lecture 1

Mathematical and Empirical Finance

Lecture 1: Fixed Income Securities andthe Term Structure of Interest Rates

Block 1, 2014–2015

Lecture 1 (Fixed Income Securities) Mathematical and Empirical Finance Block 1, 2014–2015 1 / 33

Outline of the course in lectures

1 Fixed Income Securities and the Term Structure of Interest Rates2 Mean-Variance Portfolio Theory3 Capital Asset Pricing Model4 Factor Models and Arbitrage Pricing Theory5 Derivative Pricing: Binomial Trees6 Black-Scholes Option Pricing


Outline Lecture 1

Getting started, arbitrage

Compounded interest, present value

Fixed income securities, annuities

Yield to maturity

Duration

Immunization and convexity

Yield curve and spot rate curve

Bootstrap and Nelson-Siegel

Forward rates

Floating rate bonds

Fisher-Weil duration


Getting started: arbitrage

Investment: Commitment of resources (money) to achieve later benefits(cash flows).

Distinguish physical investments, e.g. in plants (investering) from invest-ment in financial assets (belegging).

Cash flows may be obtained at different points in time: cash flow stream,often denoted as vector

x = (x0, x1, . . . , xn)

for cash flows {xi}ni=0 at equidistant time points {ti}n

i=0 in the future (t0 =0, now).

In general, cash flow streams may be uncertain; in case of fixed-incomesecurities (e.g. bonds), they are deterministic.

Cash flow streams may be affected by (dynamic) investment decisions.



Cash flow streams may be combined in a portfolio by simple addition ofthe two vectors.

An arbitrage is a portfolio that has zero cash flow (price) at t0 and positivecash flows in the future.

Example:

I Suppose someone offers you a cash flow of $110 a year from now, for aprice of $100 now. At the same time, you may borrow or deposit money atthe bank at a 7% interest rate.

I The offer implies a cash flow stream (−100, 110); the bank account impliesa cash flow stream of (−x , 1.07x) for any x ∈ R.

I Taking x = −100 (i.e. borrowing 100 from the bank) and combining the twogives combined cash flow of (0, 3): this is an arbitrage.

Arbitrage opportunities are assumed not to exist (or persist), because ev-ery investor would jump at them.

This no-arbitrage assumption implies the law of one price: differentinvestments with the same cash flows should have the same market price.



Imposing the no-arbitrage condition may lead to a unique value and henceprice of an investment. However, in the presence of uncertainty and hencerisk, it is generally not sufficient.

In that case assumptions about preferences, and in particular about thedegree of risk aversion, are typically needed.

Example:

I Someone offers you a cash flow which is either zero or $110, both withprobability 1

2 . The interest rate is 10%.I Which amount P would you be willing to pay for this?I No-arbitrage only implies P > 0 and P < 100.I Risk aversion would imply P < 50, because you prefer a risk-free cash flow

of 55 to an uncertain cash flow with the same mean.I Which price you are willing to pay depends on your degree of risk aversion.



A positive interest rate r reflects time value of money: compensation forpostponing consumption.

Simple interest would imply that investment A grows in n years to A(1+rn).

In practice interest is paid and reinvested at regular intervals, e.g. annually;so A grows in n years to Vn = A(1 + r)n. (Assuming interest rates willremain constant!)

So compounded interest leads to geometric growth.

Other compounding frequencies are possible: if interest is paid m times ayear then Vt = A(1 + r/m)mt , with t measured in years and mt ∈ N.

The limit as m→∞ is continuous compounding:

Vt = A ((1 + r/m)m)t → Aert .

This is an abstraction that is often analytically convenient.



Positive, annually compounded interest means that we have to invest 1 toreceive 1 + r > 1 a year from now.

Hence, if we need a cash flow X a year from now, then we need to invest

11 + r

X = d1X .

Here d1 < 1 is the one-year discount factor, and the result d1X is calledthe present value of X .

