- Email: [email protected]
Measuring the economic value of loan advice
Economics Letters 117 (2012) 615–618
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Measuring the economic value of loan advice Nicholas Taylor Cardiff Business School, Cardiff University, Cardiff CF10 3EU, United Kingdom
article
info
Article history: Received 21 May 2012 Received in revised form 26 July 2012 Accepted 2 August 2012 Available online 10 August 2012 JEL classification: C1 G1
abstract The purpose of this paper is to provide a method by which the economic value of loan advice can be measured. This is achieved by defining advice in terms of the forecasts of future repayments in order to produce a disutility-minimising loan strategy. The economic value of this strategy is then calculated by considering the fee one is willing to pay to avoid switching to a competing strategy during each period. Applying the methodology to UK fixed and variable mortgage rates produces performance fees that are high when advice takes account of the dynamics of repayments. Moreover, the conditions under which these fees are significantly different from zero are identified. © 2012 Elsevier B.V. All rights reserved.
Keywords: Loan advice Disutility Performance fees
1. Introduction A huge literature exists concerning the relationship between advice and decision quality (see, e.g., Bonaccio and Dalal, 2006, for a review). The general consensus that emerges from this literature is that advice tends to improve decision accuracy (ibid.). For instance, within the finance literature, the evidence suggests that supplyside brokerage analysts’ recommendations have investment value (see, e.g., Jegadeesh et al., 2004). However, less is known about the likely benefits of advice regarding the borrowing decisions of individuals. To begin to address this shortcoming we provide two contributions: a method by which the economic value of loan advice can be calculated, and an application to UK mortgage rate data. A commonly employed measure within the investment literature is the performance fee associated with conditionallyoptimised portfolios constructed using advice based on asset return forecasts; see, e.g., Rapach et al. (2010). Taking this as our knowledge starting point, we consider an individual who selects their (conditionally-optimised) source of funding using advice based on forecasts of repayments over a finite horizon. This strategy is then compared to a competing strategy and a performance fee is calculated such that the conditional expected disutility levels equate at each point in time. Applying to UK mortgage rate data, we examine (variable vs. fixed rate) mortgage advice
E-mail address: [email protected]. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.08.006
over various finite horizons (coinciding with the term over which the fixed rate applies). The results demonstrate that advice based on knowledge of a well known stochastic differential equation has economic value. 2. The proposed measure The proposed measure of the economic value of loan advice is established in two stages: the assumptions of the loan advisers are stated, and a measure of the comparative performance of two competing advisers is derived. Assumption 1. Let period t have unit length (henceforth referred to as the horizon), and let the full grid of all observation points be G = {t1 , . . . , ts }. Given this notation, a loan adviser provides advice regarding an optimal loan type for each customer in order to minimise their customer’s conditional expectation of next period disutility given by p
p
E(DUt |Ft −1 ) = E(f (Yt1 , . . . , Yts )|Ft −1 ),
t ∈ {1, . . . , T },
(1)
where Ft −1 is chosen as the σ -field generated by a set of p conditioning instruments available at time t − 1, Ytj is the loan repayment made by the customer during tj , and the function f is a p monotonic and quasi-convex disutility function dependent on Ytj such that f : [0, +∞) → R+ . Remark. The customer (referred to as the judge in the extant literature) is assumed to take the advice of the adviser.
