Monetary Economics problems
Mo7amd
4386f16hw4.pdf
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ECO 4386 Topics in Monetary Economics Fall 2016
Homework 4. Due 11/29/2016 1. Review of probability and expectations.
a. Suppose X and Y are random variables. In words, describe what expectation of X, E(X), means. b. The covariance between X and Y, COV(X,Y), is defined as:
COV(X,Y) = E[ (X-E(X))(Y-E(Y)) ]. Describe in words what it means for COV(X,Y) > 0.
c. Consider the following joint distribution for random variables X and Y:
X = 5, Y = 2 with probability .5 X = 10, Y = 8 with probability .25 X = -10, Y = 0 with probability .25
Calculate E(X) and E(Y). Calculate COV(X,Y).
2. Suppose households maximize expected utility. Let itQ be the nominal price and i tq the real price of asset i in the current period (t). Let
i 1tX + be the nominal payoff and
i 1tx + be the real payoff of the asset in the next period (t+1). Let )c(U
)c(U M
tc
1tc 1t
+ + β= or
the marginal rate of substitution between consumption at t+1 and consumption at t.
Define i t
i 1ti
1t Q X
)R1( ++ =+ where i
1tR + is the nominal rate of return on asset i, while
i t
i 1ti
1t q x
)r1( ++ =+ where i
1tr + is the real rate of return on asset i.
a. The optimal choice of household satisfies the following equation: ]xM[Eq i 1t1tt
i t ++= .
Describe in words what this equation means.
b. Rewrite the equation in part (a) in terms of real returns (i.e. )r1( i 1t++ ).
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c. Suppose one compares two assets a and b. Suppose )X(E)X(E b 1tt a
1tt ++ = but if the
assets had the same price households would prefer asset a (the “good” asset) over asset b (the “bad” asset). In equilibrium, which asset would have the lowest price? The lowest expected return?
d. Consider the risk free asset with a real payoff of .1x rf 1t =+ What is the price of the risk free asset, rftq , and what is its return
rf 1tr1 ++ ?
e. We can relate expected real return on asset i to the risk-free return by the
following equation:
)M(E
)r1,M(COV r]r[E
1tt
i 1t1t
trf 1t
i 1tt
+
++
++
+ −= .
Explain why an asset can have an expected return greater than the risk free asset? Do households view this asset as a “good” or “bad” asset relative to the risk free asset? Why?
f. Suppose an asset’s payoff is typically high when consumption is high. Would the expected return be greater than, less than, or equal to the risk free rate? Explain.
3. Consider the consumption-based asset pricing model for an indexed discount (zero
coupon) bond that matures in k periods. The optimal asset choice model implies for a k period bond:
]qM[Eq ib 1t,1k1tt
ib t,k +−+= .
a. Show that the price of the k period bond can be written as:
]M[E]M[Eq kt,tt k
1i itt
ib t,k +
= + == ∏ ,
where )c(U
)c(U M
tc
ktc k
kt,t +
+
β = is the multi-period discount factor.
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b. The yield to maturity of this k period discount bond can be written as:
)1()]rE1()rE1)(r1[()r1( ib t,k k/1ib
1ktt
ib 1tt
ib t
ib t,k γ++++=+ −++ ,
where k/1
ib t,k
ib 1t,1k1tib
t,k q )q,M(COV
1)1(
−=γ+ +−+ .
Explain in words the relationship between the yields on the k period bond and yields on one period bonds.
c. Open 4386HW4F16.xls, sheet Q3. This Excel worksheet contains monthly data
on three-month T-Bill rates and ten-year T-Bond rates. Plot the two interest rates on the same graph. Which one is generally higher? Calculate the average for the two interest rates over the sample. What is the average spread between ten-year and three-month interest rates?
What does the average spread this imply about ib t,kγ ? What does this imply about the covariance term in part b? Explain in words why households require a different yield on long-term bonds than on short-term bonds?
