fundamentals of finance - university of oulucc.oulu.fi/~jope/fof/c10/printable/fixed-income...

Fixed-income securities

Fundamentals of FinanceFixed-income securities

Jukka Perttunen

University of Oulu - Department of Finance

Fall 2017

Jukka Perttunen Fundamentals of Finance


Corporate bond

The government-issued zero-coupon bonds provide the continuously compounded risk-free discount rates above.

Time to payment 1 2 3 4 5

Continuously compounded risk-free discount rate 2.204% 2.413% 2.611% 2.792% 2.954%

A five-year corporate bond pays e40 annual coupons and repays its e1000 face value at the maturity of the bond.

Payment of the bond 40 40 40 40 1040

We set a 1.5% default risk premium on the bond.

Default risk premium of the bond 1.500% 1.500% 1.500% 1.500% 1.500%

The discount rate is the sum of the risk-free rate and the default risk premium.

Discount rate of the bond 3.704% 3.913% 4.111% 4.292% 4.454%

The discounting of the payments provides the value of bond of e976.95.

Present value of the payment 38.55 36.99 35.36 33.69 832.37︸︷︷︸976.95



Yield-to-maturity of the bond







We are able to replace the periodic discount rates with a constant yield-to-maturity of the bond:

40 e−y×1 + 40 e−y×2 + 40 e−y×3 + 40 e−y×4 + 1040 e−y×5 = 976.95 ⇒ y = 4.426%.

Yield-to-maturity of the bond 4.426% 4.426% 4.426% 4.426% 4.426%



When it comes to the value of the default risk premium, we are not necessarily able to estimate in a very precise way.

The yield-to-maturity can always be calculated, if we just have the current market price of the bond.



Sensitivity to a change in interest rates

Let us verify what happens, if the interest rates of all maturities rise by 0.5 percentage units.



The size of the default risk premium is supposed to remain unchanged.


Such being the case, the discount rates just rise by 0.5 percentage units.


Due to higher discount rates the value of the bond declines to e954.65.





Sensitivity to a change in the yield-to-maturity







Let us examine the effect of the same 0.5 percentage unit rise in the yield-to-maturity of the bond.




The bond price changes by the same amount as in the case of the interest rate rise.

It is not an analytical identity, but the change in yield definitely is in line with the change in interest rates.

The sensitivity of a bond price to a change in its yield is a good proxy of its sensitivity to changes in interest rates.



Duration

The value B of a bond can be expressed in terms of a a function of the yield-to-maturity y , where a coupon C is

paid periodically at times t1, t2 . . . tn , and the face value F is paid back at the maturity tn of the bond:

B = C e−y×t1 + C e−y×t2 + C e−y×t3 + ... + (F + C) e−y×tn .

The sensitivity of the bond price B to a change in its yield-to-maturity y is obtained by derivation:

dB

dy= −t1C e−y×t1 − t2C e−y×t2 − t3C e−y×t3 − ...− tn(F + C) e−y×tn .

Correspondingly, the change dB in the bond price can be put as

dB = −[t1C e−y×t1 + t2C e−y×t2 + t3C e−y×t3 + ... + tn(F + C) e−y×tn

]dy.

The multiplying and dividing the right-hand-side by B results

dB = −[t1

C e−y×t1

B+ t2

C e−y×t2

B+ t3

C e−y×t3

B+ ... + tn

(F + C) e−y×tn

B

]dy B.

︸︷︷︸Duration

The term in the brackets is called the duration of the bond.



Duration

The change dB in the bond price in a case of a change dy in the yield-to-maturity of the bond can be expressed

in terms of the current bond price B and the duration D of the bond:

dB = −D × dy × B.

The discrete approximation of the result appears as

∆B = −D × ∆y × B.

The duration D of a bond is the weighted average time to the payments of the bond:

D = t1C e−y×t1

B+ t2

C e−y×t2

B+ t3

C e−y×t3

B+ ... + tn

(F + C) e−y×tn

B.

The time to the each of the payments is weighted by the value-proportion of the payment.