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N. Taylor / Economics Letters 117 (2012) 615–618
Assumption 2. A selection of m loan types are available, with repayments given by p Ytj
= wt |t −1 Ytj , ′
Assumption 3. Each loan has the same maturity L, with the i-type loan repayment (that is, the ith element of Ytj ) given by the standard annuity formulae:
L→∞
Ri,tj (1 + Ri,tj )L−j+1
(1 + Ri,tj )L−j+1 − 1
= Ri,tj ,
(3)
where Ri,tj is the i-type interest rate available during tj , and L is measured in terms of the number of j-type periods. Remark. The analysis is standardised in two ways: the initial value of the loan is set to unity and its maturity is set to infinity. Remark. The maturity of the loan should be distinguished from the horizon of the loan: the former refers to the period in which the loan must be repaid, while the latter refers to the short term period over which competing loans are compared. Assumption 4. Loan repayments evolve as follows: Y tj = µ t + ϵ tj ,
(4) 1/2
1/2
Assumption 5. Two types of (imperfect) adviser exist (referred to as adviser A and adviser B), each with a different forecast of the repayment moments such that Ytj |Fk,t −1 ∼ N(µk,t |t −1 , 6k,t |t −1 ),
(5)
where µk,t |t −1 is the expected repayment vector associated with the k-type adviser, 6k,t |t −1 is the expected covariance matrix associated with the k-type adviser, and Fk,t −1 is the information set of the k-type adviser. Assumption 6. The customer of the k-type adviser employs the exponential disutility function given by f(
,...,
) = γ0 (exp(γ
p 1 Y t1
,...,γ
p 1 Yts
) − 1),
Ga,t
− E(f (Ytp1 , . . . , Ytps )| µt , 6t , w∗b,t |t −1 fixed) = 0, Gb,t
w∗k,t |t −1
where is the advice vector given by the k-type adviser constructed using Fk,t −1 ⊂ Gk,t , and δt is the maximum fee adviser A is willing to pay to avoid issuing adviser B’s advice (equivalently, it is the economic value of advice differences). Remark. If we relax the restriction on adviser A’s advice and allow this to reflect µt and 6t , then this amounts to assuming that this adviser has perfect foresight (with respect to the repayment moments). In this instance, the above will represent the cost of advice imperfection. The above equation can be solved to yield the economic value of advice differences. Given the complex nature of the conditional expectations, we propose the following numerical procedure. Proposition 1. Under Assumptions 1–6 and Definition 1, the timespecific performance fee (δt∗ ∈ R) is given by the root of the following non-linear equation: p
ϵtj = 6t νtj is an (m × 1) vector of errors, 6t is the (unique) square root of an (m × m) covariance matrix 6t , and νtj ∼ IN(0, 1).
p Yts
p
p
E(f (Yt1 + δt , . . . , Yts + δt )|Ga,t )
where µt is an (m × 1) vector of repayment means (during t),
p Yt1
p
E(f (Yt1 + δt , . . . , Yts + δt )| µt , 6t , w∗a,t |t −1 fixed)
(2)
where Ytj is an (m × 1) vector of loan repayments, and wt |t −1 is an (m × 1) vector containing an indication of the loan type selected constructed using Ft −1 (henceforth the advice vector).
Yi,tj ≡ lim
δt that solves the following equation:
(6)
− E(f (Ytp1 , . . . , Ytps )|Gb,t ) = 0,
(7a)
where E(f (Y )|G) ≈ f (E(Y |G))
+
N 1 n =1
2n!
f (2n) (E(Y |G))var(Y |G)n
n
(2j − 1), (7b)
j=1
and f (2n) (E(Y |G)) is the 2nth derivative of f (Y ) with respect to Y evaluated at E(Y |G) such that f (2n) (E(Y |G)) = γ0 γ12n exp(γ1 E(Y |G)), p Y t1
p Yts
(7c) p Yt1
p Y ts ,
with Y denoting either + δt , . . . , + δt or , . . . , and G denoting either Ga,t or Gb,t , respectively. In turn, the optimal advice vector embedded in G is given by w∗k,t |t −1 =
argmin
wk,t |t −1 ∈{0,1}
p
p
E(f (Yt1 , . . . , Yts )|Fk,t −1 ).
(7d)
where γ1 determines the risk aversion level of the customer.1
The solutions to (7a) and (7d), are obtained by numerical methods.
Given the above assumptions, we wish to assign an interpretable economic value to the expectation of the differences in disutility levels associated with the two adviser types. To this end, we consider the conditional expectation of these differences. Specifically, we penalise use of incorrect conditional expectations by considering the maximum fee adviser A is willing to pay to avoid issuing adviser B’s advice if both are provided with the next period repayment mean and covariance matrix just after their advice has been given. Thus, we have the following definition.
Proof. Taking the conditional expectation of a Taylor series expansion of the exponential disutility function in (6) about the conditional mean of Y we obtain
Definition 1. The economic value of loan advice differences, as given by the conditional performance fee, is defined as the value
E(f (Y )|G) =
∞ 1 (n) f (E(Y |G))E((Y − E(Y |G))n |G). n ! n=0
(8)
Truncating to a 2Nth-order series expansion, and invoking the conditional Gaussian assumption (see Assumption 5), leads to (7b) such that it involves nonlinear transformations of the first two conditional moments only. 3. Application: UK mortgage rates
1 Though disutility functions are generally not employed within economics, the exponential disutility function is commonly employed within other disciplines such as transport engineering; see, e.g., Cheu and Kreinovich (2007).
