- Email: [email protected]
Investment under uncertainty with a zero lower bound on interest rates
Journal Pre-proof Investment under uncertainty with a zero lower bound on interest rates George Dotsis
PII: DOI: Reference:
S0165-1765(20)30008-2 https://doi.org/10.1016/j.econlet.2020.108954 ECOLET 108954
To appear in:
Economics Letters
Received date : 19 December 2019 Accepted date : 8 January 2020 Please cite this article as: G. Dotsis, Investment under uncertainty with a zero lower bound on interest rates. Economics Letters (2020), doi: https://doi.org/10.1016/j.econlet.2020.108954. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof Title Page
Investment Under Uncertainty With a Zero Lower Bound on Interest
pro of
Rates George Dotsis
This paper examines irreversible investment decisions when the interest rate is stochastic and constrained by a zero lower bound. In contrast to the commonly found negative relationship between investment and uncertainty, it is shown that the presence of a lower bound on interest
re-
rates induces an asymmetric effect of interest rate uncertainty on investment decision. When the interest rate is low an increase in interest rate volatility decreases the value of waiting and increases investment but when the interest rate is high an increase in interest rate volatility
lP
increases the value of waiting and decreases investment.
urn a
Key words: irreversible investment decisions, zero lower bound, shadow interest rate
Jo
JEL classification: G11, G12, G31
I would like to thank Andrianos Tsekrekos, Norvald Instefjord and the participants at the 19th Annual International Real Options Conference in Athens for useful comments and suggestions. I would like to thank an anonymous referee for his valuable comments and suggestions. Any remaining errors are my responsibility alone. Contact details: Department of Economics, University of Athens, Sofokleous & Aristidou str., 5th floor, office 519. Tel: 210-36894545, email: [email protected].
1
Journal Pre-proof *Manuscript Click here to view linked References
Investment Under Uncertainty With a Zero Lower Bound on Interest
pro of
Rates
This paper examines irreversible investment decisions when the interest rate is stochastic and constrained by a zero lower bound. In contrast to the commonly found negative relationship between investment and uncertainty, it is shown that the presence of a lower bound on interest rates induces an asymmetric effect of interest rate uncertainty on investment decision. When the interest rate is low an increase in interest rate volatility decreases the value of waiting and
re-
increases investment but when the interest rate is high an increase in interest rate volatility
lP
increases the value of waiting and decreases investment.
Key words: irreversible investment decisions, zero lower bound, shadow interest rate
Jo
urn a
JEL classification: G11, G12, G31
1
Journal Pre-proof
1. Introduction In response to the 2007-8 global financial crisis central banks of advanced economies reduced
pro of
the level of short-term interest rates to near zero in an effort to stabilize financial markets and revive the economy. This has sparked renewed academic interest regarding the effects that the zero lower bound on interest rates has on fiscal and monetary policy effectiveness. In this paper I examine the impact of the zero lower bound on optimal investment rules under interest rate uncertainty.
re-
Long time ago studies in the finance literature have stressed the importance of option like characteristics in investment appraisal (see Dixit and Pindyck, 1994). When there is irreversibility and sunk costs an investor can either undertake a project today or defer and wait
lP
until more information accrues and decide whether to invest at a future time period. The option to wait is valuable because the investor can avoid unfavorable shifts with adverse effects on the value of the project. Besides cash flows, another important determinant of investment appraisal
urn a
is the level of interest rates. If cash flows are deterministic but the level of interest rate is stochastic, the option to wait and delay investment is still valuable when there is anticipated resolution of uncertainly about interest rates and therefore project's cost of capital. The contribution of this study is to show that, under certain conditions, the presence of a
Jo
lower bound on interest rates generates an asymmetric effect of interest rate volatility on investment decisions.1 To the best of my knowledge, this is a novel result. Using the shadow-rate model of Black (1995) for modeling interest rate dynamics it is shown that when the interest rate is low an increase in interest rate uncertainty reduces the value of waiting and increases 1
The lower bound exists because nominal interest rates cannot become negative since investors will choose to hold physical currency instead of an asset that pays negative interest. The lower bound can be slightly negative given the opportunity cost from holding physical currency.
