What’s Happening to Interest Rates? · Purple R143 G70 B147 Fuscia R233 G69 B140 Red R200 G30 B69 Orange 5 R238 G116 29 Dark grey R63 G69 B72 UK Base Rates since 1694 Source Bank

08/06/2015

1

What’s Happening to Interest Rates?

Andrew D Smith

[email protected]

05 June 2015

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Presentation Overview

• Interest rate history: long term and the recent past

• Using diffusion models

• Interpreting prices of interest rate options

• Supply and demand arguments for lower bounds

• Conclusions

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History of Interest Rates

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Why Interest Rates Matter for Investors

• Rates explain market prices of fixed income instruments

• Input to liability valuation calculations for insurers and pension funds

• If invested short, lower rates squeeze available balance sheet capital

• Hurdle rates for project investments

• At the apex of a dependency structure for many planning models (for

example, equity returns expressed as risk-free + risk premium)

• Interest rates spreads affect profitability of carry trades (for example,

rolling futures, swaps against liabilities discounted at bond yields.)

• A policy instrument that may be used to target inflation or FX rates.

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UK Base Rates since 1694

5

Source Bank of England, Guardian, Macrobond

05 June 2015


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UK Market Rates (2 Year Term)

6

0.0%

2.5%

5.0%

7.5%

10.0%

12.5%

15.0%

1970 75 80 85 90 95 2000 05 10 15

Gilts

LIBOR

OIS

Source: Bank of England

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International Comparison: 2 Year Gov Yields

7

0.0%

2.5%

5.0%

7.5%

10.0%

12.5%

15.0%

70 75 80 85 90 95 00 05 10 15

USD

DEM

GBP

Source: Fed, BoE, Buba

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Recent Yields on German Bonds

8

-0.25%

0.00%

0.25%

0.50%

0.75%

1.00%

2012 2013 2014 2015

5 year

4 year

3 year

2 year

1 year

Source: Buba

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UK 2-Year Spreads (Swaps over Gilts)

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-0.2%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1990 95 2000 05 10 15

LIBOR

OIS


05 June 2015

Local Volatility Diffusion Models

Calibrating Drift and Volatility Terms

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Geometric Random Walk Models

• The default model for positive economic series (such as share prices

or foreign exchange rates)

• The absolute size of an extreme percentile stress is proportional to the

current level

• This is consistent with Solvency II standard formula approach for

equities, property and the risk-free curve.

11 05 June 2015


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The Rate Fall in 2012

12

0.01%

0.10%

1.00%

10.00%

10 11 12 13 14 15

2 y

ear

gilt

spot

yie

ld (

log s

cale

)

31 July 2011, r=0.75%

31 July 2012, r = 0.07%

9σ event

1-in-1018 years

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Test for Constant Volatility

08 June 2015 13

1970 1975 1980 1985 1990 1995 2000 2005 2010

Quadra

tic V

ariation

= S

um

of S

quare

d C

hanges

The red line shows the

cumulative sum of

squared log changes.

The abrupt change of

direction in the financial

crisis shows that an

increase in (relative)

volatility accompanied

falling rates.


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What if We Don’t Take Logs?

08 June 2015 14

1970 1975 1980 1985 1990 1995 2000 2005 2010

Quadra

tic V

ariation

= S

um

of S

quare

d C

hanges

The blue line shows the

cumulative sum of

squared changes in

absolute rates. Here

the slope is decreasing;

the absolute level of

volatility has reduced

as rates have fallen.

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Displaced Random Walk Model

08 June 2015 15

1970 1975 1980 1985 1990 1995 2000 2005 2010

Quadra

tic V

ariation

= S

um

of S

quare

d C

hanges

The green line shows

the cumulative sum of

squared changes in

log(r+3.48%). This is

closer to a constant

slope, or equivalently,

the green line is as

close as we can get to

a straight line.


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What is a Diffusion Process?

• Generalise the arithmetic / geometric random walk

• Write down a plausible equation for interest rates and then fit

parameters.

• Fix a rate term, and consider rt as a Markov process

• Diffusion process – no discontinuities

• Characterised by conditional mean per unit time (drift) and

conditional variance per unit time (volatility squared)

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Diffusion Formula

17

yie

ld

time 0 1

r0 r0+μ(r0)+σ(r0)

r0+μ(r0)-σ(r0)

drt = μ(rt)dt + σ(rt)dZt

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The Cox-Ingersoll-Ross Model

18

drt = α(μ-rt)dt + ηrt½ dZt

Change in yield

Mean reversion

Ergodic Mean

Volatility

Change in

Brownian Motion

Avoids zero

(without the rt½ term, we

have the Vasiček model)

Possible parameters:

μ = 5%

α = 0.02

η = 0.20

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What is the Lower Bound for Interest Rates?