A UK 75% loan-to-value customer must choose between a variable and fixed rate mortgage. The variable rate is given by the Standard Variable Rate minus 1%, and the fixed rates have
1992
1996
2000
2004
2008
2012
Imperfection Cost 100 200 300 400 1992
1996
2000
Year
2004
2008
2008
2012
1996
2000
2004
2008
2012
1992
1996
2000
2008
2012
2012
2004
2008
2012
Imperfection Cost 100 200 300 400 1992
1996
2000
Year
2004
2008
2012
1992
1996
2000
Year
(g) Informed (24-month).
2008
0
Imperfection Cost 100 200 300 400 2004
2004
(f) Variable (60-month).
0
Imperfection Cost 100 200 300 400
2000
2012
Year
(e) Variable (36-month).
0
1996
2008
Imperfection Cost 100 200 300 400 1992
Year
(d) Variable (24-month).
2004
0
0 2004 Year
1992
2000
(c) Fixed (60-month).
Imperfection Cost 100 200 300 400
Imperfection Cost 100 200 300 400
2000
1996
Year
(b) Fixed (36-month).
0
1996
1992
2012
Year
(a) Fixed (24-month).
1992
617
0
Imperfection Cost 100 200 300 400 0
0
Imperfection Cost 100 200 300 400
N. Taylor / Economics Letters 117 (2012) 615–618
Year
(h) Informed (36-month).
(i) Informed (60-month).
Fig. 1. Imperfection costs. This figure contains the (annualised) imperfection costs (in bps) to the fixed rate only, variable rate only, and informed advice strategies. Table 1 Performance fees. Comparison
Risk aversion parameter (γ1 ) 0
1
2
3
4
34.149* [0.044] 4.549 [0.743]
26.019 [0.090] 17.613 [0.197]
18.957 [0.171] 31.748* [0.050]
11.478 [0.357] 44.319* [0.024]
7.085 [0.545] 56.359** [0.006]
56.719** [0.002] −3.682 [0.725]
34.220* [0.020] 6.099 [0.630]
21.422 [0.077] 25.554 [0.204]
11.214 [0.242] 44.995 [0.132]
0.448 [0.961] 57.130 [0.096]
76.242* [0.015] −5.767 [0.273]
41.349 [0.209] 2.559 [0.709]
19.225 [0.505] 23.618 [0.235]
3.994 [0.854] 48.374 [0.112]
−3.954 [0.854] 71.259 [0.110]
Panel A: 24-month horizon Informed vs. fixed Informed vs. variable Panel B: 36-month horizon Informed vs. fixed Informed vs. variable Panel C: 60-month horizon Informed vs. fixed Informed vs. variable
Notes: This table contains the mean (annualised) performance fees (in bps) to avoid switching from the informed advice strategy to the fixed and variable rate only strategies. * P-values associated with the null hypothesis that the performance fees equal zero are given in brackets, with rejections at the 5% level. ** P-values associated with the null hypothesis that the performance fees equal zero are given in brackets, with rejections at the 1% level.
horizons of 24, 36, and 60 months.2 While the fixed rate ‘dynamics’ over these horizons are known to all advisers with absolute certainty, only one adviser type is assumed to have knowledge of the variable rate dynamics (henceforth the informed adviser): specifically, conditional moments based on Chan et al.’s (1992) unrestricted stochastic differential equation.3 For this adviser, these moments are used as inputs into the calculation of the
2 Monthly data were obtained from the Bank of England’s Statistical Interactive Database (http://www.bankofengland.co.uk) and cover the period January 1995 to December 2011. 3 Conditional moments are calculated as follows. First, equation parameters are obtained by GMM estimation using the full sample of variable rates. Second, at each point within the sample, 100,000 sample paths of the process are generated over the
advice vector and the (annualised) performance fee, where γ0 = 0, γ1 = {0, 1, 2, 3, 4}, and N = 10. The competing advice assumes borrowing exclusively at the variable (and fixed) rate only. The imperfection costs in Fig. 1 exhibit considerable time variation, with informed advice delivering lower imperfection costs across all risk aversion levels. To formally test this observation, we exploit the time variation in fees and examine the null hypothesis that mean performance fees (that is, differences in mean imperfection costs across strategies) equal zero using a Wald test based on the Newey–West HAC estimator. The results
fixed rate horizon period. Third, the simulation averages of the time-series means and variances of these paths are used as the conditional moment measures.
