3 - lecture three - money market final

8/12/2019 3 - Lecture Three - Money Market FINAL

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BX2031:03

Personal Portfolio Management

Lecture Three

Money Market

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Learning objectives

After completion of this chapter you should be able to:

understand the relationship between interest rates and

money market security prices

value money market securities estimate holding period returns

identify the major risks faced when investing in money

market securities

understand how the yield curve is estimated

list and discuss the main term structure theories

understand a simple approach to estimating the effect

of changes in interest rates on price. 2


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Chapter outline

1 Introduction2 Money market securities

3 The valuation of money market securities

4 Risks attached to money market securities

5 Estimating the yield curve

6 Theories of term structure

7 A model of interest rates

8 Summary

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1 Introduction

The money market allows financial institutions, corporations and individuals

to meet their short-term investing and borrowing requirements.

Money market securities are short-term in nature (less than 12 months).

Securities traded in the money market include:

commercial bills promissory notes

Treasury notes.

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2 Money market securities

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Interest rates on money market instruments are quoted on a nominal annual basis.

Example: A 3.02% 7-day money market investment of $25m will pay interest

of:

nominal rate per annum = 3.02%

interest rate for 7-days = 3.02 x (7/365)

= 0.0579178082191781%

interest amount = 25m x 7/365 x 3.02%

= $14,479.45

Therefore, after 7-days, the $25m investment returns:

final payment = principal + interest

= $25m + $14 479.45

= $25,014,479.45

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It is also possible to invest in an overnight security in the money market.

Example: A $30m overnight investment will pay interest based on the

11am cash rate of 3.00% p.a.

Nominal rate per annum = 3.00%

Interest rate for 1-day = 3.00 x (1/365)

= 0.00821917808219178%

Interest amount = 30m x 1/365 x 3.00%

= $2,465.75

Therefore, after one day, the $30m investment returns:

final payment = principal + interest

= $30m + $2,465.75

= $30,002,465.75.

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The general valuation formula is:

P = price

FV = face value (or principal)

d = number of days to maturity

y = yield (nominal % per annum)

Example: The quoted yield for a 90-day dealer's bill is 3.38% per annum.

This represents a yield of 0.833424657534247% (3.38% p.a. x 90/365)

over the 90-day period.

For a $100,000 face value, the price is:

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FVP= d y1+ x

365 100

FV 100,000P= =

d y 90 3.381+ x 1+ x

365 100 365 100

= $99 173.46


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Example (cont.): The yield assumes the bill is held for the next 90-days

until maturity.

That is, the bond is purchased now for $99,173.46 and sold in 90-

days time for $100,000.

The holding period return is:

(100,000-$99,173.46)/$99,173.46 = 3.38001607374479% for 90-days.

The effective annual rate would be:

(1.00833424657534247)365/90-1 = 0.0342 or 3.42% p.a.

The continuously compounded rate of return would be:

365/90 ln(1.00833424657534247) = 0.0337 or 3.37% p.a.

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Relationship between price and yield

Example: Suppose yields are expected to decrease from 5% to 4%. A 90-

day bill could be purchased now for $98,782.14.

If the yield falls to 4%, the bill could be sold for $99,023.33.

The profit from this strategy would be $241.19.

Example: Suppose yields are expected to increase from 3% to 3.5%. A 90-

day bill could be sold (issued) now for $99,265.71.

If the yield rises to 6%, the bill could be ought back for $99,144.37.

The profit from this strategy would be $121.34.

Note: There is an inverse relationship between price and yield !!!!!

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Relationship between price and yield (cont.)

Example: Suppose yields are expected to remain unchanged at 4%.

What is the change in price of a 90-day bill over the next 30-days?

The original price of the bill at 4% p.a. and 90-days until maturity is

$99,023.33.

At the same yield and only 60-days to maturity, the price rises to

$99,346.76. The difference in price is $323.43

Note: There is an inverse relationship between price and

time to maturity !!!!

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Example: Consider a 90-day BAB with a yield of 4.5%p.a. and face value of

$100,000.

If the BAB is held for 30 days and then sold at a yield of 4.2%, what is the

holding return?

Price (d=90, y=4.5% p.a.) = $98,902.59.