Similarly to get a cash flow X in k periods from now, we need to invest(compounding frequency m, constant interest):

dk X =1

(1 + r/m)k X ,

and continuous compounding implies that present value of cash flow X int years is e−rtX .



Present value of cash flow stream (x0, x1, . . . , xn) at times t0, t1, . . . , tn,with tk = k/m, and m is the compounding frequency:

PV =n∑

k=0

xk

(1 + r/m)k .

Result can be obtained by investing each of the cash flows to get futurevalue

FV =n∑

k=0

xk (1 + r/m)n−k ,

and then discounting cash flow FV after n periods back to time 0.

With continuous compounding, PV =∑n

k=0 xk e−rtk .

In a “constant ideal bank” world, any cash flow with the same PV is equiv-alent.


Fixed-income securities, annuities

Savings deposits: varying balance and varying interest rates. With timedeposits and certificates of deposit, interest rate can be fixed for sometime, conditional on minimum deposit.

Money market instruments: loans for 1 year or less, such as commercialpaper and treasury bills.

Government securities:

I Treasury bills: sold on discount basis, so zero-coupon bonds.I Treasury notes (1–10 year) and bonds (>10 years) pay regular coupons,

fixed percentage of face value. US government bonds pay coupons twice ayear.

I Treasury strips: artificial zero-coupon bonds obtained by “stripping” couponbonds.

Corporate bonds: may involve credit (default) risk.

Callable bonds: gives issuer the right to repay face value before maturity.


Fixed-income securities, annuitiesAnnuities give a fixed cash flow over a fixed period of time, or indefinitely(perpetual annuities). They play a role in pensions and mortgages.With fixed interest (flat term structure), computations follow simply fromgeometric series, i.e., for |λ| < 1,

n∑i=0

λi =1− λn+1

1− λ.

For perpetual annuities, this leads to

P =∞∑

k=1

A(1 + r)k =

A1 + r

∞∑k=0

(1

1 + r

)k

=Ar.

For an n-period annuity, we similarly find

P =n∑

k=1

A(1 + r)k =

Ar

(1− 1

(1 + r)n

),

which can be inverted to A = r(1 + r)nP/[(1 + r)n − 1].


Fixed-income securities, annuities

Converting a loan P into an annuity (amortization) involves a repaymentscheme where the fixed amount A is composed of an increasing repay-ment part and a decreasing interest rate part.

Example: For n = 3, and P = 100 and r = 10%, we have A = 10 ×1.13/(1.13 − 1) = 40.21.

I After 1 year, 10 is paid interest, and 40.21 − 10 = 30.21 is repayment ofprincipal, so 69.79 remains;

I After 2 years, 6.98 is paid interest and 33.23 is repayment, leaving 36.56;I After 3 year, 3.66 is paid interest and 36.55 is repayment, leaving 0.


Yield to maturity

Consider bond with face value F and coupons C/m paid m times a year(so coupon rate is C/F ).

If interest r is assumed constant, and there are n remaining coupon dates,then present value of cash flows is

PV =n∑

k=1

C/m(1 + r/m)k +

F(1 + r/m)n .

Yield to maturity (YTM) λ is that value of r such that PV equals the marketprice P of the bond, i.e.,

P =n∑

k=1

C/m(1 + λ/m)k +

F(1 + λ/m)n .

Hence the YTM is the internal rate of return of the bond (see Chapter 2).

λ must be obtained numerically; in Excel, the function IRR can be used.


Yield to maturity

If the interest rate really would be constant and fixed, then no-arbitrageimposes λ = r .

In practice, the interest rate varies with maturity.

The YTM of a zero coupon bond (C = 0) is simply the interest rate forlending and borrowing over period of T = n/m years. This will be definedas the spot rate.

The YTM of a coupon bond may be seen as weighted average of suchspot rates, and hence as a single summary of the spot rate curve.

If a bond trades at par (P = F ), then the YTM equals coupon rate C/F .

