Forecasting Interest RatesForecasting Interest Rates
Structural ModelsStructural Models
Structural ModelsStructural Models
Structural models are an attempt to Structural models are an attempt to determine causal relationships between determine causal relationships between various economic variables: various economic variables: Exogenous variables: Taken as givenExogenous variables: Taken as givenEndogenous Variables: Explained by the modelEndogenous Variables: Explained by the model
Exogenous Model Endogenous
Example: DemandExample: Demand
Demand:Demand:Exogenous: Income (I), Price (P)Exogenous: Income (I), Price (P)Endogenous: Quantity Demanded (D)Endogenous: Quantity Demanded (D)
Exogenous
Income
PriceModel
Endogenous
Quantity Demanded
D = D( I, P)
Example: DemandExample: Demand
The basic model The basic model suggests that as suggests that as prices fall, quantity prices fall, quantity demanded risesdemanded rises For a given level of For a given level of
income and income and preferences, if P=$12, preferences, if P=$12, Q = 300.Q = 300.
If price falls to $8 If price falls to $8 (again, for a fixed level (again, for a fixed level of income and of income and preferences), Q =400preferences), Q =400
0
4
8
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28
0 100 200 300 400 500
Quantity
Pri
ce (
$)
Example: DemandExample: Demand
As income increases, As income increases, demand increases.demand increases. For a given level of For a given level of
income and income and preferences, if P=$12, preferences, if P=$12, Q = 300.Q = 300.
If Income rises, Q=400 If Income rises, Q=400 at a price of $12at a price of $12
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4
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32
0 100 200 300 400 500
Quantity
Pri
ce (
$)
Example: SupplyExample: Supply
Supply:Supply:Exogenous: Costs (C), Price (P)Exogenous: Costs (C), Price (P)Endogenous: Quantity Supplied (S)Endogenous: Quantity Supplied (S)
S = S(C, P)S = S(C, P)
Example: SupplyExample: Supply
The basic model The basic model suggests that as suggests that as prices rise, quantity prices rise, quantity supplied increasessupplied increases
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24
0 100 200 300 400 500
Quantity
Pri
ce (
$)
Example: SupplyExample: Supply
As costs rise, supply As costs rise, supply fallsfalls
Qs = S(C,P)Qs = S(C,P)
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0 100 200 300 400 500
Quantity
Pri
ce (
$)
EquilibriumEquilibrium
Qd = D(I,P)Qd = D(I,P) Qs = S(C,P)Qs = S(C,P) In Equilibrium, Qs = QdIn Equilibrium, Qs = Qd
P* = P(I,C)P* = P(I,C) Q* = Q(I,C)Q* = Q(I,C)
Note that Price is no Note that Price is no longer exogenous, it is longer exogenous, it is explained!explained!
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0 100 200 300 400 500
QuantityP
rice
($)
Using Models to ForecastUsing Models to Forecast
In the previous example, we ended up with a In the previous example, we ended up with a price equationprice equation
P = P(C,I) P = P(C,I) The next step would be to estimate the modelThe next step would be to estimate the model
P = a(C) + b(I) (where a and b are constants)P = a(C) + b(I) (where a and b are constants) Now, note that the following implies:Now, note that the following implies:
P’ = a(C’) + b(I’) (‘ indicates a future value)P’ = a(C’) + b(I’) (‘ indicates a future value) Therefore, to forecast Price:Therefore, to forecast Price:
Forecast Costs (C’)Forecast Costs (C’) Forecast Income (I’)Forecast Income (I’) Insert into the estimated price equation to get P’Insert into the estimated price equation to get P’
Interest Rate ModelsInterest Rate Models(Real Interest Rates)(Real Interest Rates)
Economic models look at how optimizing Economic models look at how optimizing behavior by households and firms behavior by households and firms translates into the supply and demand for translates into the supply and demand for credit.credit.Firms choose capital investment projects to Firms choose capital investment projects to
maximize shareholder value (Demand)maximize shareholder value (Demand)Households choose consumption/savings to Households choose consumption/savings to
maximize utility (Supply)maximize utility (Supply)Supply = Demand defines the equilibrium Supply = Demand defines the equilibrium
interest rateinterest rate
Household SavingsHousehold Savings
Without an active financial markets, Without an active financial markets, household consumption is restricted to household consumption is restricted to equal current incomeequal current income
With capital markets, the With capital markets, the present value of lifetime consumption must equal the present value of lifetime income (assuming (assuming all debts are eventually repaid)all debts are eventually repaid)
A Simple ExampleA Simple Example
Suppose that your current income is equal to Suppose that your current income is equal to $50,000 and you anticipate next year’s income $50,000 and you anticipate next year’s income to be $60,000. The current interest rate is 5%.to be $60,000. The current interest rate is 5%.