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N. Taylor / Economics Letters 117 (2012) 615–618
in Table 1 show that informed advice generally delivers positive performance fees, with a range of −5.8 bps to 76.2 bps. For extreme levels of risk aversion these fees are significantly above zero (though this result weakens as the horizon increases). However, for a particular risk aversion level and horizon, there are no occasions when informed advice delivers significant fees with respect to both types of competing advice—a result that implies that informed advice delivers limited benefits in this context. 4. Conclusions Time variation in the proposed measure enables use of a simple statistical test such that one can examine the null hypothesis that advice strategies have equal value. Applying the methodology to UK mortgage rates suggests that the costs of following fixed only and variable only strategies can be high (particularly during certain periods). By contrast, the informed strategy delivers
lower imperfection costs, though the statistical significance of relative performance is restricted to customers with particular risk preferences and horizons. The challenge of future research is to demonstrate that loan advice has value under all conditions. References Bonaccio, S., Dalal, R., 2006. Advice taking and decision-making: an integrative literature review, and implications for the organizational sciences. Organizational Behaviour and Human Decision Processes 101, 127–151. Chan, K., Karolyi, A., Longstaff, F., Sanders, A., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227. Cheu, R., Kreinovich, V., 2007. Exponential disutility functions in transportation problems: a new theoretical justification. Technical Report UTEP-CS-07-04. Jegadeesh, N., Kim, J., Krische, S., Lee, C., 2004. Analyzing the analysts: when do recommendations add value? Journal of Finance 59, 1083–1124. Rapach, D., Strauss, J., Zhou, G., 2010. Out-of-sample equity premium prediction: combination forecasts and links to the real economy. Review of Financial Studies 23, 821–862.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Measuring the economic value of loan advice Nicholas Taylor Cardiff Business School, Cardiff University, Cardiff CF10 3EU, United Kingdom
article
info
Article history: Received 21 May 2012 Received in revised form 26 July 2012 Accepted 2 August 2012 Available online 10 August 2012 JEL classification: C1 G1
abstract The purpose of this paper is to provide a method by which the economic value of loan advice can be measured. This is achieved by defining advice in terms of the forecasts of future repayments in order to produce a disutility-minimising loan strategy. The economic value of this strategy is then calculated by considering the fee one is willing to pay to avoid switching to a competing strategy during each period. Applying the methodology to UK fixed and variable mortgage rates produces performance fees that are high when advice takes account of the dynamics of repayments. Moreover, the conditions under which these fees are significantly different from zero are identified. © 2012 Elsevier B.V. All rights reserved.
Keywords: Loan advice Disutility Performance fees
1. Introduction A huge literature exists concerning the relationship between advice and decision quality (see, e.g., Bonaccio and Dalal, 2006, for a review). The general consensus that emerges from this literature is that advice tends to improve decision accuracy (ibid.). For instance, within the finance literature, the evidence suggests that supplyside brokerage analysts’ recommendations have investment value (see, e.g., Jegadeesh et al., 2004). However, less is known about the likely benefits of advice regarding the borrowing decisions of individuals. To begin to address this shortcoming we provide two contributions: a method by which the economic value of loan advice can be calculated, and an application to UK mortgage rate data. A commonly employed measure within the investment literature is the performance fee associated with conditionallyoptimised portfolios constructed using advice based on asset return forecasts; see, e.g., Rapach et al. (2010). Taking this as our knowledge starting point, we consider an individual who selects their (conditionally-optimised) source of funding using advice based on forecasts of repayments over a finite horizon. This strategy is then compared to a competing strategy and a performance fee is calculated such that the conditional expected disutility levels equate at each point in time. Applying to UK mortgage rate data, we examine (variable vs. fixed rate) mortgage advice
E-mail address: [email protected]. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.08.006
over various finite horizons (coinciding with the term over which the fixed rate applies). The results demonstrate that advice based on knowledge of a well known stochastic differential equation has economic value. 2. The proposed measure The proposed measure of the economic value of loan advice is established in two stages: the assumptions of the loan advisers are stated, and a measure of the comparative performance of two competing advisers is derived. Assumption 1. Let period t have unit length (henceforth referred to as the horizon), and let the full grid of all observation points be G = {t1 , . . . , ts }. Given this notation, a loan adviser provides advice regarding an optimal loan type for each customer in order to minimise their customer’s conditional expectation of next period disutility given by p
p
E(DUt |Ft −1 ) = E(f (Yt1 , . . . , Yts )|Ft −1 ),
t ∈ {1, . . . , T },
(1)
where Ft −1 is chosen as the σ -field generated by a set of p conditioning instruments available at time t − 1, Ytj is the loan repayment made by the customer during tj , and the function f is a p monotonic and quasi-convex disutility function dependent on Ytj such that f : [0, +∞) → R+ . Remark. The customer (referred to as the judge in the extant literature) is assumed to take the advice of the adviser.