Relationship between bond price and YTM is downward sloping, see Fig-ures 3.3 and 3.4: higher yields mean lower price (given the same cashflows).

The steepness of this curve is measure of interest rate sensitivity of thebond.


Duration

In a constant interest rate world, with no default, bonds carry no risk.

In practice the interest rate varies over time, and is different for differentmaturities, as represented by the term structure of interest rates.

This means that value of bonds is exposed to interest rate risk (eventhough cash flows are risk-free!).

Duration is a measure of interest rate sensitivity, or equivalently of ex-posure to interest rate risk.

If a bond has cash flows ck at times tk = k/m, k = 1, . . . , n (coupons andprincipal), and has YTM λ, then the Macaulay duration is:

D =n∑

k=1

wk tk , wk =ck/(1 + λ/m)k∑nj=1 cj/(1 + λ/m)j

.


Duration

Example: 4-year bond, annual compounding, coupon rate 5%, priceP = 90.064.

YTM λ = 8%.

Duration calculations:

t CF d PV w D

0 −90.064 1.0001 5 0.926 4.630 0.051 0.0512 5 0.857 4.287 0.048 0.0953 5 0.794 3.969 0.044 0.1324 105 0.735 77.178 0.857 3.428

sum 90.064 1.000 3.707

We find D = 3.707 < 4. In general, D < n/m = time to maturity.


Duration

Definition of duration does not involve sensitivity. However,

dPdλ

=d

dλ

n∑k=1

ck

(1 + λ/m)k =n∑

k=1

−(k/m)ck

(1 + λ/m)k+1 =−DP

(1 + λ/m)= −DMP,

where DM = D/(1 + λ/m) is the modified duration.

Because

DM = − 1P

dPdλ,

this is a semi-elasticity, providing a first-order approximation to percentagechanges in P caused by interest rate changes:

∆PP≈ −DM∆λ.

“Modification” factor not needed in case of continuous compounding.


Duration

Suppose we have portfolio of k different bonds: we buy ni bonds with pricePi , so total portfolio value is P =

∑ki=1 niPi .

If all bonds have same YTM, then it follows from definition of duration thatportfolio duration is

D =k∑

i=1

wiDi , wi =niPi

P.

In practice different bonds have different yields (non-flat yield curve). Canbe resolved by more sensible definition of duration (Fisher-Weil).



Suppose we need cash flow stream (x1, . . . , xn) at times (t1, . . . , tn) tomeet cash obligations.

Interest risk can be perfectly eliminated by buying xk/100 zero-couponbonds with maturities tk , k = 1, . . . , n.

This perfect cash flow matching is not always feasible or practical.

Alternative is to set up portfolio of (at least) two bonds, such that portfoliohas same present value and duration as cash flow stream.

The type of immunization leads to positions that are, to first-order approx-imation, insensitive to interest rate risk.

It is clear that of the two bonds, one should have smaller duration, and theother larger duration than the cash flow stream.



Suppose that cash flow stream has present value P and duration D; andtwo bonds have values P1 and P2, and durations D1 and D2, respectively.Then numbers n1 and n2 of bonds should solve

n1P1 + n2P2 = P,

n1P1D1 + n2P2D2 = PD,

so that

n1 =P(D − D2)

P1(D1 − D2),

n2 =P(D − D1)

P2(D2 − D1).

To remain insensitive to interest rate risk, frequent rebalancing is needed.

Same caveat of common YTM to aggregate durations applies.



Duration approximates interest rate sensitivity based on first-order Taylorseries expansion of bond price function P = P(λ).

Better approximation is obtained from second-order Taylor series expan-sion:

∆P ≈ dPdλ

∆λ+12

d2P

dλ2 (∆λ)2

= −DMP∆λ+12

CP(∆λ)2,

where C is the convexity:

C =1P

d2P

dλ2

With continuous compounding, C =∑n

k=1 wk t2k , with same wk as in defi-

nition of duration. With compounding frequency m, small adjustments areneeded, see p. 66.