4. The consumption based asset pricing can be applied to stock prices. In class, we showed that the consumption based asset pricing model implies real stock prices are given by:
)qM( Elim]dM[Eq s ktkt,ttk 1k
ktkt,tt
s t ++
∞→
∞
= ++ += ∑ ,
where ktd + is the real dividend in time t+k and kt,tM + is the multi-period stochastic discount factor for t+k. a. Describe in words what the above equation implies are the determinants of stock prices. b. Suppose 0)qM( Elim s ktkt,ttk =++∞→ . Then real stock prices can be written as the
present value of expected future dividends:
∑ ∞
=
+
+ =
1k ks
t,k
ktts t )r1(
dE q ,
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where s t,kr is the k-period discount rate for stocks. We can, in turn, write the discount rate for stocks as: )1)(r1()r1( s t,k
ib t,k
s t,k γ++=+ ,
where ibt,kr is the real interest rate on a k-period indexed bond. The term
s t,kγ
reflects the difference between the required return on stocks versus the required return on indexed bonds. Finally, we can write the “premium” on stocks as:
k/1
ktkt,t
ktkt,ts t,k )dM(E
)d,M(COV 1)1(
−=γ+
++
++ .
If s t,kγ >0, how do households view the relative riskiness of stocks versus bonds?
Relate in words the relationship between s t,kγ and )d,M(COV ktkt,t ++ . Explain
why if s t,kγ > 0 , i.e. )d,M(COV ktkt,t ++ < 0, investors tend to view stocks as a less attractive asset (everything else equal) than bonds. c. Excel spreadsheet Q4 has data on actual annual prices and dividends for SP500 (or equivalent) from 1871 to 2012. (Note: we actually use nominal prices and nominal dividends but this will not affect calculations based on the price-dividend ratio). Define the gross returns on stocks to be:
s 1t
t s
s t q
dq )r1( t
−
+ =+ ,
where stq is the price of the stock and td is the dividend payment. Use the excel spread sheet to calculate the gross returns from 1872-2012. What is the average geometric return for the SP500 from the period 1872-2012? Hint: find the natural logarithm on gross returns ( )r1ln( st+ ) in each year and then average over the sample period.
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d. Consider the case where the discount rate (required return) for stocks is a
constant. Then we can write stock prices in terms of its price dividend ratio:
∑ ∞
=
+
+ =
1k ks
tktt
t
s t
)r1(
)d/d(E
d q
,
where )d/d(E tktt + is one plus the expected accumulated dividend growth from t to
t+k.
If the discount rate on stocks is truly constant and households have rational expectations, then calculating the present value of dividends using actual data on dividends rather than expectations of dividends should be approximately equal to the true stock price (the errors in expectations get averaged out). Because we do not have data for dividends off to infinity, we start with actual value of price-dividend ratio in 2012 and work backwards:
actualsels
d q
d q
=
2012
2012
mod
2012
2012
)1(
1 mod
1
11 mod
s
el
t
s t
t
t el
t
s t
r
d q
d d
d q
+
+
=
+ ++
for t = 1871,…,2011.
We set rs to be the geometric average return on stocks over the sample calculated in part c. Excel spreadsheet Q4 has data on actual annual prices and dividends for SP500 (or equivalent) from 1871 to 2012. (Note: we actually use nominal returns and nominal dividends but this will not affect calculations based on the price-dividend ratio. Use the data in the excel spreadsheet as well as the formula above to calculate the model’s predictions for the price-dividend ratio. Plot the actual price-dividend ratio and the model price-dividend ratio. Which one is more volatile (has more “wiggles”)? Is it possible to explain most of the variability in the price-dividend ratio (and hence stock prices) by changes in expectations of future dividends? If not, what else could explain movements in the price-dividend ratio?
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5. The consumption based asset pricing model ties all asset returns to properties of consumption. Suppose marginal utility is given by:
σ− ++ = 1t1tc c)c(U .