In the absence of financial markets, your In the absence of financial markets, your consumption stream would be $50,000 this year consumption stream would be $50,000 this year and $60,000 next year.and $60,000 next year.
C = Y (Current Consumption = Current Income)C = Y (Current Consumption = Current Income)
C’ = Y’ (Future Consumption = Future Income)C’ = Y’ (Future Consumption = Future Income)
Consumption PossibilitiesConsumption Possibilities
0102030405060708090
100
0 10 20 30 40 50 60 70 80 90 100
Current Consumption (000s)
Futu
re C
onsu
mpt
ion
(000
s)
Now, Add Financial MarketsNow, Add Financial Markets
You can alter your current consumption by taking out a You can alter your current consumption by taking out a loan or putting money in the bankloan or putting money in the bank
C = $50,000 + (Borrowing/Lending)C = $50,000 + (Borrowing/Lending)
Loans must be repaid with interest next year. Deposits Loans must be repaid with interest next year. Deposits earn interest (for simplicity assume that these rates are earn interest (for simplicity assume that these rates are the same)the same)
C’= $60,000 – (1.05)(Borrowing/Lending)C’= $60,000 – (1.05)(Borrowing/Lending)
Y (Current Income)
Y’ (Future Income)
Now, Add Financial MarketsNow, Add Financial Markets
We can combine these two conditions to get the We can combine these two conditions to get the following: following:
)1(
'
)1(
'
r
YY
r
CC
In the previous example, we had
)05.1(
000,60$000,50$
)05.1(
'
CC
Consumption PossibilitiesConsumption Possibilities
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100 110 120
Current Consumption (000s)
Fu
tue
r C
on
su
mp
tio
n (
00
0s)
Borrowing
Lending
The budget constraint indicates all the possible ways to consume your lifetime wealth (PV of lifetime income)
Consumption PossibilitiesConsumption Possibilities
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100 110 120
Current Consumption (000s)
Fu
tue
r C
on
su
mp
tio
n (
00
0s)
$112,500
$107,142
Slope = $112,500/$107,142 = 1.05 = (1+ r)
This is the relative price of future consumption in terms of current consumption
Optimal BehaviorOptimal Behavior
Households need a way to “Rank” Households need a way to “Rank” consumption/savings choices. This is done with consumption/savings choices. This is done with a Utility Functiona Utility Function
U(C, C’) = Total UtilityU(C, C’) = Total Utility
Utility functions only have two restrictionsUtility functions only have two restrictions More of everything always better (total utility is More of everything always better (total utility is
increasing in consumption)increasing in consumption) The more you have, the less its worth (As The more you have, the less its worth (As
consumption increases, marginal utility decreases)consumption increases, marginal utility decreases)
Optimal BehaviorOptimal Behavior Given the possibilities, households choose an optimal Given the possibilities, households choose an optimal
solution solution
Marginal Benefit = Marginal Cost
Increase in Happiness From Spending an Extra $ Today
(Marginal Utility)
=
Decrease in Happiness From Spending an Extra $ Tomorrow
(Marginal Utility)
(1+r)
Optimal Consumption Optimal Consumption
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100 110 120
Savings = $20,000
Suppose that at an interest rate of 5%, you choose to save $20,000. Note that tomorrow’s consumption is now $60,000 + $20,000(1.05) = $81,000
SavingsSavings
0123456789
0 10 20 30 40 50
Savings ($)
Inte
rest
Rat
e (%
)
Optimal BehaviorOptimal Behavior
Marginal Utility
At C = $30,000=
Marginal Utility
At C’ = $81,000(1.05)
We know this decision is optimal. We know this decision is optimal. Therefore, we can say that:Therefore, we can say that:
Optimal BehaviorOptimal Behavior
Suppose that interest rates increase to Suppose that interest rates increase to 7%.7%.
Marginal Utility
At C = $30,000<
Marginal Utility
At C’ = $81,000(1.07)
We need to alter consumption a bit to re-balance this equation!! (We need to raise today’s marginal utility and lower tomorrow’s!!)
