Trailing Stop-Loss and Re-Entry Strategies in Europe

A stop-loss order is a method that can be used by investors to limit downside risk and can be explained by investors’ loss aversion. Loss aversion refers to the widely studied psychological phenomenon where expected losses have a greater impact on the investors’ preferences than expected gains. The use of stop-loss rules allows for this asymmetric profile. The stop loss hypothesis, also, states that higher returns can be obtained by limiting the downside risk of long positions. A stop-loss order can be established at a percentage of the asset value that grows over time as it rises, stopping if the asset value starts to go down (trailing stop-loss order). In this work, it is used trailing stop-loss orders (with a difference to the asset market value of 3%) with long positions in financial markets. After being stopped out in a down market, the re-entry rule is to buy the asset again as soon as the market grows by at least 3%. Whenever the portfolio is sold out by applying the trailing stop-loss rule, the investment is held in cash until the trigger is set off. In this article the rules are applied in Germany, France, Spain, and Greece during the financial crisis, to make an empirical validation of trailing stop-loss rules and re-entry rules for these markets. It was found that the followed strategy presents mixed results, proving to beat the buy-and-hold strategy in some scenarios of rising and falling prices. But the most important result is to generate fewer losses in very volatile scenarios such as Greece in 2014.

I. INTRODUCTION
Technical trading rules are commonly used as a method to generate buying/selling signals in financial markets (Taylor, 2014). There is a large set of technical trading rules, some of them based on quite simple procedures and others based on sophisticated statistical and numerical methodologies.
The performance of technical trading rules can vary over time and over market environments, which is consistent with the adaptive market hypothesis (Taylor, 2014;Lo, 2004). The main idea behind this belief is that the same trading rule in different environments can lead to opposite results (Lo, 2004, Kim et al., 2011Taylor, 2014;Neely et al., 2014). Hence, the performance of technical trading rules can only be empirically validated in some particular markets or particular market environments.
During the past years, Europe has been hit by an unprecedented financial crisis. However, the crisis was not reflected equally in all countries. Typically, the northern countries were less affected (for instance Germany and France) than the countries in Southern Europe (such as Spain and Greece) (Islam, 2016;Gros, et al., 2014). So, I decided to empirically test a set of technical trading rules in the crisis environment, but in these two types of financial markets with opposite behaviors.
In this work, I use different technical trading rules to generate buying and selling signals. For the selling signal, the main idea is to study the use of trailing stop-loss orders. The buying signal is, mostly, a re-entry strategy (after the first entrance). Accordingly, after being stop-out by the market, it's needed to establish a rule that will enable a reentry into the market. The rules used to generate the buy/sell signals were chosen because they are likely to promote a successful investment strategy during a financial crisis environment.
In this article, I apply trailing stop-loss rules and re-entry strategies in some European markets during the financial crisis, namely Germany, France, Spain, and Greece. These strategies were employed in a one-year investment period between 2012 and 2015, to make an empirical validation of trailing stop-loss rules and re-entry strategies in two of the least affected and two of the most affected European countries.
In section II, I make a brief presentation of the theory behind the stop-loss hypothesis, and re-entry strategies. In section III it presents the data and methodology, and finally, in section IV, I show the results, which suggest that trailing stop-loss rules should be considered in investment strategies applied to these European financial markets. Section V concludes.