6 6 6 6



Duration

The change dB in the bond price in a case of a change dy in the yield-to-maturity of the bond can be expressed

in terms of the current bond price B and the duration D of the bond:

dB = −D × dy × B.

The discrete approximation of the result appears as

∆B = −D × ∆y × B.

The duration D of a bond is the weighted average time to the payments of the bond:

D = t1C e−y×t1

B+ t2

C e−y×t2

B+ t3

C e−y×t3

B+ ... + tn

(F + C) e−y×tn

B.

The time to the each of the payments is weighted by the value-proportion of the payment.

6 6 6 6

The numerator of each of the weights (value-proportions) represents the present value of the payment, whereas the

denominator equals to the sum of the present values of all payments.

The present values are obtained by the yield-to-maturity of the bond as the discount rate.



Duration of the bond










Let us turn back to the initial situation before the rise in the interest rates.

The duration of the bond is

38.27

976.95× 1 +

36.61

976.95× 2 +

35.03

976.95× 3 +

33.51

976.95× 4 +

833.54

976.95× 5 = 4.62.



Predicting by duration










On the basis of the duration of the bond, we expect a 0.5 percentage unit change in yield to change the bond price by

∆B = −4.62× 0.005× e976.95 = −e22.57.

We expect the bond price to decline to

B = e976.95− e22.57 = e954.38 (≈ e954.64).



Duration of a bond portfolio

The duration of a portfolio of bonds is the value-weighted average of the durations of the bonds in the portfolio:

Dp =n∑

i=1

wiDi .

Suppose that our bond portfolio contains the following positions in three bonds with annual coupons:

Bond Position Face value Coupon Time-to-maturity Yield-to-maturity Value Duration

1 Asset 1000 40 5 4.426% 976.95 4.62

2 Asset 1000 40 10 4.622% 942.85 8.38

3 Liability 1000 50 10 5.326% 987.30 2.86

2907.10

The value of the liability bond #3 can be considered negative from our point of view, and thus the current value

of the portfolio is

Bp = e976.95 + e942.85− e987.30 = e932.50.

However, when calculating the duration of the portfolio, we use the gross value of e2907.10 of the portfolio, and

a negative value of duration for the liability bond:

Dp =976.95

2907.10× 4.62 +

942.85

2907.10× 8.38 +

987.30

2907.10× (−2.86) = 3.30.



Duration of a bond portfolio

With a 0.5 percentage unit change in the yields, the values of the bonds appear as

Bond Position Face value Coupon Time-to-maturity Yield-to-maturity Value

1 Asset 1000 40 5 4.926% 954.63

2 Asset 1000 40 10 5.122% 904.25

3 Liability 1000 50 10 5.826% 973.29

The value of the portfolio is now

Bp = e954.63 + e904.25− e973.29 = e885.59.

The change in the value of the portfolio is

∆Bp = e885.59− e932.50 ≈ −e47.

On the basis of the duration, we expected a decline of

∆Bp = −3.30× 0.005× e2907.10 ≈ −e48.



Summary of interest rates

Short-term interest ratesQuoted

rate

Simple

rate

Continuously

compounded rate

Discount

factor

3-month Euribor

6-month Euribor

12-month Euribor

2.124%

2.190%

2.332%

2.124%

2.190%

2.332%

2.118%

2.178%

2.305%

0.99472

0.98917

0.97721

Swap rates

2y/6m swap rate

3y/6m swap rate

2.587%

2.783%

2.587%

2.783%

0.95014

0.92074

Six-month forward rates

0.5× 1.0 forward rate





2.447%

2.808%

2.851%

3.143%

3.193%

2.447%

2.808%

2.851%

3.143%

3.193%

2.432%

2.788%

2.831%

3.119%

3.168%

0.97721

0.96368

0.95014

0.93544

0.92074



Discount rates and forward rates

We have solved the periodic discount factors of our exemplary interest rate data.

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0

Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074

In order to be able to evaluate the sensitivity of a floating-rate bond price price to changes in interest rates, we

transform the discount factors to the corresponding continuously compounded discount rates:

dT = e−rT ⇒ r = −1

Tln dT .