2
Journal Pre-proof
investment, while when the interest rate is high an increase in interest rate uncertainty increases the value of waiting. The intuition of this result is simple. When the interest rate is low the value
pro of
of waiting is small since the firm faces more downside risk than upside risk because the benefits from a realization of lower interest rates are truncated by the lower bound. Note that this result is absent in models that do not take into account the zero lower bound constraint. Ingersoll and Ross (1992) and Carmona and Leon (2007) examine optimal investment decision when interest rates follow the square root process and find that an increase in interest rate volatility delays
re-
investment.2
2. Investment Decision with Shadow Interest Rates Following Black (1995), suppose that the observed risk-free interest rate rt0 is the maximum of
lP
the shadow rate, st0 , and a lower bound rL:
rt0 max( st0 , rL )
(1)
urn a
The interest rate is equal to the shadow rate whenever the shadow rate is above the lower bound or otherwise equal to the lower bound.
3
The shadow rate follows a one-factor Ornstein–
Uhlenbeck diffusion proposed by Vasicek (1977): (2)
Jo
dst ( st )dt dZt
3
Black (1995) suggests that the short term interest rate has option like characteristics since it can be viewed as a call option on the shadow interest rate with a strike price equal to the lower bound. Recent studies find that shadow rate models are more informative about the monetary policy stance when interest rates are at the lower bound (see, for example, Wu and Xia (2016).
3
Journal Pre-proof
where st is the shadow interest rate at time t, Zt is a standard Wiener process, κ is the speed of mean reversion, θ is the long run mean σ is the volatility. Suppose that at time t (t0 or t1) a firm
pro of
can undertake a project that costs K and generates a riskless cashflow equal to C one period later. The investment opportunity at time t0 is equal to I t0 P(t0 , t1 ) C K if the firm decides to invest immediately and at time t1 is equal to I t1 P(t1 , t2 ) C K if the firm decides to wait for
tj one period. The term P(ti , t j ) EtQi exp max( su , rL )du is the price of a discount bond at ti
re-
time i that matures at time j. The discounted expected value of the investment opportunity at time t0 is given by:
(3)
lP
P(t0 , t1 ) Qt0 I t1 P(t0 , t1 ) tQ0 max P(t1 , t2 ) C K , 0
The expectation is taken with respect to the risk-neutral probability measure Q. where the long-
, given that the market price of risk λ is constant. Note that
urn a
run mean transforms to
the discount bond prices under shadow rate dynamics do not admit a closed form solution.4 Figure I plots the current and the discounted expected value of the investment opportunity, calculated via Monte Carlo simulation, with parameters C=1, K=0.9, κ=0.2, θ* =2%, rL =0% and σ = 0.01 or σ = 0.05 and t1-t0 = t2-t1 = 1 year. I assume that λ = 0 in order to examine the impact
Jo
of volatility that arises solely from the diffusion component. The speed of mean revision implies a half-life of approximately 2.3 years. This is in line with many empirical studies that have found that short-term interest rates are highly persistent. The horizontal axis plots the shadow rate at time t0 with values ranging from the -20% to 20%. The dotted line plots the project value if the 4
Gorovoi and Linetsky (2004) provide an approximation solution of bond prices based on Weber-Hermite parabolic cylinder functions.
4
Journal Pre-proof
project is undertaken immediately. Note that the internal rate of return (IRR) of the investment is approximately equal to 12%. The option values in the numerical example are given by max
pro of
(Discounted Expected Investment Value - Current Investment Value, 0). The blue line plots the discounted expected value of the investment opportunity when the volatility of the interest rate is 5% and the red line plots the discounted expected value of the investment opportunity when the volatility of the interest rate is 1%. Figure I illustrates that even when the current investment value is negative the expected investment value is positive because
re-
there is a possibility that the interest rate may fall below the IRR in the next period. Figure I shows that the time value of the option does not always increase with volatility. When the current interest rate is low the increase in volatility decreases the present value of the
lP
expected investment opportunity, while when the current interest rate is high the present value of the expected investment opportunity increases. This is because the presence of the lower bound generates an asymmetric effect of interest rate volatility that depends on the current level of
urn a
interest rates. When the current rate is low, the presence of the lower bound truncates the benefits from a realization of a lower future rate, while leaving unaltered the downside risk from an increase in rates.