BAB held for 30 days then sold as 60 day BAB, yield = 4.2% pa.

Price (d=60, y=4.2% p.a.)= $99,314.32.

HRt= ln (Pt/Pt-1) = ln($99,314.32/$98,902.59) = 0.0042

(0.42% continuously compounding for 30 days).

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The continuously compounding rate per annum is calculated by multiplying

the 30-day rate by the proportion of the year remaining:

Continuously compounding rate per annum = 0.00415434376568409 x

365 / 30 = 0.0505 (or 5.05% p.a. continuously compounding per

annum).

Alternatively, the discrete rate of return for the period is:

HRt = ($99 314.32 - $98 902.59) / $98 902.59 = 0.00416298501384049

(or 0.42% p.a. for 30 days).

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The nominal rate per annum is calculated by multiplying the 30-day discrete

rate by the proportion of the year remaining.

Nominal rate per annum = 0.00416298501384049 x 365 / 30 = 0.0506 (or

5.06% p.a. nominal per annum).

Time component (vary time to maturity, keep yield fixed)

Price = $100,000/(1+0.045 x 60/365) = $99,265.71

Time component = ($99,265.71 - $98,902.59) / $98,902.59 =

0.00367149131281607 (or 4.47% p.a.).

Yield component = (0.00416298501384049 - 0.00367149131281607)

= 0.000491493701024418 (or 0.60% p.a.).

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Sensitivity of money market security price to changes in yield

Elasticity measures the percentage change in security price given a 1%

change in the yield.

That is:

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dP (1+y)

Elasticity= xd(1+y) P


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Example: Price of bill

Example: Percentage change in yield

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FV 100,000P= =

d y 90 3.51+ x 1+ x

365 100 365 100

= $99 144.37

d(1+y) (1+y )-(1+y )21=(1+y) (1+y )1

(0.040-0.035) x 90/365 =

90 4.001+ x

365 100

= 0.122232785549377%


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Example: Price change (note: elasticity here is the negative of the time tomaturity, i.e. -1)

The actual decrease in price given the increase in yields from 3.5%

p.a. to 4% p.a. is:

$99,144.37 - $99,023.33 = -$121.04

In this case, approximation is quite close, with a percentage error of

only 0.12% per annum.

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d(1+y)dP=P x x Elasticity

(1+y)

= $99 144.37 x 0.00122232785549377 x -1

= -$121.19


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Textbook page 160

Solve problems 14, 15, 16, 19, 21 & 23

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Problems


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For money market securities the major risks are:

interest rate risk

default risk

inflation risk

foreign exchange risk

marketability risk

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Interest rate risk:

is uncertainty as to the future value of the money market security

is due to changes in yields.

Example of interest rate risk

Imagine an investor purchases a dealers bill with face value of $100,000,

a yield of 4% per annum, time to maturity of 90 days and a price of

$99,023.33.

If the bill is held until maturity, the return on the bill is the nominal yield

of 4% per annum.

What if the investor sells the bill when the bill has 30 days left to

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Example (cont.):

The price obtained for the bill at 30 days to maturity is determined by the30-day yield quoted at that time. Say that the yield at 30 days to maturity

is 3%. In this case the price is:

This would give a yield of 4.49% per annum for the 60-day

investment period

If the yield were to increase to 5% the price would be:

The yield over the 60-day period is 3.49% per annum.

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100 000P= $99 754.03

30 3.001+ x

365 100

100 000P= $99 590.7230 5.00

1+ x365 100


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Default risk

Not so important for Treasury notes, which are government securities

with little likelihood of default.

The risk of default can be significant when investing in corporate

securities such as promissory notes.

Rating agencies such as Moodys, Standard and Poors, and Dunn and

Bradstreet provide a service to investors by grading different types of

debt.

Moodys provides short-term debt ratings where the short-term

ratings refer to the issuers ability to meet all short-term obligations.

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Inflation risk

This risk relates to the underlying rate of inflation.

The greater the rate of inflation, the smaller the real rate of return.

Yields are quoted in nominal terms, but investors care about real rate of

return.

Hence, the larger the relative rate of inflation, for fixed nominal yield,

the smaller the real rate of return.

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Inflation risk (cont.)

