forecasting interest rates structural models. structural models are an attempt to determine causal...

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)


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

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Quantity

Pri

ce (

$)


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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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)


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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Quantity

Pri

ce (

$)


As costs rise, supply As costs rise, supply fallsfalls

Qs = S(C,P)Qs = S(C,P)

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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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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


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100

120

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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)


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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

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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 (%

)


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:


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

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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


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


Consumption and WealthConsumption and Wealth

10

57.6

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50

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70

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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


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


Current Spending: Current Spending: $30,000$30,000

Savings: $20,000Savings: $20,000


10

57.6

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50

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S = -$20,000

(Person A)

S = $20,000

(Person B)


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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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

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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

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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

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Inte

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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

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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

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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)

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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

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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

forecasting interest rates structural models. structural models are an attempt to determine causal...

Documents

p slide

model p

price p exogenous

price p endogenous

price equation p

current income c

income increases

future income slide