This can be done by raising future consumption and lowering current consumption.
Optimal Consumption Optimal Consumption
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100 110 120
Savings = $30,000
Suppose that at an interest rate of 7%, you choose to save $30,000. Note that tomorrow’s consumption is now $60,000 + $30,000(1.07) = $92,100
Aggregate SavingsAggregate Savings
0123456789
10
0 10 20 30 40 50
Savings ($)
Inte
rest
Rat
e (%
)
S
Optimal BehaviorOptimal Behavior
Suppose you alter your consumption to Suppose you alter your consumption to C = $20,000 (S = $30,000) , C’ = $92,000C = $20,000 (S = $30,000) , C’ = $92,000
Marginal Utility
At C = $20,000=
Marginal Utility
At C’ = $92,100(1.07)
The new consumption pattern is also optimal!!
Again, assume that the interest rate is 5%, Again, assume that the interest rate is 5%, consider two individualsconsider two individuals
Person APerson A
Current income: $10,000
Anticipated future income: $50,000
Wealth: $57,619Wealth: $57,619
Person BPerson B
Current Income: $50,000
Anticipated Future income: $8,000
Wealth: $57,619Wealth: $57,619
Consumption and WealthConsumption and Wealth
10
57.6
0
50
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Consumption and WealthConsumption and Wealth
With capital markets, consumption is not With capital markets, consumption is not determined by current income, but by wealth determined by current income, but by wealth (present value of lifetime income)(present value of lifetime income)
These two individuals, having the same wealth, These two individuals, having the same wealth, should choose the same consumptionshould choose the same consumption
Again, assume that the interest rate is 5%, Again, assume that the interest rate is 5%, consider two individualsconsider two individuals
Person APerson A
Current income: $10,000Current income: $10,000
Anticipated future Anticipated future income: $50,000income: $50,000
Wealth: $57,619Wealth: $57,619
Current Spending: Current Spending: $30,000$30,000
Savings: -$20,000Savings: -$20,000
Person BPerson B
Current Income: $50,000Current Income: $50,000
Anticipated Future Anticipated Future income: $8,000income: $8,000
Wealth: $57,619Wealth: $57,619
Current Spending: Current Spending: $30,000$30,000
Savings: $20,000Savings: $20,000
Consumption and WealthConsumption and Wealth
10
57.6
0
50
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
S = -$20,000
(Person A)
S = $20,000
(Person B)
Consumption and WealthConsumption and Wealth
With capital markets, consumption is not With capital markets, consumption is not determined by current income, but by wealth determined by current income, but by wealth (present value of lifetime income)(present value of lifetime income)
These two individuals, having the same wealth, These two individuals, having the same wealth, should choose the same consumption.should choose the same consumption.
For a given level of wealth, savings declines with For a given level of wealth, savings declines with income growthincome growth
Aggregate SavingsAggregate Savings
0123456789
10
0 10 20 30 40 50
Savings ($)
Inte
rest
Rat
e (%
)
S
From the previous example, a rise in income growth might reduce savings from 20 to 10.
S’
A Quantitative ExampleA Quantitative Example
22
11
12
11
,
)1(
..
1-1
21
YSrc
YScts
ccMax
cc
Step #1: AffordabilityStep #1: Affordability
22
11
)1( YSrc
YSc
Recall that the two constraints can be reduced to one constraint by eliminating ‘S’
rY
Yr
cc
112
12
1
Step #2: OptimalityStep #2: Optimality
1-1 ,
12
11
21
ccccU
Increase in Happiness From Spending an Extra $ Today
(Marginal Utility)
=
Decrease in Happiness From Spending an Extra $ Tomorrow
(Marginal Utility)
(1+r)
Step #2: OptimalityStep #2: Optimality
1-1 ,
12
11
21
ccccU
Marginal Utilities are just the derivatives!!
rcc 1)()( 21
Marginal Utility Today Marginal Utility Tomorrow
Characterizing the SolutionCharacterizing the Solution
1
21)1(
c
cr
Note that the interest rate is independent of the absolute level Note that the interest rate is independent of the absolute level of consumption. (The interest rate is stationary)of consumption. (The interest rate is stationary)
The long run mean is determined (primarily) by betaThe long run mean is determined (primarily) by beta The Variance is determined by sigma The Variance is determined by sigma Current and Future consumption can be found by inserting the Current and Future consumption can be found by inserting the
above restriction into the wealth constraintabove restriction into the wealth constraint
US Interest RatesUS Interest Rates
%6.4025.198.