• The lower bound is an artefact of the model we choose.

08 June 2015 19

Model Interest Rate Lower Bound

Geometric random walk Zero

Cox-Ingersoll-Ross Zero (instant rate)

Strictly positive (positive terms)

Arithmetic random walk Unbounded

Vasiček Unbounded

Displaced geometric random walk -3.48%

Interest Rate Option Pricing Models

Black, Bachelier and Related Models

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Implied Volatility of Interest Caps and Floors

08 June 2015 21

0%

10%

20%

30%

40%

50%

60%

70%

80%

0% 2% 4% 6% 8% 10%

strike

5 y

ear

lognorm

al im

plie

d v

ola

tilit

y

Implied volatility flattens for large strikes.

Implied volatility explodes as strikes approach zero

As at 29 May 2015

Source: Bloomberg


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1-Year Ahead Percentiles of GBP LIBOR

22

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 Jan-10 Jan-12 Jan-14

95%-ile

Mean

5%-ile


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What is the Lower Bound for Interest Rates?

• The lower bound is an artefact of the model we choose.

08 June 2015 23

Model Interest Rate Lower Bound

Bank of England (Shimko method

of extrapolating Black /

lognormal implied volatility)

Zero

Bachelier (normal implied

volatility); Johnson curves

Unbounded

Displaced geometric random

walk (Piterbarg DDSV LMM)

-3.48% or other estimated

constant


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Extending Displaced Geometric RW’s

24

• First published 1906.

• Reprinted 2009.

• Norman Lloyd Johnson (1917-2004)

qualified as FIA in 1949.

• William Palin Elderton (1877–1962) was

President of the Institute of Actuaries 1932–

1934.

• Johnson distribution family includes normal

and displaced lognormal distributions.

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What do the Examiners Say? (CT8, Apr 2012, Q 6)

25

(i) Write down a stochastic differential equation for the short rate rt for the

Vasiček model.

(ii) State the type of process of which the Vasiček model is a particular example.

(iii) Solve the stochastic differential equation in (i).

(iv) State the distribution of rt for t given.

(v) Derive the expected value and the second moment of rt for t given.

(vi) Outline the main drawback of the Vasiček model.

Examiners’ Model Solution to part (vi):

“The process may become negative which is undesirable in a nominal interest

rate model”

[1]

[1]

[5]

[1]

[3]

[1]

05 June 2015

Supply and Demand Considerations

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Could Interbank Rates become Negative?

• Short government bond yields have become negative in several

currencies

• LIBOR (inter-bank unsecured borrowing) has remained positive

• Typically settled as two cash flows (deposit, followed by redemption +

interest)

• No particular administrative issues with the interest rate becoming

negative – it would mean banks had to pay other banks to look after

their money.

• Negative swap rates also theoretically feasible.

• What about negative perpetuity yields?

27 05 June 2015


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Are Negative Interest Rates Logical?

• Investors can choose to hold physical banknotes.

• It could be argued that the option to hold banknotes should keep

market interest rates above zero.

• But this ignores:

– Cost of storage

– Risk of theft

– Damage from floods, fire

– Cost of moving cash, especially across borders

– Legal tender issues

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Conclusions

05 June 2015


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Lessons Learned from Recent Rate Moves

• Negative nominal rates are already here for a few economies, and

could spread further.

• Model output is a function not only of input parameters but also the

chosen model structure. Consider several alternative models.

• If a single model says negative rates (or rates above 100%) can’t

happen, this does not mean these rates can’t happen.

• Think about fundamental supply and demand in the broader economy

• Interpreting derivative prices requires a whole host of subsidiary

assumptions.

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05 June 2015

31

Expressions of individual views by members of the Institute and Faculty of

Actuaries and its staff are encouraged.

The views expressed in this presentation are those of the presenter.

Questions Comments

Documents

What’s Happening to Interest Rates? · Purple R143 G70 B147 Fuscia R233 G69 B140 Red R200 G30 B69 Orange 5 R238 G116 29 Dark grey R63 G69 B72 UK Base Rates since 1694 Source Bank