Continuously compounded discount rate 2.178% 2.305% 2.466% 2.557% 2.670% 2.753%

The exemplary interest rate data provides us also the six-month forward rates in terms of a simple rate as well as

in terms of a continuously compounded rate.

Continuously compounded forward rate 2.432% 2.788% 2.831% 3.119% 3.168%

Simple forward rate 2.447% 2.808% 2.851% 3.143% 3.193%



Floating-rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074



We suppose that the coupon rate of a floating-rate bond is just reset, and the first period’s coupon rate equals to

the current six-month Euribor rate of 2.190%.

First period’s fixed coupon rate 2.190%

The expected coupon rate of any later period always equals to the six-month forward rate of the period.

Future period’s expected coupon rate 2.447% 2.808% 2.851% 3.143% 3.193%

The periodic coupon payments are determined by the annual coupon rates for the e1000 principal over each of the

six-month periods.

Expected coupon payment 10.95 12.24 14.04 14.26 15.72 15.97

Finally, at the maturity of the bond, the principal of e1000 is paid back.

Principal repayment 1000.00



Valuation of the floating-rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074







The discounting of the coupon payments and the repayment of the principal shows that the bond trades at par.

Present value of the total payment 10.83 11.93 13.53 13.54 14.70 935.44︸︷︷︸1000

The first period’s coupon rate together with the expected future period’s coupon rates cover the required rate of

return of the bond, which is set by the time value of money alone, in this case of a bond without any default risk.

A risk-free floating-rate bond always trades at par at the time of the reset of the coupon rate.



Single-period valuation of the floating-rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074








As we know that the bond is to trade at par, again,

at the end of the first six-month period, we are able

to ignore any later payments, and value the bond in

terms of a single-period bond with just one coupon

payment and the terminal value of e1000.

Time to payment 0.5


First period’s fixed coupon payment 10.95

End-of-period value of the bond 1000.00

End-of-period total value 1010.95

Present value 1000.00︸︷︷︸1000



Scenario with rising interest rates

Let us a consider a scenario, where the continuously compounded discount rates of all maturities rise by 50 basis

points (0.5 percentage units) immediately after the reset of the first period’s coupon rate.

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


We are able to calculate the discount factors corresponding to the changed discount rates.

Discount factor 0.98670 0.97234 0.95649 0.94069 0.92381 0.90702

When it comes to the continuously compounded forward rates, if the underlying continuously compounded discount

rates rise by the same amount at the both ends of the forward period, the forward rate rises by the same amount.


The simple forward rates can easily be transformed from the continuously compounded froward rates.




Valuation of the floating-rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98670 0.97234 0.95649 0.94069 0.92381 0.90702



Despite of the change in the interest rates the first period’s coupon rate remains at it’s already fixed level of 2.190%.


The remaining future period’s expected coupon rates are defined by the changed simple forward rates.


The periodic coupon payments together with the repayment of the principal result to the bond price of e997.50.



Present value of the total payment 10.80 14.36 15.85 15.80 16.87 923.81︸︷︷︸997.50



Single-period valuation of the floating-rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98670 0.97234 0.95649 0.94069 0.92381 0.90702








The same result can be achieved by simply valuing

the coupon payment of the first period, together with

the terminal value of e1000, now by the changed

discount rate or the discount factor.

Time to payment 0.5





Present value 997.50︸︷︷︸997.50



Duration of the floating rate bond

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98670 0.97234 0.95649 0.94069 0.92381 0.90702








Alternatively, we may apply the duration of the

floating-rate bond, which is equal to the time to

the next reset of the coupon rate, in this case

0.5 years:

∆B = −0.5× 0.005× 1000 = −2.50.

Time to payment 0.5





Present value 997.50︸︷︷︸997.50



Interest rate swap

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074



Three-year swap rate 2.783% 2.783% 2.783%

Let us consider a three-year swap which pays the fixed swap rate and receives the floating six-month Euribor rate.