In Figure I the expected investment value becomes smaller than the current NPV after a
Jo
certain level of current rates. The break-even rate is the current rate at which the option to wait is zero (discounted expected investment value = current investment value). At this rate the option is sufficiently in-the-money and the benefit of waiting is exactly offset by the cost from undertaking immediately the positive NPV project. For rates below the break-even rate, the expected investment value falls below the current value and for rates above the break-even rate
5
Journal Pre-proof
the expected investment value exceeds the current value. When the current rate is below the break-even it is optimal to undertake the project at its current NPV because the cost of waiting
pro of
from foregone interest exceeds any benefits from the possibility of a future decrease in the interest rate.
[Insert Figure I Here]
Figure II plots the break-even rate for various levels of volatility (ranging from 1% to 18%). The break-even rate is an inverted U-shaped function of volatility. As volatility increases the
re-
option becomes sufficiently in-the-money at higher interest rates. The increase in volatility decreases the benefits of waiting and the decision to invest in the project will be undertaken at relatively higher interest rates. For very high levels of volatility (e.g., 10%) an additional
lP
increase in volatility decreases the break-even rate and the option becomes sufficiently in-themoney at lower levels of interest rates.5 The inverted U-shaped relationship is the result of the combined impact that volatility has on current as well as on expected investment value. At low
urn a
levels of volatility the decrease in the expected investment value dominates and that generates an upward path on break-even rates, while the decrease in the current investment value dominates at high levels of volatility and that generates a downward path on break-even rates.
Jo
[Insert Figure II Here]
3. Conclusion
This paper shows that the presence of a lower bound on interest rates can produce a positive relationship between interest rate volatility and investment. When interest rates are low, an
5
Note that an annual volatility of 10% is unrealistically high and rarely observed. Using the extracted time series of the US shadow rate from Wu and Xia (2016), the volatility of the shadow rate has been historically between 0.5% to 2% with the exception of the 1980-1982 period when volatility increased significantly and reached almost 10%.
6
Journal Pre-proof
increase in rate volatility may in fact induce firms to invest in positive NPV projects rather than waiting. This result is relevant especially for the case of low-risk projects where the risk-free
pro of
rate comprises the largest component of the hurdle rate.
References
Black, F., 1995. Interest Rates As Options, Journal of Finance, 50, 1371–1376.
re-
Carmona, J., and A. Leon, 2007. Investment Option Under CIR Interest Rates. Finance Research Letters, 4,242–253.
Dixit, A. and R. Pindyck, 1994. Investment Under Uncertainty, Princeton University Press,
lP
Princeton, NJ.
Gorovoi, V., and V. Linetsky, 2004. Black’s Model of Interest Rates as Options, Eigenfunction Expansions, and Japanese Interest Rates, Mathematical Finance, 14, 49–78.
urn a
Ingersoll, J. E. and S. A. Ross. 1992. Waiting to Invest: Investment and Uncertainty. Journal of Business, 65, l-29.
McDonald, R. and D. Siegel, 1986. The Value of Waiting to Invest, Quarterly Journal of Economics, 101, 707–28.
Vasicek, O., 1977. An Equilibrium Characterization of the Term Structure, Journal of Financial
Jo
Economics, 5, 177-188.
Wu, J. C., and F. D. Xia, 2016. Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound, Journal of Money, Credit, and Banking, 48, 253-291.
7
lP
re-
pro of
Journal Pre-proof
Jo
urn a
Figure I: The figure plots the current investment value (dotted line) and the discounted expected value of the investment opportunity (red and blue lines) if the investment decision is postponed for one period, when C=1, K=0.9, κ=0.2, θ* =2%, rL =0 and σ = 0.01 or σ = 0.05 and Δt=1 year.
8
re-
pro of
Journal Pre-proof
Jo
urn a
lP
Figure II: The figure plots the break-even rate as a function of volatility for C=1, K=0.9, κ=0.2, θ* =2%, rL =0 and option maturity=1 year.