Yield curve and spot rate curve

In practice, bonds with different maturities have different YTM. Even bondswith same maturity but different coupon rate have different YTM.

Yield curve plots yields against maturity; typical shape is upward sloping.

When yields are constructed from zero coupon bonds, they are referred toas spot rates.

With compounding frequency m, we have for zero coupon bond prices withmaturity t :

Pt = 100dt =100

(1 + st/m)mt ,

so

st = m

[(100Pt

)1/mt

− 1

],

With continuous compounding, this simplifies to Pt = e−st t100, so that

st = − ln(Pt/100)/t.



Zero coupon government bonds are not available for maturities higher thana year.

However, artificial zero coupon bonds (strips) or discount factors may beobtained by combinations of coupon bonds.

This method of recursively obtaining discount factors and hence the spotrate curve is known as stripping or bootstrapping.

Assuming annual coupon payments (m = 1), and that the coupon bondwith maturity t has coupon payment Ct and price Pt , we have:

I P1 = d1(100 + C1), so d1 = P1/(100 + C1);I P2 = d1C2 + d2(100 + C2), so d2 = (P2 − d1C2)/(100 + C2);I In general,

dt =

(Pt − Ct

t−1∑i=1

di

)/(100 + Ct ).


Bootstrap and Nelson-SiegelExample:

I Prices of 1-year zero-coupon bond, 2-year 4% coupon bond, and 3-year 3%coupon bond are

P1 = 97.087, P2 = 100.969, P3 = 98.077.

I Bootstrap leads to discount factors d1 = 0.97087,

d2 = (100.969− 0.97087× 4)/104 = 0.93351,

d3 = (98.077− (0.97087 + 0.93351)× 3)/103 = 0.89673.

I This leads to the following spot rates:

s1 = 100/97.087− 1 = 3%,

s2 = (100/93.351)1/2 − 1 = 3.5%,

s3 = (100/89.673)1/3 − 1 = 3.7%.

I Using IRR function in Excel, yields to maturity are

y1 = 3%, y2 = 3.49%, y3 = 3.69%.



If (liquid) bond prices are not available at regular intervals, then the boot-strap method will fail. Various inter- and extrapolation methods exist.

A first option would be to assume that the spot rate curve st is a polynomialof degree k , say st =

∑ki=0 ai t i , and to determine the coefficients ai by

minimizing the sum of squared differences between model-implied bondprices and market bond prices.

A related idea was proposed by Nelson and Siegel (1987). They assumedthat for some coefficients β1, β2, β3 and λ,

st = β1 + β21− e−t/λ

t/λ+ β3

(1− e−t/λ

t/λ− e−t/λ

).

Here β1 is the asymptote (as t →∞), β1 +β2 is the intercept (as t → 0),so both together determine the level and slope of the term structure. Theparameters β3 and λ influence the curvature. Some examples are givenon the next slide


Nelson-Siegel curves, β1 = 8%, β2 = −5%

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

0 5 10 15 20 25 30

beta3=0, lambda=5

beta3=0, lambda=1.5

beta3=5, lambda=5


Forward rates and expectations hypothesis

Forward rate is interest rate for borrowing and lending from dates t1 to t2,under terms agrees upon today.

Compare 100, invested for two years:

I two-year zero-coupon bonds yields 100(1 + s2)2;I one-year zero-coupon bond, payoff invested against forward rate from 1 to

2, gives 100(1 + s1)(1 + f1,2).I hence no-arbitrage imposes f1,2 = (1 + s2)2/(1 + s1)− 1.

More generally, with compounding frequency m:

fi,j = m

{[(1 + sj/m)j

(1 + si/m)i

]1/(j−i)

− 1

},

whereas in case of continuous compounding

ft1,t2 =st2 t2 − st1 t1

t2 − t1.



Forward curve is plot of ft,t+∆ against t , for fixed ∆ (typically ∆ = 1/m).