Define ρ+
=β 1
1 , where ρ is the time rate of preference and 1t,c
t
1t g1 c
c +
+ += where
1t,cg + is the growth rate of consumption in time period t+1. This implies that stochastic discount factor is a function of the growth rate of consumption:
σ−
+
σ−
++ + ρ+
+ =
β=
β =
)1( )g1(
c c
)c(U )c(U
M 1t,c t
1t
tc
1tc 1t .
We can approximate the risk free real interest rate as:
1t,ctt gEr +σ+ρ= + a constant.
a. If consumption growth is expected to increase what should happen to real interest rates?
b. Worksheet Q5 has data on nominal interest rate on three-month treasury bills, inflation (percent increase in the consumption deflator for non-durable consumption) and real growth in nondurable consumption. The spreadsheet also has set the value of 0.2=σ . In a new column, create a measure of the real interest rate that is the nominal interest rate in time t minus inflation in time period t+1 ( 1tt
actual t Rr +π−= ). In
another column, create an approximate value for the real interest rate based on the consumption based model, actual1t,c
elmod t gr +σ= . Compare, in a chart, the actual
real interest rate and the one implied by the model. Which one is more volatile? Do the same thing, this time with value of 0.5=σ . What happens to the volatility of elmodtr as σ gets larger?
c. Suppose we pick the value of σ so that )r(VAR)r(VAR elmodt actual t ≈ . In your
spreadsheet, try different values of σ until the variances of actual and model interest rates are approximately equal. What is this value of σ ?
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6. Consider the returns on stocks relative to bonds. Worksheet Q6 contains data on annual nominal returns on stocks and annual real returns on 1-year bonds.
a. Using the inflation (pi(t)) in nondurable consumption, calculate real returns on
stocks and bonds. Plot the real returns on stocks and returns on bond over the full sample. Use Excel to calculate the mean (average) and variance of real stock returns over the full sample from 1954 to 2012.
b. We can approximate the expected real returns on stocks and bonds with the following equations:
)r,g(COVr)r(VAR5.)r(E s 1t
c 1t
rf 1t
s 1t
s 1t +++++ σ+=+
)r,g(COVr)r(VAR5.)r(E b 1t c
1t rf
1t b
1t b
1t +++++ σ+=+ .
What does the model predict should happen to the expected real returns on stocks and (keeping their variances constant) if the covariance between consumption and their returns increases?
c. The worksheet also contains data on consumption growth (nondurable consumption and total consumption). Use the Excel command COVAR to calculate )r,g(COV s 1t
c 1t ++ and )r,g(COV
b 1t
c 1t ++ over the full sample for nondurable
consumption. Which one is greater? Is this consistent qualitatively with which asset has greater expected returns?
d. Taking the two equations in part b and substituting for rf1tr + , one can write the equity premium as: )]r(VAR)r(VAR[5.)]r,g(COV)r,g(COV[)r(E)r(E b 1t
s 1t
b 1t
c 1t
s 1t
c 1t
b 1t
s 1t ++++++++ −−−σ=− .
Assuming )r,g(COV)r,g(COV b 1t c
1t s
1t c
1t ++++ > , what happens to the equity premium if σ increases? Explain in words why?
e. Suppose we wanted to find the value of σ that reconciled the historical equity
premium we see in the data. That involves taking the equation in part d. and solving for σ :
)]r,g(COV)r,g(COV[ )]r(VAR)r(VAR[5.)r(E)r(E
b 1t
c 1t
s 1t
c 1t
b 1t
s 1t
b 1t
s 1t
++++
++++
− −+−
=σ .
Using the full sample averages, variances, and covariances previously calculated, find the value of σ needed for the model to fit actual data on stock and bond returns and nondurable consumption growth. How does this value compare to that found in Q5 part c.? Is it possible for the consumption based asset pricing
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model (with power utility) to fit both the volatility of short term interest rates and the equity premium with the same value of σ ?
f. Answer part e. again, this time dropping the years 2008 through 2012 from your calculations. Is the equity premium larger after dropping 2008-2012? Compare the value of σ for the full sample with that of the sample that drops 2008 - 2012? Does adding 2008 - 2012 lessen or exacerbate the equity premium puzzle? Explain why.