616
N. Taylor / Economics Letters 117 (2012) 615–618
Assumption 2. A selection of m loan types are available, with repayments given by p Ytj
= wt |t −1 Ytj , ′
Assumption 3. Each loan has the same maturity L, with the i-type loan repayment (that is, the ith element of Ytj ) given by the standard annuity formulae:
L→∞
Ri,tj (1 + Ri,tj )L−j+1
(1 + Ri,tj )L−j+1 − 1
= Ri,tj ,
(3)
where Ri,tj is the i-type interest rate available during tj , and L is measured in terms of the number of j-type periods. Remark. The analysis is standardised in two ways: the initial value of the loan is set to unity and its maturity is set to infinity. Remark. The maturity of the loan should be distinguished from the horizon of the loan: the former refers to the period in which the loan must be repaid, while the latter refers to the short term period over which competing loans are compared. Assumption 4. Loan repayments evolve as follows: Y tj = µ t + ϵ tj ,
(4) 1/2
1/2
Assumption 5. Two types of (imperfect) adviser exist (referred to as adviser A and adviser B), each with a different forecast of the repayment moments such that Ytj |Fk,t −1 ∼ N(µk,t |t −1 , 6k,t |t −1 ),
(5)
where µk,t |t −1 is the expected repayment vector associated with the k-type adviser, 6k,t |t −1 is the expected covariance matrix associated with the k-type adviser, and Fk,t −1 is the information set of the k-type adviser. Assumption 6. The customer of the k-type adviser employs the exponential disutility function given by f(
,...,
) = γ0 (exp(γ
p 1 Y t1
,...,γ
p 1 Yts
) − 1),
Ga,t
− E(f (Ytp1 , . . . , Ytps )| µt , 6t , w∗b,t |t −1 fixed) = 0, Gb,t
w∗k,t |t −1
where is the advice vector given by the k-type adviser constructed using Fk,t −1 ⊂ Gk,t , and δt is the maximum fee adviser A is willing to pay to avoid issuing adviser B’s advice (equivalently, it is the economic value of advice differences). Remark. If we relax the restriction on adviser A’s advice and allow this to reflect µt and 6t , then this amounts to assuming that this adviser has perfect foresight (with respect to the repayment moments). In this instance, the above will represent the cost of advice imperfection. The above equation can be solved to yield the economic value of advice differences. Given the complex nature of the conditional expectations, we propose the following numerical procedure. Proposition 1. Under Assumptions 1–6 and Definition 1, the timespecific performance fee (δt∗ ∈ R) is given by the root of the following non-linear equation: p
ϵtj = 6t νtj is an (m × 1) vector of errors, 6t is the (unique) square root of an (m × m) covariance matrix 6t , and νtj ∼ IN(0, 1).
p Yts
p
p
E(f (Yt1 + δt , . . . , Yts + δt )|Ga,t )
where µt is an (m × 1) vector of repayment means (during t),
p Yt1
p
E(f (Yt1 + δt , . . . , Yts + δt )| µt , 6t , w∗a,t |t −1 fixed)
(2)
where Ytj is an (m × 1) vector of loan repayments, and wt |t −1 is an (m × 1) vector containing an indication of the loan type selected constructed using Ft −1 (henceforth the advice vector).