II. LITERATURE OVERVIEW
A stop-loss order is a method that can be used by investors to limit their downside risk. The use of stop-loss orders is explained by investors' loss aversion (Montier, 2007;Zhu & Zhang, 2010;Tooth, 2014). Loss aversion refers to the widely studied psychological phenomenon which implies that losses have a more significant impact on investors' preferences than gains. Risk aversion is mainly caused by loss aversion (Zhu & Zhang, 2010). Therefore, investors have asymmetric preferences on risk: they would like to limit the loss while maintaining the potential return. The use of stop-loss rules allows for this asymmetric profile.
The research on the efficacy of stop-loss strategies is scarce and mostly focused on U.S. stock market (Klement, 2013). Table I summarize the principal theoretical findings on this  subject.   TABLE I: PRINCIPAL THEORETICAL FINDINGS  Study  Conclusions Lei & Li (2009) The importance of stop-loss strategies comes from risk reduction and not from return enhancement. Clare et al. (2013) Simple stop-loss strategies do not add value to a portfolio.
Klement (2013) Trailing stop-loss strategies can enhance absolute and risk-adjusted returns.
Tooth (2014) Stop-loss rules on maxmeasures portfolio selection strategy do not add value to a portfolio.
Kaminski & Lo (2014) Simple stop-loss strategies lead to low returns, but, if it used in longer sampling frequencies, the returns will be enhanced. Source: Researcher, 2022.
According to Lei & Li (2009), the stop-loss rules help investors to reduce risk by automatically selling losing assets. They conclude that those rules allow investors to reduce investment risk although not contribute to an increased return. Clare et al. (2013) compared some technical trading rules within the S&P500 index, giving special attention to moving average (MA) and stop-loss trading rules, concluding that simple stop-loss strategies do not add value to a portfolio.
By extending previous works on stop-loss rules by adding additional rules for re-entering the market, Klement (2013, pp. 44) states that "the value of stop-losses can be assessed only when re-entry rules are considered jointly with stop-loss rules". By analyzing the joint effect of stop-loss and re-entry rules, the author found that stop-loss rules reduce volatility and excessive losses, besides the evidence on return improvement is mixed.
Tooth (2014) tested whether stop-loss rules could limit downside risk and improve long-term portfolio growth. By applying stop-losses to the maxmeasures portfolio selection strategy, and bootstrapped backtest of arbitrary strategies on the S&P500 between 1969 and 2012, the author concluded that the use of such rules does not add value to a portfolio. Kaminski & Lo (2014) proposed a simple analytical framework to evaluate the add-up value by stop-loss rules on the expected return and volatility of a portfolio. Focusing on daily futures price data, they provided empirical evidence that, with longer sampling frequencies, some stop-loss policies boost the expected return while reducing volatility.
The studies presented above, despite having some contradictory results, allow considering that the simultaneous use of stop-loss and re-entry rules could lead to both, risk reduction and return enhancement.
A stop-loss order is an attempt to shield the investment strategy, namely by not allowing the investment value to fall below a specific value specified in advance. The stop-loss order can protect a position previously taken into the market, with a fixed floor (simple stop-loss order).
In a spot market, in the event of a bear market, investors will not take long positions, so, stop-loss orders are mostly used when the direction of the market is unclear (Tooth, 2014;Shefrin & Statman, 1985), like during a crisis. The determination of the floor can vary but usually depends on the investors' ability to accept losses. Typically, the stop is established with two factors taken into consideration: the amount of money that will be lost if the stop is achieved and the asset spot price volatility (Teweles et al., 1987;Tooth, 2014). Tooth (2014) considered that a 7-8% stop-loss has become largely accepted in financial markets, after the seminal work of O'Neil (1988).
The simple stop-loss order settles on a straightforward proposition: a floor (F) is established, usually at some level below the purchase price, which is the minimum value the portfolio can take. The initial investment is fully applied to the risky asset. The floor value is compared with the portfolio value, , over time. In this comparison, two different situations may happen: 1. > the application in the stock remains unchanged. 2. > the stock is immediately sold out.
Commonly the floor is settled at a level below the purchase price, for long positions taken on the risky asset. The primary objective is to avoid significant losses by establishing a previous loss limit based on the initial invested capital (simple stop-loss) The major issue surrounding simple stoploss orders is that it is impossible to obtain a profit with its use unless the market grows overall the investment periods.
When an investor is loss averse, his/her anxiety overgrows in the event of a negative return. Sometimes the investor can sell the position too soon, resulting in a loss. This early selling position motivated the loss aversion, implying that the investor will not benefit from a recovery of the market, after their short-term loss. If one such investor uses trailing stoploss orders instead, he/she can benefit from an increasing market, having the added "emotional comfort" benefit of knowing that the investment has a floor below which it will not go down (Klement, 2013).
Additionally, the protection can be held with a moving floor, usually, a percentage of the asset value that grows over time as the asset rises, stopping at the maximum price level when it starts to drop down (trailing stop-loss order). The order will allow the floor to increase over time as the market grows, enabling profit if the growth lasts for some (but not necessarily all) of the investment period. Once again, the percentage is established according to the investor's capacity of accepting losses.
In this work, stop-loss orders with long positions in financial markets will be used. The goal is to apply the methodology in spot markets which, often, do not allow taking short sell positions. Thus, only long market positions will be taken. The stop-loss hypothesis states that by limiting the downside of long positions, higher returns will be achieved (Tooth, 2014;O'Neil, 1988). Hence, these orders are particularly relevant in a crisis environment and should be more effective in countries with financial problems (like Spain and Greece) over richer countries (such as France and Germany).
As stated by Klement (2013) most of the previous work on stop-loss trading rules did not allow re-entry into the market. By avoiding a re-entry, the profit of such a strategy is severely restricted as it is not possible to profit from later recoveries on the market. That is why it is so important to have a rule that enables a re-entry into the market. In a downturn of the market, the asset is immediately sold, but with the re-entry rule it can be rebought in a recovery.
As has been done by Klement (2013), I decided to apply a similar approach to test trailing stop-loss and re-entry rules simultaneously. The methodology employs a procedure based on cumulative past returns, the sell (trailing stop-loss) and buy (re-entry) signals are: where is the cumulative return on the asset, since the last portfolio recomposition, γ is the maximum percentage amount that the cumulative return can go down, and δ is the minimum percentage the market must recover before reentering the market. After being stopped out in a down market, the re-entry rule that was applied is straightforward: to rebuy the asset as soon as the market grows by, at least, δ. This re-entry rule was selected because it will permit gains even if the market does not grow much over the sample period. Given that the sample period goes between 2012-2015, during a financial crisis, the existence of significant rises in prices is not expected to happen.

