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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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2 Money market securities
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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2 Money market securities
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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2 Money market securities
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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3 The valuation of money market securities
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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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3 The valuation of money market securities
dP (1+y)
Elasticity= xd(1+y) P
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Sensitivity of money market security price to changes in yield
Example: Price of bill
Example: Percentage change in yield
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3 The valuation of money market securities
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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Sensitivity of money market security price to changes in yield
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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3 The valuation of money market securities
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
23
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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4 Risks attached to money market securities
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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
maturity?25
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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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4 Risks attached to money market securities
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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4 Risks attached to money market securities
(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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5 Estimating the yield curve
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5 Estimating the yield curve
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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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5 Estimating the yield curve
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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5 Estimating the yield curve
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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5 Estimating the yield curve
6.47 5.89y(2.5) 5.89 (2.5 2)
3 2
= 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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6 Theories of term structure
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