duration of the cashflow
• With help of the modified duration of the liabilities a combination of bonds can be
found with the same modified duration
• There is more than one solution, because an optimal is dependent on your market
views or otherwise convexity has to be calculated to try to find an optimum
• Theoretically you could also match the liabilities with help of zerobonds but because of
low liquidity (thus a high cost price!) this isn’t an acceptable answer
99

• A graphical depiction of the cash flows
100

• Calculate the future cash flows
101
needed cash flow [EUR] 100.000,00
growth rate of cash flows 2,50%
needed_cash_flow*(1+growth_rate)^(time_to_maturity)
time to maturity cash flow of liabilities zero-rate yields present value
16 148.451 1,75% 112.468,57
17 152.162 1,80% 112.355,28
18 155.966 1,90% 111.146,11
19 159.865 2,00% 109.736,27
20 163.862 2,10% 108.134,05
PV 553.840,27

• Calculate the present value of the cash-flows
102
needed cash flow [EUR] 100.000,00
growth rate of cash flows 2,50%
cash_flow*(1+zero_rate_yields)^-time_to_maturity
time to maturity cash flow of liabilities zero-rate yields present value
16 148.451 1,75% 112.468,57
17 152.162 1,80% 112.355,28
18 155.966 1,90% 111.146,11
19 159.865 2,00% 109.736,27
20 163.862 2,10% 108.134,05
PV 553.840,27

• Given these zerobond-rates an amount of
Euro 553.840,27 is needed today to buy this
early retirement scheme
103

• Calculate the weights of the present value of
the cashflows to calculate the average yield
104
time to maturity cash flow of liabilities zero-rate yields present value weight yield contribution
16 148.451 1,75% 112.468,57 20,31% 0,36%
17 152.162 1,80% 112.355,28 20,29% 0,37%
18 155.966 1,90% 111.146,11 20,07% 0,38%
19 159.865 2,00% 109.736,27 19,81% 0,40%
20 163.862 2,10% 108.134,05 19,52% 0,41%
PV 553.840,27 100,00% 1,91%

• Calculate the weights of the present value of
the cashflows to calculate the mac duration
time to maturity cash flow of liabilities zero-rate yields present value weight yield contribution mac_duration duration contribution
16 148.451 1,75% 112.468,57 20,31% 0,36% 16,00 3,25
17 152.162 1,80% 112.355,28 20,29% 0,37% 17,00 3,45
18 155.966 1,90% 111.146,11 20,07% 0,38% 18,00 3,61
19 159.865 2,00% 109.736,27 19,81% 0,40% 19,00 3,76
20 163.862 2,10% 108.134,05 19,52% 0,41% 20,00 3,90
PV 553.840,27 100,00% 1,91% 17,98
105

• Calculate with help of the average yield and
mac duration the modified duration of the
liabilities to calculate the target duration of
the fixed-income portfolio
• Modified duration of the liabilities:
18/(1+0,0191) = 17,6
106

• Look for a combination or one bond with the
same target modified duration
ticker coupon coupon date time to maturity yield to maturity Clean Price modified duration
BGB 3 jun/34 19,6 2,02% 115,70 17,51
107

• Calculate the nominal value (hence assumption of no accrued interest)
nominal 553.840,27 / 115,70 * 100 = 478.665,83
• Nominal value is usually in multiples of Euro 1,000 or Euro 10,000
• Thus we would buy either Euro 480,000 nominal or Euro 475,000
108

• Other combinations are also possible with at least two bonds, or even a
complete portfolio of several bonds with an average modified duration of
17,6
• An optimal can also be calculated by using convexity
• The portfolio although invested in Belgian Government Bonds isn’t
diversified into other countries or businesses (credits)
• There is still a reinvestment risk of the coupon cash flows
109

Interest Rate Risk
• Any questions left?
110

Interest Rate Risk
Suggested reading and used as source/reference:
• “Bond Markets, Analysis and Strategies”, Frank
Fabozzi, Prentice Hall International Editions
(1996)
• “Fixed Income Securities, Tools for Today’s
Markets”, Bruce Tuckman, Wiley Finance (2002)
• “Bond Market Securities”, Moorad Choudhry,
Prentice Hall (2001)
111

Interest Rate Risk
Other suggested reading:
• “Managing Financial Risk”, Charles W.
Smithson, McGraw-Hill (1998)
• “Debt, the first 5000 years”, David Graeber,
Melvillehouse (2011)
112

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