III. III. DATA AND METHODOLOGY
To simulate the investment portfolio strategies, I used some European markets, namely, Germany and France (two of the countries least affected by the financial crisis in Europe) and Spain and Greece (two of the countries most affected by the financial crisis in Europe) on the sample period of 2012-2015.
In this simulation stock indexes were used instead of stocks, meaning that I am testing the market (market risk) and not a specific stock (with specific risk). Thus, I focused on well-known indexes of Germany (Dax 30), France (CAC 40), Spain (IBEX 35), and Greece (ASE).
Daily returns were calculated over a one-year holding period. Whenever the portfolio was sold out by the application of the stop-loss rule, the investment was held in cash until the trigger was set off and the asset was repurchased as follows: i) The position is stopped out by the trailing stop-loss order, at a 3% level; Sell signal.
ii) After being stopped out if the markets grow at least 3% a re-entry on the market is purchased. Buy signal.
Our methodology employs a procedure in which the investor updates their portfolio, after the initial investment, only if: i) the position is stopped out by the trailing stop-loss order. ii)after being stopped out, the markets grow at least 3%. The choice of 3% (a much lower percentage than the typical 7-8% accepted in this type of work - Teweles et al., 1987;Tooth, 2014) concerns the fact that Europe is facing an unprecedented financial crisis, which justifies lowering the accepted risk limit. The use of 3% for all markets, contradicts previous studies that advocate that the rules depend, in addition to investor risk aversion, on the level of market volatility (Teweles et al., 1987;Tooth, 2014). However, this choice was made to improve the comparability. That means that the rules are not flexible or adjusted to the performance of the market which can lead to poor performance. However, this approach will guarantee a methodology that can easily be used by any investor as it does not require advanced technical knowledge.
Transaction costs, in financial markets, are expenses incurred when buying or selling an asset. Trading costs might be so high that strategies using high-frequency trading rules, such as stop-loss orders, can prove unprofitable (Keim & Madhavan, 1995). Thus, the inclusion of transaction costs in this research is essential to establish the consistency of the results. Transaction costs include commissions, exchange, and other fees, bid/ask spreads, market impact costs, and taxes. They are usually segmented into direct and indirect costs. By direct costs, commissions, exchange, and other fees are referred to, while the remaining are indirect costs. Since the latter is difficult to quantify and there is no consensus in the literature on how to determine them, they are not often considered.
"Despite the increasingly prominent role of transaction costs in both practice and research, estimates of transaction costs are not always available or, where available, subject to considerable expense and error." (Lesmond et al., 1999(Lesmond et al., , pp. 1114. Therefore, the quantification of trading costs is critical for consideration in this study and, by consequence, to the reliability of my findings. Moreover, the impact of transaction costs depends on the asset involved in the transaction. I have therefore established to use the transaction costs charged by one online broker, ActivoBank, as it is easily reachable on the internet and because it works with the major stock exchanges in the world. The transaction costs that the broker charges can be found in Table II. The use of these transaction costs means that it was not considered indirect costs like several other researchers, such as Fifield et al. (2005), for instance.
The hypotheses to be tested are: Hypothesis 1) the use of trailing stop-loss orders with a reentry strategy will lead to portfolio returns that will exceed the returns of a buy and hold strategy on the same asset, in a rising market.
Hypothesis 2) the use of trailing stop-loss orders with a reentry strategy will lead to portfolio returns that will exceed the returns of a buy and hold strategy on the same asset, in a drawdown market.
Hypothesis 3) the use of trailing stop-loss orders and reentry strategy will lead to higher portfolio returns in the financial markets of the countries less affected by the crisis.