9
PII: DOI: Reference:
S0165-1765(20)30008-2 https://doi.org/10.1016/j.econlet.2020.108954 ECOLET 108954
To appear in:
Economics Letters
Received date : 19 December 2019 Accepted date : 8 January 2020 Please cite this article as: G. Dotsis, Investment under uncertainty with a zero lower bound on interest rates. Economics Letters (2020), doi: https://doi.org/10.1016/j.econlet.2020.108954. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof Title Page
Investment Under Uncertainty With a Zero Lower Bound on Interest
pro of
Rates George Dotsis
This paper examines irreversible investment decisions when the interest rate is stochastic and constrained by a zero lower bound. In contrast to the commonly found negative relationship between investment and uncertainty, it is shown that the presence of a lower bound on interest
re-
rates induces an asymmetric effect of interest rate uncertainty on investment decision. When the interest rate is low an increase in interest rate volatility decreases the value of waiting and increases investment but when the interest rate is high an increase in interest rate volatility
lP
increases the value of waiting and decreases investment.
urn a
Key words: irreversible investment decisions, zero lower bound, shadow interest rate
Jo
JEL classification: G11, G12, G31
I would like to thank Andrianos Tsekrekos, Norvald Instefjord and the participants at the 19th Annual International Real Options Conference in Athens for useful comments and suggestions. I would like to thank an anonymous referee for his valuable comments and suggestions. Any remaining errors are my responsibility alone. Contact details: Department of Economics, University of Athens, Sofokleous & Aristidou str., 5th floor, office 519. Tel: 210-36894545, email: [email protected].
1
Journal Pre-proof *Manuscript Click here to view linked References
Investment Under Uncertainty With a Zero Lower Bound on Interest
pro of
Rates
This paper examines irreversible investment decisions when the interest rate is stochastic and constrained by a zero lower bound. In contrast to the commonly found negative relationship between investment and uncertainty, it is shown that the presence of a lower bound on interest rates induces an asymmetric effect of interest rate uncertainty on investment decision. When the interest rate is low an increase in interest rate volatility decreases the value of waiting and
re-
increases investment but when the interest rate is high an increase in interest rate volatility
lP
increases the value of waiting and decreases investment.
Key words: irreversible investment decisions, zero lower bound, shadow interest rate
Jo
urn a
JEL classification: G11, G12, G31
1
Journal Pre-proof
1. Introduction In response to the 2007-8 global financial crisis central banks of advanced economies reduced
pro of
the level of short-term interest rates to near zero in an effort to stabilize financial markets and revive the economy. This has sparked renewed academic interest regarding the effects that the zero lower bound on interest rates has on fiscal and monetary policy effectiveness. In this paper I examine the impact of the zero lower bound on optimal investment rules under interest rate uncertainty.
re-
Long time ago studies in the finance literature have stressed the importance of option like characteristics in investment appraisal (see Dixit and Pindyck, 1994). When there is irreversibility and sunk costs an investor can either undertake a project today or defer and wait
lP
until more information accrues and decide whether to invest at a future time period. The option to wait is valuable because the investor can avoid unfavorable shifts with adverse effects on the value of the project. Besides cash flows, another important determinant of investment appraisal
urn a
is the level of interest rates. If cash flows are deterministic but the level of interest rate is stochastic, the option to wait and delay investment is still valuable when there is anticipated resolution of uncertainly about interest rates and therefore project's cost of capital. The contribution of this study is to show that, under certain conditions, the presence of a
Jo
lower bound on interest rates generates an asymmetric effect of interest rate volatility on investment decisions.1 To the best of my knowledge, this is a novel result. Using the shadow-rate model of Black (1995) for modeling interest rate dynamics it is shown that when the interest rate is low an increase in interest rate uncertainty reduces the value of waiting and increases 1
The lower bound exists because nominal interest rates cannot become negative since investors will choose to hold physical currency instead of an asset that pays negative interest. The lower bound can be slightly negative given the opportunity cost from holding physical currency.
