1

On the significance of Sraffa's reswitching: some long-standing financial puzzles

and their joint resolution

Mike OsborneSheffield Business School, Sheffield Hallam University

Keynes Seminar

Post Keynesian Study Group

Robinson College, Cambridge

15 February 2011

On the significance of Sraffa's reswitching: some long-standing financial puzzles and their joint resolution

Outline

Methodology (multiple interest rates)

An alternative route to the results

Contemporary relevance (capital budgeting)

Meaning (bond market analysis)

3

Puzzles

1. The multiple interest rate puzzle: What is the use and meaning of all interest rates in the time value of money equation?

4

p ci

(1 r) i1

n

5

p ci

(1 r) i1

n

p c1

(1 r1) c2

(1 r1)2 c3

(1 r1)3 c4

(1 r1)4

p c1

(1 r2) c2

(1 r2)2 c3

(1 r2)3 c4

(1 r2)4

p c1

(1 r3) c2

(1 r3)2 c3

(1 r3)3 c4

(1 r3)4

p c1

(1 r4 ) c2

(1 r4 )2 c3

(1 r4 )3 c4

(1 r4 )4

6

Examples of the number of interest rates

• Retail loan of five years with 60 monthly payments- 60 monthly interest rates

• Retail mortgage of 25 years with 300 monthly payments- 300 monthly interest rates

• Govt bond of 30 years duration with semi-annual coupons- 60 semi-annual interest rates

• In extreme cases, when analysing a portfolio of instruments between payment dates, the number of rates can rise to several thousand interest rates.

7

‘Now it is true that an equation of the nth degree has n roots of one sort or another, and that therefore the general equation for the definition of a rate of interest can also have n solutions, where n is the number of " years " concerned. Indeed, if we adopt continuous compounding, as in strict theory we should, the theoretical number of solutions is infinite! Nevertheless, in the type of payments series with which we are most likely to be concerned, it is extremely probable that all but one of these roots will be either negative or imaginary, in which case they will have no economic significance.’

Boulding, K. 1936b. Time and investment: A reply, Economica, 3(12)

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Questions about the multiple interest rate puzzle

• Where are the roots located?Each root is a value of zj= (1+rj)

• Do all values of zj or rj have use?

• Do all values of zj or rj have meaning?

9

The absolute values of the interest rates |rj| embedded in the zeros zj = (1+rj) of a typical 4th order TVM equation

+ i

- i

Real ax is

r2

r3

r4

r1 z1

z2

z3

z4

‘If we accept without proof the so-called fundamental theorem of algebra that every equation f(x)=0, where is a polynomial in x of given degree n and the coefficients a1,a2, …,an are given real or complex numbers, has at least one real or complex root, and take into consideration that all computations with complex numbers are carried out with the same rules as with rational numbers, then it is easy to show that the polynomial f(x) can be represented (and in only one way) as a product of first-degree factors where a, b,…,l are real or complex numbers.’ Furthermore: ‘Multiplying out the expression and comparing the coefficients of the same powers of x, we see immediately that

which are Viete’s formulas.’

Aleksandrov, A., Kolmogorov, A. & Lavrent’ev, M. 1969. Mathematics: Its content, methods and meaning, NY, Dover, 1999 reprint (Vol.1, pp. 271-272)

The Fundamental Theorem of Algebra

a1 a b c ... l,

a2 ab ac ... kl,

a3 abc abd ...,

..................................

an abc...l

x 4 a1x 3 a2x 2 a3x a4 0

(x a)(x b)(x c)(x d) 0

x 4 (a bc d)x 3 (abac ad bc bd)x 2 (abc abd acd cdb)x abcd 0

12

1 bi

(1 r) i1

n

1

(1 r)n0

The special form of a TVM polynomial

A particular manifestation of Viete’s formulas in the context of the time value of money equation

13

1 bi

(1 r) i1

n

1

(1 r)n0

ji rb

The special form of a TVM polynomial

The special relationship between the coefficients and interest rates of the special form

A particular manifestation of Viete’s formulas in the context of the time value of money equation

14

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

15

1

c1

p

(1 r)

c2

p

(1 r)2

c3

p

(1 r)3

c4

p 1

(1 r)4 1

(1 r)40

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

16

1

c1

p

(1 r)

c2

p

(1 r)2

c3

p

(1 r)3

c4

p 1

(1 r)4 1

(1 r)40

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

jrp

c

p

c

p

c

p

c14321

17

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

jrp

c

p

c

p

c

p

c14321

j

i

r

cp

1

1

c1

p

(1 r)

c2

p

(1 r)2

c3

p

(1 r)3

c4

p 1

(1 r)4 1

(1 r)40

18

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

jrp

c

p

c

p

c

p

c14321

j

i

r

cp

1

Every orthodox TVM equation has a ‘complex twin’

1

c1

p

(1 r)

c2

p

(1 r)2

c3

p

(1 r)3

c4

p 1

(1 r)4 1

(1 r)40

19

p c1

(1 r ) c2

(1 r )2 c3

(1 r )3 c4

(1 r )4

Every orthodox TVM equation has a ‘complex twin’

P c1

(1 R) c2

(1 R)2 c3

(1 R)3 c4

(1 R)4

(1 Ra j ) (1 r j )

(1 R)(1m j ) (1 r j )

p

p m j

p

p

a j(1 R)n

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Puzzles

1. The multiple interest rate puzzle: What is the use and meaning of all interest rates in the time value of money equation?