IV. RESULTS
To be able to compare the markets understudy, a summary of the descriptive statistics of the annual stock indexes is presented in Table III. Although Spain and Greece were more affected by the financial crisis than Germany and France, it appears that the market that diverges the most, from the average behavior, is the Greek. The latter presents the highest and lowest annual growth rates in all periods under analysis.
Despite the drop felt in all the examined markets, in 2014, the most stable market was the German, followed by the French. The analysis of the maximum, minimum, and average values, as well as the standard deviation, of the stock indexes, confirms that Germany and France markets are the ones with the greatest stability and the smallest annual price fluctuation.
Even though the Spanish market presents a positive difference in its behavior compared to the Greek, it still presents a marked difference in terms of price stability and growth compared to the German and French markets.
The results of the application of the defined strategy, for the buying and selling rules in each of the markets, are presented in Tables IV, V, VI, and VII, for Germany, France, Spain, and Greece, respectively.
As the goal was to test the stop-loss rule empirically, it was necessary to establish whether stop-losses were indeed effective at limiting downside risk. Thus, the tables present the number of holding days, the number of days in and out of the market, and an evaluation of the return achieved in each trade (buying and selling).    For each market, the periods in which the index was in the portfolio (rising market) are presented, subdivided into periods with gains and losses. In this regard, the periods in which the index was in the portfolio are more in the markets of Southern Europe than of Northern Europe. However, when compared, the average number of days on the market is substantially lower. This fact suggests that the multiple entries and exits in southern markets are driven by price volatility, and do not necessarily represent gains. This fact is confirmed by the analysis of holding periods with gains and losses. Indeed, the periods with gains are higher in the German and French markets. Also, the number of days out of the market confirms that the entries and exits in Spain and Greece do not offset the days that the index is in the portfolio.
The comparison of the rate of return, maximum, minimum, and average values of each transaction, show very volatile results in the two Southern European markets.
However, in 2014, which was a particularly bad year for all of the studied countries, in Southern European countries, the lowest number of market entries (and consequently the greater number of days out of the market) results in lower losses. These results, point out to the conclusion that the value of stop-loss strategies may come mainly from risk reduction rather than return achievement.
For a better performance comparison, Table VIII presents the portfolio return and the Buy-and-Hold (B/H) strategy return. This comparison can demonstrate if this strategy can beat the market. The volatility and Sharpe's ratio are also displayed in the table. This ratio (S) is one of the most widely used risk measuring tools. Through this method one can compute the excess portfolio return received per unit of volatility using the following equation: where represents the return on the portfolio, the riskfree rate and the standard deviation of the return on the portfolio. An increase in number will bring an increase in return per unit of risk. The strategy presented in this article only presented better results than the buy-and-hold strategy in 2015, and for all markets under analysis. However, the positive difference is more marked in the Northern European markets. These results lead to the conclusion that the strategy used here can beat the market in periods of growth (Germany and France) and in periods of decrease (Spain and Greece).
A note should be made for the year of 2014 and the results obtained in the Greek market. For this year the results of the current strategy also beat the buy-and-hold strategy. It should be noted that it was a year in which all the four markets performed poorly, so the result obtained mainly derives from the implementation of the strategy of maintaining the investment in cash for a substantial part of the year. This result confirms the strategy's ability to avoid losses.
Through Sharpe's ratio, one can compute the expected return adjusted by the level of risk. The portfolio returns, adjusted for the risk, in the countries less affected by the crisis, still show better performances than the ones of the countries more affected.

