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

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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

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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

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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!

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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?

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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

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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)

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Interest Rate Risk

A few graphs to illustrate interest rate risk

• 2yrs European Swaprate (10/20/14)

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The credit crisis in another perspective!

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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)

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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

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http://www.investopedia.com/terms/i/interestraterisk.asp

Interest Rate Risk

• First a little example to illustrate interest rate risk

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• 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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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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This is Bello’s investment strategy:

• Every 12 months he invests in a term deposit at the bank

• He reinvests accrued interest

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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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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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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

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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!

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Interest Rate Risk

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

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• 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 areno 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.

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• 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.

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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!

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• 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)

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Time Anna’s account[EUR]

Bello’s account[EUR]

0 1000 1000

0,5 1005 1010 *)

1 ? 1020

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*) 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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Overview based on market expectations:

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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?

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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

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An overview of some possible results of Anna’s investment policy

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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

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10

face_value

coupon

Pay off scheme of a bullet bond

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• How to price a bond:

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Note from the pricing formula:

• There is implicit an assumption that all income is reinvested against the same yield, which is seldom the case.

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Other measures to quantify price risks

• Yield to maturity

• Macaulay Duration

• Modified Duration

• Convexity

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Let’s have a closer look at our pricing formula of bond:

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• 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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• From the pricing formula it becomes clear that most of the price is determined by the present value of the nominal

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• 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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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

Note the yields we use are yield to maturity (ytm)

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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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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

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Another important risk measure is Macaulay Duration which is the first derivative of the pricing formula.

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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

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• 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

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

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Modified duration is used as an approximation to calculate the

return of a bond when interest rates change.

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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%

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• 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!)

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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.=

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

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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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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…

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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..

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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

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• 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).

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• 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

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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 CurveYield[%]

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 ?

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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.

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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 - 1R6x12 = 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

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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.

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• 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%

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• 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%

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• 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%

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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!

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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

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The bank offered these rates, with annual increasing interest rates:

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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?

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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

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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

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Derive the zero curve

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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

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Calculate the par yield curve

Year 1 = 1% (or equals the zero rate)

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• 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%

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• How to calculate for year t:

Σ((1+y(1))^-1) + ((1+y(2))^-2) + .. + ((1+y(t))^-t)

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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

(1+zerorate(3))^-3

add the

discountfactors

100*(1-((1+0,015)^-3))/2,9219

The resulting par yield curve in the last column

Interest Rate RiskAs a starting point we return to our par bond-yieldcurve

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

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• 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?

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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 CurveYield[%]

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

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• 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

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BondMaturity

[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

Here is our bond universum which we can use to select the correct bonds for the hedge

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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.)

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• 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.

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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)

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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. abarbell leads to a higher reinvestment risk, everyquarter you will have to roll into another bond, ifthe interest rates are higher you could make aprofit.. But you could also lose in market-to-market-value in the long bond.

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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!

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

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• 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.

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Here are some zerobond-rates:

(derived swap-rates as of Nov. 10, 2014)

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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)

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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


• 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?

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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

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• A graphical depiction of the cash flows

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• Calculate the future cash flows

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needed cash flow [EUR] 100.000,00

growth rate of cash flows 2,50%

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

needed_cash_flow*(1+growth_rate)^(time_to_maturity)


• Calculate the present value of the cash-flows

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needed cash flow [EUR] 100.000,00

growth rate of cash flows 2,50%

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

cash_flow*(1+zero_rate_yields)^-time_to_maturity


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

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• Calculate the weights of the present value of the cashflows to calculate the average yield

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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

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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


• 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

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• Look for a combination or one bond with the same target modified duration

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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


• 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

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• 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

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• Any questions left?

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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)

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Other suggested reading:

• “Managing Financial Risk”, Charles W. Smithson, McGraw-Hill (1998)

• “Debt, the first 5000 years”, David Graeber, Melvillehouse (2011)

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Interest rate risk teacher edition

Economy & Finance

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