Fixed leg paid

Fixed coupon rate 2.783% 2.783% 2.783%

Coupon payment / principal repayment 27.83 27.83 1027.83

Present value of the total payment 27.20 26.44 946.36︸︷︷︸(−)1000Floating leg received



Coupon payment / principal repayment 10.95 12.24 14.04 14.26 15.72 1015.97




Single-period valuation of the floating leg

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.96368 0.95014 0.93544 0.92074



Three-year swap rate 2.783% 2.783% 2.783%

Fixed leg paid







We may replace the floating leg with the corresponding single-period bond.



Interest rate swap

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0

Continuously compounded discount rate 2.178% 2.305% 2.557% 2.753%

Discount factor 0.98917 0.97721 0.95014 0.92074

Fixed leg paid







In further analysis we do not need all those rates anymore.




Let us a consider our scenario, where the continuously compounded discount rates of all maturities rise by 50 basis

points (0.5 percentage units) immediately after the reset of the first period’s coupon rate.

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


We have already calculated the discount factors corresponding to the changed discount rates.

Discount factor 0.98670 0.97234 0.94069 0.90702




Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98670 0.97234 0.94069 0.90702

The coupon rate of the fixed leg remains at the already contracted level of 2.783%.

Fixed leg paid



The rise in the discount rates, however, declines the present value of the future payments.

Present value of the total payment 27.06 26.18 932.26︸︷︷︸(−)985.50

The floating leg coupon rate remains fixed as well, but the present value is declined.

Floating leg received



Present value 1000.00︸︷︷︸997.50




Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98670 0.97234 0.94069 0.90702

Fixed leg paid



Present value of the total payment 27.06 26.18 932.26︸︷︷︸(−)985.50Floating leg received



Present value 1000.00︸︷︷︸997.50

The initial value of the swap with a nominal principal of e1000 was e1000 − e1000 = e0.

After a 50 basis points rise in interest rates the value of the swap is e997.50 − e985.50 = e12.



Duration of the fixed leg

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.95014 0.92074

Fixed leg paid



Present value of the total payment 27.20 26.44 946.36︸︷︷︸(−)1000

Let us turn back to the initial situation before the rise in the interest rates.

We are able to calculate the yield-to-maturity of the fixed leg of the swap:

27.83 e−y×1 + 27.83 e−y×2 + 1027.83 e−y×3 = 1000 ⇒ y = 2.745%.

Correspondingly, the duration of the fixed leg of the swap is

27.83 e−0.02745×1

1000× 1 +

27.83 e−0.02745×2

1000× 2 +

1027.83 e−0.02745×3

1000× 3 = 2.92.



Valuation of the swap

Time to payment 0.5 1.0 1.5 2.0 2.5 3.0


Discount factor 0.98917 0.97721 0.95014 0.92074

Fixed leg paid



Present value of the total payment 27.20 26.44 946.36︸︷︷︸D = 2.92Floating leg received



Present value 1000.00︸︷︷︸D = 0.5

The duration of the floating leg is known to be

equal to the time to the next reset of the coupon

rate, it is D = 0.5.

In our scenario of 50 basis points rise in interest rates we expect

– the value of the fixed leg paid to change by −2.92× 0.005× 1000 = −e14.60,

– the value of the floating leg received to change by −0.5× 0.005× 1000 = −e2.50,

– the value of the contract to end up at the level of (−e1000 + e1000) + e14.10− e2.50 = e12.10.



Neutralizing the duration of the bond portfolio

For the portfolio of three bonds we had

Bp = e976.95 + e942.85− e987.30 = e932.50.

Dp =976.95

2907.10× 4.62 +

942.85

2907.10× 8.38 +

987.30

2907.10× (−2.86) = 3.30.

For the swap with a e1000 nominal principal we have

Bs = e1000− e1000 = e0.

Ds =1000

2000× 0.50 +

1000

2000× (−2.92) = −1.21.

With 3.96 swap contracts we are able to neutralize the duration of the bond portfolio:

n × (−1.21)× e2000 + 3.30× 2907.10 = 0 ⇒ n = 3.96.

With a 0.5 percentage unit change in the yields, we expect the values to change by

∆Bp = −3.30× 0.005× e2907.10 ≈ −e48,

∆Bs = 3.96×[− (−1.21)× 0.005× e2000

]≈ e48.


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