V. CONCLUDING REMARKS
Hypothesis 1) the use of trailing stop-loss orders and reentry strategy lead to portfolio returns higher than the buyand-hold strategy on the same asset, in a rising market.
Hypothesis 2) the use of trailing stop-loss orders and reentry strategy lead to portfolio returns that exceed the returns of a buy-and-hold strategy on the same asset, in a drawdown market.
Hypothesis 3) the use of trailing stop-loss orders and reentry strategy lead to higher portfolio returns in the financial markets of the countries less affected by the crisis.
The use of trailing stop-loss orders and re-entry strategy were effective at limiting downside risk, and the re-entry rule was also effective to gain upside potential, but that did not occur in all scenarios. This conclusion can be achieved by the mixed results found in different market conditions. Through Sharpe's ratio also allow concluding that portfolio returns, in the countries less affected by the crisis, still show better performances than the ones of the countries more affected.
It was found that the followed strategy presents mixed results, proving to beat buy-and-hold strategy in some scenarios of rising and falling prices. But the most important result is to generate fewer losses in very volatile scenarios such as Greece in 2014.


Private Saving in Ghana: The Combined Efforts of Financial Development, Interest Rates, and Inflation

Understanding private saving behaviour of the citizenry is crucial for informed policy decisions in an economy. The reason is not far-fetched; economies gain sustained growth from investment through saving. However, it does appear much of the available literature is on aggregate private savings rather than on private saving behaviour. This study updates the literature by looking at the combined effects of financial development, interest rate and inflation on private saving behaviour among Ghanaians. Data for the study were obtained from the World Bank Development Indicators between 1980 and 2019. We employed Johansen Cointegration Test; attempts are made to ascertain the existence of a long-run relationship among variables using the Vector Autoregressive (VAR) Model. The study confirms a significant positive relationship between private saving behaviour and financial sector development. This partly explains the relevance of deepening the financial sector through reductions in costs of performing transactions and initiating contracts to encourage private saving through improved propensity to save by the old and to rake in new entrants. A reliance on macroeconomic variables to forecast the behaviour of private saving enjoins policy decision makers to consider the implications of their decisions for private saving. Among the recommendations are, lower borrowing costs across the economy, resulting in increased investment and consumption spending, and hence economic recovery only in times of stagnation.

INTRODUCTION
Saving has enjoyed prominence in both theoretical and empirical analyses of macroeconomics and policy design across globe (Le Blanc, Porpiglia, Teppa, Zhu & Ziegelmeyer, 2016;Bonfinger & Ries, 2017;Kapounek, Korab & Deltuvaite, 2016;Thu, 2018). This in part is due to its ability to accumulate capital for investment and economic growth, and also the elusiveness of its precise measurement. And so, this paper describes saving as an essential feature in every economy that needs a considerable policy attention. Households' saving behaviour is tainted mostly with irrationality and needs various levels of engagements whether in the academia, policymaker circles or particularly at national governments' deliberations. It is usual in most situations to want to believe that global economic wellbeing is a function of the interplay of several economic units, of which household saving behaviour ranks favourably but independent of the reasons for their respective behaviour. Thus, aggregate savings is regardless of an individual's or household's rational behaviour; instead, what matters here is the saving behaviour motivators; what informs the household's intent to saving or dissave? Note that private or household saving is used in the sense of domestic saving to evoke explanations for its fluctuating behaviour (see Figure  1).
Interestingly, 'Saving' and Savings' have had a loose application among economists. Whereas 'Saving' is a flow concept, 'Savings' is a stock item at a given time. Thus, this paper will frequently refer saving as a flow concept whose behaviour is shrouded in ambiguities.