Often called the Fisher effect, inflation risk is defined as follows:

This can be rearranged to give a definition of the real rate of return:

Thus, if the nominal rate is held constant and the inflation rate rises, the

real rate of return must fall.

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(1 + y) = (1 + )(1 + r)

where

y = nominal yield

= expected inflation rate

r = real rate of return

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Exchange rate risk

is relevant for foreign securities (e.g. Eurocurrencies).

Marketability or liquidity risk

is caused by thin trading

liquidity premium may be built into price.

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Yield curve is the relationship between yield and time to maturity.

It is assumed that securities differ only in terms of their time to maturity.

If the yield curve and the time to maturity is known for a money market

security, the security yield can be read from an appropriate yield curve.

The yield curve is used for pricing bonds not currently traded.

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There are many techniques to estimate yield curves.

Examples include:

non-linear model

quadratic approximation

piece-wise linear model.

Non-linear model

where:

a1, a2, a3, a4= parameters to be estimated (by NLLS)

tj= time to maturity of the zero-coupon security

exp(.) = exponential function

y(tj) = nominal yield per annum.

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jj j 3 421y(t )=(a +a t )*exp(a t )+a


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Quadratic model (spline smoothing)

where:

a, b, c = parameters to be estimated (by OLS)

tj= time to maturity of the money market security.

Piecewise linear model

where

y1is the observed yield at time t1(nearest the required estimate).

Note: The piecewise linear model goes through consecutive yields. This

ensures that observed yields are part of the yield curve.

It is mathematically the simplest model to apply.

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2 1jj 1 1

2 1

y -yy(t )=y + (t -t )

t -t

2

j j jPV(t )=a+bt +ct


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Piecewise linear model

Example: Given observed 2-month and 3-month yields of 5.89% p.a. and

6.47% p.a., use the piecewise approach to estimate the yield curve for a

maturity of 2.5 months (i.e. y(2.5)).

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6.47 5.89y(2.5) 5.89 (2.5 2)

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= 6.18% p.a.


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These theories provide an explanation for the shape and predictive power of

the yield curve.

The four basic theories of the term structure are:

the expectations theory

liquidity premium

Segmentation

preferred habitat.

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The expectations theory

Longer term rates are simply combinations of shorter term rates.

Investors are hence indifferent between holding long and short-term

securities.

(1+ 0R2) = (1+ 0R1)(1+E(1R2))

or (1+E(1R2)) = (1+ 0R2)/(1+ 0R1)

where aRbis the yield observed at time a with maturity at time b where a>0, this indicates a future yield.

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The expectations theory (cont.)

Example: Say the current 1-month rate is 4% p.a. and the 3-month rate is

3% p.a. The rate for the following 2-month period is:

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31+0.03x

12-1 x 6 = 0.0249 or 2.49%

11+0.04x

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The liquidity premium

Investors are compensated for holding a security that does not match the

investors preferred investment horizon.

It is assumed that most investors prefer short-term to longer-term

investments.

(1+ 0R2) = (1+ 0R1)(1+E(1R2)(1+1lp2))

where 1lp2 is the liquidity premium (lp) required to attract an investor

over the second period, rather than just the first period.

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Segmentation

Short rate is not related to the long rate in any direct manner as these

rates are assumed to be set in different markets subject to different

supply and demand effects.

The implicit forward rate is not necessarily equal to the expected spotrate.

Preferred habitat

Premium (PR) is required to attract investors away from their preferredinvestment term.

The implicit forward rate is not necessarily equal to the expected spot

rate.

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Textbook page 161

Solve problems 27 & 28

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Problems


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Simplest model assumes random walk (one-factor)

dr = a r dt + s r dz

where

dr is the instantaneous change in the interest rate

a is the instantaneous drift

r is the interest rate

dt is the instantaneous change in time

s is the instantaneous std. dev. of interest rates

dz is a Wiener process (mean 0, std dev. of 1).

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7 A model of interest rates


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Money market securities are priced using simple-interest formula.

The price of money market securities is inversely related to both the yield andtime to maturity.

Risks of money market securities include:

interest rate risk

default risk inflation risk

exchange rate risk

liquidity risk.

Four competing theories of term structure:

expectations

liquidity premium

segmented

preferred habitat.

8 Summary

3 - lecture three - money market final

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