2. The reswitching puzzle: When two techniques of production are compared, reswitching is when one technique is cheapest at a low interest rate, switches to being more expensive at a higher rate, and then reswitches to being cheapest at yet higher rates

Reswitching - Samuelson (1966) model

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c L1(1 r) L2(1 r)2 L3(1 r)3

C L1(1 R) L2(1 R)2 L3(1 R)3

Time period Labor inputs for technique A Labor inputs for technique B

1 (last period) L1 = 0 L1 = 6

2 (two periods ago) L2 = 7 L2 = 0

3 (three periods ago) L3 = 0 L3 = 2

cA L2(1 r)2

C A L2(1 R)2

cB L1(1 r) L3(1 r)3

C B L1(1 R) L3(1 R)3

c cB

cAL1

L2

(1 r) 1 L3

L2

(1 r)

C C B

C AL1

L2

(1 R) 1 L3

L2

(1 R)

c 6

7(1 r) 1 2

7(1 r)

C 6

7(1 R) 1 2

7(1 R)

c 6

7(1 r) 1 2

7(1 r)

C 6

7(1 R) 1 2

7(1 R)

Reswitching - Samuelson (1966) model

c 2

7

R r1 R r2

(1 R)

C 8

72

7

(R 0)(R 2)

(1 R)

0 2R

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Reswitching

c L1

L2

(1 r) 1 L3

L2

(1 r)

C L1

L2

(1 R) 1 L3

L2

(1 R)

(1 Ra j ) (1 r j )

c L3

L2

a j(1 R)

a j (r j R)

Reswitching - Sraffa-Pasinetti (1966) model

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wa 1 0.8(1 r)

20(1 r)8

wb 1 0.8(1 r)

(1 r)25 24

w wb

wa

(1 r)25 24

20(1 r)8

w (1 r)25 24

20(1 r)8

W (1 R )25 24

20(1 R )8

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20 0.25

25

W w R ri

1

25

20(1 R )8

W 1.25 1

20

R ri 1

3

R ri

4

25

(1 R )8

Reswitching - Sraffa-Pasinetti (1966) model

w (1 r)25 24

20(1 r)8

W (1 R )25 24

20(1 R )8

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Puzzles

1. The multiple interest rate puzzle: What is the use and meaning of all interest rates in the time value of money equation?

2. The reswitching puzzle: When two techniques of production are compared, reswitching is when one technique is cheapest at a low interest rate, switches to being more expensive at a higher rate, and then reswitches to being cheapest at yet higher rates

3. The capital budgeting puzzle: NPV versus IRR.Academics recommend NPV while many practitioners use IRR

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NPV I 0 ci

(1 r)i1

n

0 I0 ci

(1 IRR) i1

n

Capital budgeting

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NPV versus IRR

($20)

($10)

$0

$10

$20

$30

$40

$50

$60

$70

$80

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%

Rate of Interest

NPV

A

B

IRRA

IRRB

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The pitfalls of IRR

1. multiple values of IRR exist

2. the two criteria can produce inconsistent rankings of mutually exclusive projects

3. a non-constant yield curve affecting the cost of capital prevents use of the IRR criterion

4. by itself, IRR does not indicate whether a project is ‘borrowing’ or ‘lending’

Textbook description of IRR pitfalls:Brealey, R., Myers, S. & Allen, F. 2009.Principles of Corporate Finance, 9th ed., McGraw-Hill

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Reswitching Corporate finance

NPV I 0 ci

(1 r)i1

n

0 I0 ci

(1 IRR) i1

n

C L3

L2

a j(1 R)

(1 Ra j ) (1 r j )

a j (r j R)

C * L1

L2

(1 R) 1 L3

L2

(1 R)

C L1

L2

(1 r) 1 L3

L2

(1 r)

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Reswitching Corporate finance

NPV I 0 ci

(1 r)i1

n

0 I0 ci

(1 IRR) i1

n

(1 r)(1m j ) (1 IRR j )

n

j

j r

am

I

NPV

)1(0

(1 r a j ) (1 IRR j )

C L3

L2

a j(1 R)

(1 Ra j ) (1 r j )

a j (r j R)

C * L1

L2

(1 R) 1 L3

L2

(1 R)

C L1

L2

(1 r) 1 L3

L2

(1 r)

32

Puzzles1. The multiple interest rate puzzle: What is the use and

meaning of all interest rates in the time value of money equation?