2
Journal Pre-proof
investment, while when the interest rate is high an increase in interest rate uncertainty increases the value of waiting. The intuition of this result is simple. When the interest rate is low the value
pro of
of waiting is small since the firm faces more downside risk than upside risk because the benefits from a realization of lower interest rates are truncated by the lower bound. Note that this result is absent in models that do not take into account the zero lower bound constraint. Ingersoll and Ross (1992) and Carmona and Leon (2007) examine optimal investment decision when interest rates follow the square root process and find that an increase in interest rate volatility delays
re-
investment.2
2. Investment Decision with Shadow Interest Rates Following Black (1995), suppose that the observed risk-free interest rate rt0 is the maximum of
lP
the shadow rate, st0 , and a lower bound rL:
rt0 max( st0 , rL )
(1)
urn a
The interest rate is equal to the shadow rate whenever the shadow rate is above the lower bound or otherwise equal to the lower bound.
3
The shadow rate follows a one-factor Ornstein–
Uhlenbeck diffusion proposed by Vasicek (1977): (2)
Jo
dst ( st )dt dZt
3
Black (1995) suggests that the short term interest rate has option like characteristics since it can be viewed as a call option on the shadow interest rate with a strike price equal to the lower bound. Recent studies find that shadow rate models are more informative about the monetary policy stance when interest rates are at the lower bound (see, for example, Wu and Xia (2016).
3
Journal Pre-proof
where st is the shadow interest rate at time t, Zt is a standard Wiener process, κ is the speed of mean reversion, θ is the long run mean σ is the volatility. Suppose that at time t (t0 or t1) a firm
pro of
can undertake a project that costs K and generates a riskless cashflow equal to C one period later. The investment opportunity at time t0 is equal to I t0 P(t0 , t1 ) C K if the firm decides to invest immediately and at time t1 is equal to I t1 P(t1 , t2 ) C K if the firm decides to wait for
tj one period. The term P(ti , t j ) EtQi exp max( su , rL )du is the price of a discount bond at ti
re-
time i that matures at time j. The discounted expected value of the investment opportunity at time t0 is given by:
(3)
lP
P(t0 , t1 ) Qt0 I t1 P(t0 , t1 ) tQ0 max P(t1 , t2 ) C K , 0
The expectation is taken with respect to the risk-neutral probability measure Q. where the long-
, given that the market price of risk λ is constant. Note that
urn a
run mean transforms to
the discount bond prices under shadow rate dynamics do not admit a closed form solution.4 Figure I plots the current and the discounted expected value of the investment opportunity, calculated via Monte Carlo simulation, with parameters C=1, K=0.9, κ=0.2, θ* =2%, rL =0% and σ = 0.01 or σ = 0.05 and t1-t0 = t2-t1 = 1 year. I assume that λ = 0 in order to examine the impact
Jo
of volatility that arises solely from the diffusion component. The speed of mean revision implies a half-life of approximately 2.3 years. This is in line with many empirical studies that have found that short-term interest rates are highly persistent. The horizontal axis plots the shadow rate at time t0 with values ranging from the -20% to 20%. The dotted line plots the project value if the 4
Gorovoi and Linetsky (2004) provide an approximation solution of bond prices based on Weber-Hermite parabolic cylinder functions.
4
Journal Pre-proof
project is undertaken immediately. Note that the internal rate of return (IRR) of the investment is approximately equal to 12%. The option values in the numerical example are given by max
pro of
(Discounted Expected Investment Value - Current Investment Value, 0). The blue line plots the discounted expected value of the investment opportunity when the volatility of the interest rate is 5% and the red line plots the discounted expected value of the investment opportunity when the volatility of the interest rate is 1%. Figure I illustrates that even when the current investment value is negative the expected investment value is positive because
re-
there is a possibility that the interest rate may fall below the IRR in the next period. Figure I shows that the time value of the option does not always increase with volatility. When the current interest rate is low the increase in volatility decreases the present value of the
lP
expected investment opportunity, while when the current interest rate is high the present value of the expected investment opportunity increases. This is because the presence of the lower bound generates an asymmetric effect of interest rate volatility that depends on the current level of
urn a
interest rates. When the current rate is low, the presence of the lower bound truncates the benefits from a realization of a lower future rate, while leaving unaltered the downside risk from an increase in rates.