In case of continuous compounding, this becomes

lim∆→0

ft,t+∆ = st + t × dst

dt,

so the difference between forward and spot curves is maturity times slopeof the spot curve.

Note that the intercept of the spot curve st0 is sometimes called the shortrate. This is not zero! (Error on p. 80.)

In the example sketched before, s1 = 3%, s2 = 3.5% and s3 = 3.7%imply f1,2 = 4.00% and f2,3 = 4.10%.



Typical shape of spot rate curve is upward sloping: rate increases withmaturity. Exceptions are possible, but usually short-lived.

Possible explanations of shape of term structure:

I Expectations hypothesis: risk-neutrality, so expected two-year return onone-year bonds equals return on two-year bonds:

(1 + s1)(1 + s′1) = (1 + s2)2 = (1 + s1)(1 + f1,2),

where s′1 is market expectation about s1 next year. Hence s′1 = f1,2, andmore generally

s′j−1 = f1,j .

Cannot explain persistently upward sloping term structure.I Liquidity preference: longer-maturity bonds less “liquid”, hence larger return

required;I Market segmentation: different agents have different time horizon.


Floating rate bonds

Floating rate bonds have coupons that vary with the spot rate. So withannual compounding, first coupon (in a year’s time) equals the current 1-year spot rate times the face value, C = s1F . After each year, the couponis reset.

Exactly a year before maturity (i.e., at the last reset date), value of thebond is

F + C1 + s1

= F1 + s1

1 + s1= F .

By the same argument, value of the bond at all reset dates is equal to facevalue.

Duration of floating rate bond equals the period between reset dates (oneyear in case of annual compounding).

Floating rate bond may be seen as combination of fixed-coupon bond plusfixed-for-floating swap; hence such swaps can be used to decrease dura-tion and hence interest rate sensitivity of bond portfolio.



If term structure is not flat, then uncertainty about future interest ratescannot be reduced to changes in YTM.

This leads to alternative definition of duration: Fisher-Weil duration.

Let {sti}ni=1 be current continuously compounded spot rate curve. Con-

sider a parallel shift in spot rate curve:

(st1 , . . . , stn ) 7→ (st1 + λ, . . . , stn + λ).

This changes the present value of a cash flow from

P(0) =n∑

i=1

exp(−sti ti)xi

to

P(λ) =n∑

i=1

exp(−[sti + λ]ti)xi ≈ P(0) +dP(λ)

dλ

∣∣∣∣λ=0× λ.



UsingdP(λ)

dλ=

n∑i=1

−ti exp(−[sti + λ]ti)xi ,

and dti = exp(−sti ti), we have

1P(0)

dP(λ)

dλ

∣∣∣∣λ=0

= −n∑

i=1

wi ti = −DFW , wi =dti xi∑nj=1 dtj xj

.

This leads to∆PP≈ −DFW ∆s,

with ∆P = P(λ)− P(0) and ∆s = λ.

Of course, not all interest rate risk is in terms of parallel shifts; there is alsoa risk that the slope or curvature of the spot rate curve changes. Moreadvanced risk measures can be constructed for that.



Using spot rate curve and FW duration, we may reconsider immunization,without needing instruments with the same YTM.

Example:I Suppose we wish to hedge against interest rate risk in two-year 4% coupon

bond, using position in one- and three-year zero coupon bonds.I The data are as in the example on slide 24, so P = 100.969 (coupon bond)

P1 = 97.087, and P3 = 89.673 (zeros).I Durations are D = 1.9615, D1 = 1 and D3 = 3.I The number of bonds follow from the two equations on slide 20, yielding

n1 =P(D − D3)

P1(D1 − D3)= 0.5400,

n3 =P(D − D1)

P3(D3 − D1)= 0.5413.

I If spot rate curve increases by 1%, then values change to P = 99.008 andn1P1 +n3P3 = 99.012, so the hedged position changes by less than 1 cent.Similarly if spot rate curve decreases by 1%.


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