1)1( 1
r
In the US, real consumption growth averages 2.5% per yearIn the US, real consumption growth averages 2.5% per year Beta is assumed to equal .98, sigma equals 1Beta is assumed to equal .98, sigma equals 1
Suppose that US consumption growth increases to 3.5%........
%6.5035.198.
1)1( 1
r
Capital InvestmentCapital Investment
Investment refers to the purchase of new capital equipment by the private sector
Firms only invest in projects that add to Firms only invest in projects that add to shareholder value. Therefore, they invest in shareholder value. Therefore, they invest in positive net present value projects.positive net present value projects.
Present Value of Lifetime Profits > CostPresent Value of Lifetime Profits > Cost
A Numerical ExampleA Numerical Example Consider an investment project that generates $25/year in Consider an investment project that generates $25/year in
profits. It has an initial cost of $100. The current interest profits. It has an initial cost of $100. The current interest rate is 5%. Is this project worthwhile?rate is 5%. Is this project worthwhile?
Present Present ValueValue == $25$25 ++
$25$25(1.05)(1.05)
$25$25
(1.05)(1.05)++ 22
$25$25
(1.05)(1.05) 33++ ++...
Year 0 Year 1 Year 2 Year 3
Present Present ValueValue ==
$25$25
.05.05== $500$500 > $100$100
CostCost
A Numerical ExampleA Numerical Example
An alternative way of asking the same question An alternative way of asking the same question is: Does this project generate a sufficient internal is: Does this project generate a sufficient internal rate of return given the firm’s cost of capital rate of return given the firm’s cost of capital (5%)?(5%)?
Internal Internal Rate of Rate of ReturnReturn
=$25$25
$100$100
Investment Cost
Annual $ Return
= ..2525 >>
Given the 5% market interest rate, any project that generates an internal rate of return of at least 5% is profitable
.05.05
Defining ProductionDefining Production
A production function defines total output for A production function defines total output for given supplies of the factors of production given supplies of the factors of production (Capital, Labor and Productivity)(Capital, Labor and Productivity)
Y = F(K, L, A)Y = F(K, L, A)
OutputOutput CapitalCapitalLaborLabor
ProductivityProductivity
Production should exhibit diminishing marginal returns. That is, as capital increases (holding other factors fixed), its contribution to production decreases
Production (Holding Employment Production (Holding Employment Fixed)Fixed)
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0 200 400 600 800 1000
Capital ($)
Pro
fits
($)
/Yr
F(K,L,A)
$25
$100
$100
$10
Internal Rate of Return = 25%
Internal Rate of Return = 10%
Internal Rates of ReturnInternal Rates of Return
0
5
10
15
20
25
30
0 200 400 600 800 1000
Capital ($)
Ret
urn
(%)
Given the market interest rate of 5%, the first 5 investment projects are profitable.
Investment DemandInvestment Demand
It is assumed that labor and capital are It is assumed that labor and capital are complimentscompliments. That is, when employment rises, . That is, when employment rises, the productivity of capital increases as well. the productivity of capital increases as well.
Therefore, as a rise in employment should Therefore, as a rise in employment should increase the demand for capital and, hence, the increase the demand for capital and, hence, the demand for loansdemand for loans
Further, any technological improvement should Further, any technological improvement should also raise the demand for investmentalso raise the demand for investment
A Rise in Investment DemandA Rise in Investment Demand
0
5
10
15
20
25
30
35
0 200 400 600 800 1000
At a market interest rate of 10%, a productivity improvement might increase investment demand from $400 to %500
ttt
tktt t
I
Ikktosubject
IPLAkr
Maxt
1
1
0
1
1
This is the Production Function
A Numerical Example
This is the Cost of Investment
New investment increases the capital stock
To get the internal rate of return, take the derivative of production with respect to ‘K’ and divide by the price of capital.