In Figure I the expected investment value becomes smaller than the current NPV after a
Jo
certain level of current rates. The break-even rate is the current rate at which the option to wait is zero (discounted expected investment value = current investment value). At this rate the option is sufficiently in-the-money and the benefit of waiting is exactly offset by the cost from undertaking immediately the positive NPV project. For rates below the break-even rate, the expected investment value falls below the current value and for rates above the break-even rate
5
Journal Pre-proof
the expected investment value exceeds the current value. When the current rate is below the break-even it is optimal to undertake the project at its current NPV because the cost of waiting
pro of
from foregone interest exceeds any benefits from the possibility of a future decrease in the interest rate.
[Insert Figure I Here]
Figure II plots the break-even rate for various levels of volatility (ranging from 1% to 18%). The break-even rate is an inverted U-shaped function of volatility. As volatility increases the
re-
option becomes sufficiently in-the-money at higher interest rates. The increase in volatility decreases the benefits of waiting and the decision to invest in the project will be undertaken at relatively higher interest rates. For very high levels of volatility (e.g., 10%) an additional
lP
increase in volatility decreases the break-even rate and the option becomes sufficiently in-themoney at lower levels of interest rates.5 The inverted U-shaped relationship is the result of the combined impact that volatility has on current as well as on expected investment value. At low
urn a
levels of volatility the decrease in the expected investment value dominates and that generates an upward path on break-even rates, while the decrease in the current investment value dominates at high levels of volatility and that generates a downward path on break-even rates.
Jo
[Insert Figure II Here]
3. Conclusion
This paper shows that the presence of a lower bound on interest rates can produce a positive relationship between interest rate volatility and investment. When interest rates are low, an
5
Note that an annual volatility of 10% is unrealistically high and rarely observed. Using the extracted time series of the US shadow rate from Wu and Xia (2016), the volatility of the shadow rate has been historically between 0.5% to 2% with the exception of the 1980-1982 period when volatility increased significantly and reached almost 10%.
6
Journal Pre-proof
increase in rate volatility may in fact induce firms to invest in positive NPV projects rather than waiting. This result is relevant especially for the case of low-risk projects where the risk-free
pro of
rate comprises the largest component of the hurdle rate.
References
Black, F., 1995. Interest Rates As Options, Journal of Finance, 50, 1371–1376.
re-
Carmona, J., and A. Leon, 2007. Investment Option Under CIR Interest Rates. Finance Research Letters, 4,242–253.
Dixit, A. and R. Pindyck, 1994. Investment Under Uncertainty, Princeton University Press,
lP
Princeton, NJ.
Gorovoi, V., and V. Linetsky, 2004. Black’s Model of Interest Rates as Options, Eigenfunction Expansions, and Japanese Interest Rates, Mathematical Finance, 14, 49–78.
urn a
Ingersoll, J. E. and S. A. Ross. 1992. Waiting to Invest: Investment and Uncertainty. Journal of Business, 65, l-29.
McDonald, R. and D. Siegel, 1986. The Value of Waiting to Invest, Quarterly Journal of Economics, 101, 707–28.
Vasicek, O., 1977. An Equilibrium Characterization of the Term Structure, Journal of Financial
Jo
Economics, 5, 177-188.
Wu, J. C., and F. D. Xia, 2016. Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound, Journal of Money, Credit, and Banking, 48, 253-291.
7
lP
re-
pro of
Journal Pre-proof
Jo
urn a
Figure I: The figure plots the current investment value (dotted line) and the discounted expected value of the investment opportunity (red and blue lines) if the investment decision is postponed for one period, when C=1, K=0.9, κ=0.2, θ* =2%, rL =0 and σ = 0.01 or σ = 0.05 and Δt=1 year.
8
re-
pro of
Journal Pre-proof
Jo
urn a
lP
Figure II: The figure plots the break-even rate as a function of volatility for C=1, K=0.9, κ=0.2, θ* =2%, rL =0 and option maturity=1 year.
9