Yi,tj ≡ lim
δt that solves the following equation:
(6)
− E(f (Ytp1 , . . . , Ytps )|Gb,t ) = 0,
(7a)
where E(f (Y )|G) ≈ f (E(Y |G))
+
N 1 n =1
2n!
f (2n) (E(Y |G))var(Y |G)n
n
(2j − 1), (7b)
j=1
and f (2n) (E(Y |G)) is the 2nth derivative of f (Y ) with respect to Y evaluated at E(Y |G) such that f (2n) (E(Y |G)) = γ0 γ12n exp(γ1 E(Y |G)), p Y t1
p Yts
(7c) p Yt1
p Y ts ,
with Y denoting either + δt , . . . , + δt or , . . . , and G denoting either Ga,t or Gb,t , respectively. In turn, the optimal advice vector embedded in G is given by w∗k,t |t −1 =
argmin
wk,t |t −1 ∈{0,1}
p
p
E(f (Yt1 , . . . , Yts )|Fk,t −1 ).
(7d)
where γ1 determines the risk aversion level of the customer.1
The solutions to (7a) and (7d), are obtained by numerical methods.
Given the above assumptions, we wish to assign an interpretable economic value to the expectation of the differences in disutility levels associated with the two adviser types. To this end, we consider the conditional expectation of these differences. Specifically, we penalise use of incorrect conditional expectations by considering the maximum fee adviser A is willing to pay to avoid issuing adviser B’s advice if both are provided with the next period repayment mean and covariance matrix just after their advice has been given. Thus, we have the following definition.
Proof. Taking the conditional expectation of a Taylor series expansion of the exponential disutility function in (6) about the conditional mean of Y we obtain
Definition 1. The economic value of loan advice differences, as given by the conditional performance fee, is defined as the value
E(f (Y )|G) =
∞ 1 (n) f (E(Y |G))E((Y − E(Y |G))n |G). n ! n=0
(8)
Truncating to a 2Nth-order series expansion, and invoking the conditional Gaussian assumption (see Assumption 5), leads to (7b) such that it involves nonlinear transformations of the first two conditional moments only. 3. Application: UK mortgage rates
1 Though disutility functions are generally not employed within economics, the exponential disutility function is commonly employed within other disciplines such as transport engineering; see, e.g., Cheu and Kreinovich (2007).
A UK 75% loan-to-value customer must choose between a variable and fixed rate mortgage. The variable rate is given by the Standard Variable Rate minus 1%, and the fixed rates have
1992
1996
2000
2004
2008
2012
Imperfection Cost 100 200 300 400 1992
1996
2000
Year
2004
2008
2008
2012
1996
2000
2004
2008
2012
1992
1996
2000
2008
2012
2012
2004
2008
2012
Imperfection Cost 100 200 300 400 1992
1996
2000
Year
2004
2008
2012
1992
1996
2000
Year
(g) Informed (24-month).
2008
0
Imperfection Cost 100 200 300 400 2004
2004
(f) Variable (60-month).
0
Imperfection Cost 100 200 300 400
2000
2012
Year
(e) Variable (36-month).
0
1996
2008
Imperfection Cost 100 200 300 400 1992
Year
(d) Variable (24-month).
2004
0
0 2004 Year
1992
2000
(c) Fixed (60-month).
Imperfection Cost 100 200 300 400
Imperfection Cost 100 200 300 400
2000
1996
Year
(b) Fixed (36-month).
0
1996
1992
2012
Year
(a) Fixed (24-month).
1992
617
0
Imperfection Cost 100 200 300 400 0
0
Imperfection Cost 100 200 300 400
N. Taylor / Economics Letters 117 (2012) 615–618
Year
(h) Informed (36-month).
(i) Informed (60-month).