2. The reswitching puzzle: When two techniques of production are compared, reswitching is when one technique is cheapest at a low interest rate, switches to being more expensive at a higher rate, and then reswitches to being cheapest at yet higher rates

3. The capital budgeting puzzle: NPV versus IRR.Academics recommend NPV while many practitioners use IRR

4. The bond market puzzle: Is there an accurate formula for the interest elasticity of the price of a bond (duration)?

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Reswitching Corporate finance Bond finance

NPV I 0 ci

(1 r)i1

n

0 I0 ci

(1 IRR) i1

n

(1 r)(1m j ) (1 IRR j )

n

j

j r

am

I

NPV

)1(0

(1 r a j ) (1 IRR j )

C L3

L2

a j(1 R)

(1 Ra j ) (1 r j )

a j (r j R)

C * L1

L2

(1 R) 1 L3

L2

(1 R)

C L1

L2

(1 r) 1 L3

L2

(1 r)

n

ii

r

cp

1 )1(

n

ii

R

cP

1 )1(

34

p

pM r

35

Reswitching Corporate finance Bond finance

NPV I 0 ci

(1 r)i1

n

0 I0 ci

(1 IRR) i1

n

(1 r)(1m j ) (1 IRR j )

n

j

j r

am

I

NPV

)1(0

(1 r a j ) (1 IRR j )

C L3

L2

a j(1 R)

(1 Ra j ) (1 r j )

a j (r j R)

C * L1

L2

(1 R) 1 L3

L2

(1 R)

C L1

L2

(1 r) 1 L3

L2

(1 r)

n

ii

r

cp

1 )1(

n

ii

R

cP

1 )1(

(1 R)(1m j ) (1 r j )

p

p m j

a j(1 R)n

(1 Ra j ) (1 r j )

36

p

pM r

p

pMD r

MD* MD

(1 r)

p

pMD

r

(1 r)

By construction:

MD = the weighted average maturity of the cash flows, where the weight for each maturity is the ratio of the present value of the relevant cash flow to the present value (price) of the bond

37

p

pM r

p

pMD r

MD* MD

(1 r)

p

pMD

r

(1 r)

(1 R)(1m j ) (1 r j )

(1 R)(1m1) (1 r1)

m1 r

(1 R)

By construction:

MD = the weighted average maturity of the cash flows, where the weight for each maturity is the ratio of the present value of the relevant cash flow to the present value (price) of the bond

38

p

pM r

p

pMD r

MD* MD

(1 r)

p

pMD

r

(1 r)

(1 R)(1m j ) (1 r j )

(1 R)(1m1) (1 r1)

m1 r

(1 R)

p

pMD m1

12

mmp

p n

j

By construction:

MD = the weighted average maturity of the cash flows, where the weight for each maturity is the ratio of the present value of the relevant cash flow to the present value (price) of the bond

39

p

pM r

p

pMD r

MD* MD

(1 r)

p

pMD

r

(1 r)

(1 R)(1m j ) (1 r j )

(1 R)(1m1) (1 r1)

m1 r

(1 R)

p

pMD m1

12

mmp

p n

j

The product of the (n-1) unorthodox mark-ups, mj for j =2 to n, is approximately the weighted average maturity of the cash flows, where the weight for each maturity is the ratio of the present value of the relevant cash flow to the present value (price) of the bond

40

Puzzles1. The multiple interest rate puzzle: What is the use and meaning of

all interest rates in the time value of money equation?

2. The reswitching puzzle: When two techniques of production are compared, reswitching is when one technique is cheapest at a low interest rate, switches to being more expensive at a higher rate, and then reswitches to being cheapest at yet higher rates

3. The capital budgeting puzzle: NPV versus IRR.Academics recommend NPV while many practitioners use IRR

4. The bond market puzzle: Is there an accurate formula for the interest elasticity of the price of a bond (duration)?

41

Puzzles1. The multiple interest rate puzzle: What is the use and meaning of

all interest rates in the time value of money equation?

2. The reswitching puzzle: When two techniques of production are compared, reswitching is when one technique is cheapest at a low interest rate, switches to being more expensive at a higher rate, and then reswitches to being cheapest at yet higher rates

3. The capital budgeting puzzle: NPV versus IRR.Academics recommend NPV while many practitioners use IRR

4. The bond market puzzle: Is there an accurate formula for the interest elasticity of the price of a bond (duration)?

5. Can multiple interest rate analysis be made dynamic & stochastic? Does the Black-Scholes equation have a complex twin?