Characterizing the SolutionCharacterizing the Solution
1
L
k
P
Ar
k
From the Demand side, we see that the interest rate is influenced by:
•Productivity (A)
•Price of Capital (P)
•Relative Factor Supplies (K, L)
Capital Market EquilibriumCapital Market Equilibrium
A capital market A capital market equilibrium is an interest equilibrium is an interest rate that clears the rate that clears the market (i.e.,savings market (i.e.,savings equals investment)equals investment) r*= 10%, r*= 10%, S* = I*= 300S* = I*= 300
0
4
8
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16
20
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Inte
rest
Rat
e S
I
Example: Oil Price ShocksExample: Oil Price Shocks
Two oil price shocks occurred in the 1970’s. The first Two oil price shocks occurred in the 1970’s. The first (1973) was widely considered permanent while the (1973) was widely considered permanent while the second (1979) was considered more temporarysecond (1979) was considered more temporary
First Oil Price ShockFirst Oil Price Shock
A rise in energy prices permanently lowers incomes
0
4
8
12
16
20
0 100 200 300 400 500
Inte
rest
Rat
e S
I
1
21)1(
c
cr
With both current and future consumption falling, savings does not change
First Oil Price ShockFirst Oil Price Shock
A rise in energy prices permanently lowers productivity
0
4
8
12
16
20
0 100 200 300 400 500
Inte
rest
Rat
e S
IA drop in productivity lowers investment demand
1
L
k
P
Ar
k
Interest rates should fall
Second Oil Price ShockSecond Oil Price Shock
A temporary rise in oil prices temporarily lowers income and consumption (c1 falls)
0
4
8
12
16
20
0 100 200 300 400 500
Inte
rest
Rat
e S
I
1
21)1(
c
cr
To “buffer” some of the loss in income, savings drops
Second Oil Price ShockSecond Oil Price Shock
A temporary drop in productivity has a negligible impact on capital investment projects
0
4
8
12
16
20
0 100 200 300 400 500
Inte
rest
Rat
e S
I
Investment remains unchanged
1
L
k
P
Ar
k
Interest rates rise
Example: Oil Price ShocksExample: Oil Price Shocks
Real interest rates fell in (1973), but increased Real interest rates fell in (1973), but increased in 1979.in 1979.
-10
-5
0
5
10
15
201/
1/19
65
1/1/
1968
1/1/
1971
1/1/
1974
1/1/
1977
1/1/
1980
1/1/
1983
1/1/
1986
1/1/
1989
1/1/
1992
1/1/
1995
1/1/
1998
1/1/
2001
InflationRealNominal
Government Deficits and Interest Government Deficits and Interest RatesRates
Last year, the government borrowed Last year, the government borrowed roughly $450 billion from financial markets. roughly $450 billion from financial markets. Should this have an impact on real Should this have an impact on real interest rates?interest rates?
Nominal Interest Rates & InflationNominal Interest Rates & Inflation
i = r + Inflation?i = r + Inflation?Wealth effects (Higher inflation lowers the Wealth effects (Higher inflation lowers the
purchasing power of lifetime wealth)purchasing power of lifetime wealth)The Darby effect (The government taxes The Darby effect (The government taxes
nominal income)nominal income)Expected vs. Actual inflationExpected vs. Actual inflation
Nominal Interest Rates & the FedNominal Interest Rates & the Fed
The Federal Reserve has two potentially The Federal Reserve has two potentially offsetting effects on the nominal interest offsetting effects on the nominal interest rate: rate:
Liquidity EffectLiquidity EffectAnticipated Inflation effectAnticipated Inflation effect
Forecasting Nominal Interest RateForecasting Nominal Interest Rate
Any Interest rate equation could potentially Any Interest rate equation could potentially have any of the following variables:have any of the following variables: Income GrowthIncome GrowthProxies for productivityProxies for productivityRelative price of capitalRelative price of capitalGovernment DeficitsGovernment Deficits Inflation RatesInflation RatesMonetary Policy VariablesMonetary Policy Variables
Mehra ModelMehra Model
43
2111
1
15.04.
07.23.37.)(37.
34.ln12.61.71.22.
tt
tttt
ttttt
ii
iii
RFRyRFRi
If you are currently in time ‘t’ and would like to If you are currently in time ‘t’ and would like to make a forecast of TBill rates in time ‘t+1’. What make a forecast of TBill rates in time ‘t+1’. What would you need?would you need?
Mehra ModelMehra Model
To Forecast the TBill rate, you need:To Forecast the TBill rate, you need:A Forecast for price (to calculate the inflation A Forecast for price (to calculate the inflation
rate)rate)A Forecast of Federal Reserve PolicyA Forecast of Federal Reserve PolicyA Forecast of GDP (to calculate income A Forecast of GDP (to calculate income
growth)growth)Past history of TBill Rates and InflationPast history of TBill Rates and Inflation
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