Fig. 1. Imperfection costs. This figure contains the (annualised) imperfection costs (in bps) to the fixed rate only, variable rate only, and informed advice strategies. Table 1 Performance fees. Comparison
Risk aversion parameter (γ1 ) 0
1
2
3
4
34.149* [0.044] 4.549 [0.743]
26.019 [0.090] 17.613 [0.197]
18.957 [0.171] 31.748* [0.050]
11.478 [0.357] 44.319* [0.024]
7.085 [0.545] 56.359** [0.006]
56.719** [0.002] −3.682 [0.725]
34.220* [0.020] 6.099 [0.630]
21.422 [0.077] 25.554 [0.204]
11.214 [0.242] 44.995 [0.132]
0.448 [0.961] 57.130 [0.096]
76.242* [0.015] −5.767 [0.273]
41.349 [0.209] 2.559 [0.709]
19.225 [0.505] 23.618 [0.235]
3.994 [0.854] 48.374 [0.112]
−3.954 [0.854] 71.259 [0.110]
Panel A: 24-month horizon Informed vs. fixed Informed vs. variable Panel B: 36-month horizon Informed vs. fixed Informed vs. variable Panel C: 60-month horizon Informed vs. fixed Informed vs. variable
Notes: This table contains the mean (annualised) performance fees (in bps) to avoid switching from the informed advice strategy to the fixed and variable rate only strategies. * P-values associated with the null hypothesis that the performance fees equal zero are given in brackets, with rejections at the 5% level. ** P-values associated with the null hypothesis that the performance fees equal zero are given in brackets, with rejections at the 1% level.
horizons of 24, 36, and 60 months.2 While the fixed rate ‘dynamics’ over these horizons are known to all advisers with absolute certainty, only one adviser type is assumed to have knowledge of the variable rate dynamics (henceforth the informed adviser): specifically, conditional moments based on Chan et al.’s (1992) unrestricted stochastic differential equation.3 For this adviser, these moments are used as inputs into the calculation of the
2 Monthly data were obtained from the Bank of England’s Statistical Interactive Database (http://www.bankofengland.co.uk) and cover the period January 1995 to December 2011. 3 Conditional moments are calculated as follows. First, equation parameters are obtained by GMM estimation using the full sample of variable rates. Second, at each point within the sample, 100,000 sample paths of the process are generated over the
advice vector and the (annualised) performance fee, where γ0 = 0, γ1 = {0, 1, 2, 3, 4}, and N = 10. The competing advice assumes borrowing exclusively at the variable (and fixed) rate only. The imperfection costs in Fig. 1 exhibit considerable time variation, with informed advice delivering lower imperfection costs across all risk aversion levels. To formally test this observation, we exploit the time variation in fees and examine the null hypothesis that mean performance fees (that is, differences in mean imperfection costs across strategies) equal zero using a Wald test based on the Newey–West HAC estimator. The results
fixed rate horizon period. Third, the simulation averages of the time-series means and variances of these paths are used as the conditional moment measures.
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in Table 1 show that informed advice generally delivers positive performance fees, with a range of −5.8 bps to 76.2 bps. For extreme levels of risk aversion these fees are significantly above zero (though this result weakens as the horizon increases). However, for a particular risk aversion level and horizon, there are no occasions when informed advice delivers significant fees with respect to both types of competing advice—a result that implies that informed advice delivers limited benefits in this context. 4. Conclusions Time variation in the proposed measure enables use of a simple statistical test such that one can examine the null hypothesis that advice strategies have equal value. Applying the methodology to UK mortgage rates suggests that the costs of following fixed only and variable only strategies can be high (particularly during certain periods). By contrast, the informed strategy delivers
lower imperfection costs, though the statistical significance of relative performance is restricted to customers with particular risk preferences and horizons. The challenge of future research is to demonstrate that loan advice has value under all conditions. References Bonaccio, S., Dalal, R., 2006. Advice taking and decision-making: an integrative literature review, and implications for the organizational sciences. Organizational Behaviour and Human Decision Processes 101, 127–151. Chan, K., Karolyi, A., Longstaff, F., Sanders, A., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227. Cheu, R., Kreinovich, V., 2007. Exponential disutility functions in transportation problems: a new theoretical justification. Technical Report UTEP-CS-07-04. Jegadeesh, N., Kim, J., Krische, S., Lee, C., 2004. Analyzing the analysts: when do recommendations add value? Journal of Finance 59, 1083–1124. Rapach, D., Strauss, J., Zhou, G., 2010. Out-of-sample equity premium prediction: combination forecasts and links to the real economy. Review of Financial Studies 23, 821–862.