Statement of the Problem
Although there have been extensive studies on saving, the linkage has mostly been to micro-indicators at the blindside of a macro-linkage and has therefore left a literature gap to fill. The need for this current study is threefold; to show that household saving gravitate towards aggregate saving at the macroeconomic level. This is because the financial sector flourish on positive saving. Saving is a function of financial development, described as improving the quality and quantity of financial services with a comparatively low cost of doing business. This low cost of doing business has influenced the saving behaviour of the average depositor. Güngör, Çiftçioğlu, and Balcılar (2014) posit that financial development has a positive composite index influence on private saving behaviour of the household. Financial development is also regarded as a deliberate macroeconomic policy that focuses on accumulating liquid assets, acquiring investment information, and strategically allocating resources, leading to a positive saving behaviour (Levine, 1997;Paun, Musetescu, Topan, & Danuletiu, 2019). Altaee and Al-Jafari (2019) reveal that financial development does not just propel domestic savings in the short-run but has a long-lasting significant positive effect on the savings culture of the household. There is therefore the need to bring to the fore the relevance of financial sector development in mobilising savings through low contractual costs and affordable business fees. Second, the household saving behaviour is divergently positioned. de Melo and Tybout (1986) and Hondroyiannis Lolos and Papapetrou (2005) hold contrary views on the positive saving-financial development relationship. They appear to agree with the earlier position of Griffin (1972) who laid bare the elusiveness and unpredictability of private saving. Regardless of the deepening level of the financial sector, the average household saving behaviour is not always linear. These unsettled academic brawls need an examination. Although negative (low) interest rates regime is preferred in times of recession, saving is traditionally known to inversely relate with interest rates. High deposit interest rates motivate households to want to increase deposits which could be used for investment purposes for economic growth. Paleyo-Romero (2020) found that low (negative) interest rates regime has both a contractionary effect on saving (substitution effect) and an expansionary effect (income effect). While the substitution effect lowers the reward on private saving, the zeal to maintain financial income causes an expansion in private deposits. And finally, this study also became inevitable due to the need to show that national policy on inflation could improve the saving magnitude. Thus, consumers are biased in their expectations because the individual does not expect equal changes in incomes as in price levels. Cited empirical studies associate a positive link between private saving and unanticipated inflation (Juster & Wachtel, 1972). However, Dash and Kumar (2018), Gashe (2017) showed contrary results from their studies. These findings are inconclusive and therefore need a further verification.
This current study rides on the backdrop of inadequate empirical studies pointing to outcomes of different macroeconomic policy decisions on private saving in Ghana, except those on aggregate savings tainted with irrationality of the household. If ever there are studies, policy decisions on the joint role of financial development, interest rate, and inflation in determining private saving behaviour have not had a deserving recognition in the empirical literature. Some authorities in economics outside Ghana have effortfully linked household saving behaviour to psychological, economic, and demographic factors (see Crespo-Cuaresma, Fidrmuc & Hake, 2014;Fidrmuc, Hake & Stix, 2013;in Kapounek et al., 2016;Basabose, 2020).

Hypothesis Development
There is a form of symbiosis between household saving and financial development. This paper therefore hypothesises that financial sector development affects private saving positively. This is because, a well-functioning financial market is adequate grounds for encouraged saving. Likewise, improved household saving strengthens the activities of financial markets and therefore there is need to want to know the weight and direction of the relationship between saving and financial development through hypothesis building. Even with real interest rates, we do know that a higher propensity to save eventually lowers interest rates which again is linked to demographic factors (see Bonfinger & Ries, 2017). This study fills a literature gap by effortfully identifying the saving-interest rates nexus. Are positive or negative interest rates panacea for improved saving? We hypothesise that there is a direct relationship between saving and positive interest rates. The economy flourishes in a stable financial system and interest rates regime. These two variables working for improved saving will grow the economy as expected. There is a hypothesised positive link between saving and inflation. Mathematically, we have the following hypotheses: H 0 : Financial sector development has a positive effect on private saving H 1 : Financial sector development has no effect on private saving H 0 : Interest rates have a positive effect on private saving H 1 : Interest rates have no effect on private saving H 0 : Inflation has no effect on private saving H 1 : Inflation has a positive effect on private saving

The Trend of Saving in Ghana
In the period 2006 to 2011, Ghana registered severe instability of the private saving trend. Thus, from a five-year average of 6.54 (2001)(2002)(2003)(2004)(2005) to an average decline rate of 1.78 between 2006 and 2010 (see Figure 1). This severe instability could have partly been attributed to the global economic crunch of 2007/2008 with its spill-over effects on the global economy. Despite these developments, private savings rose appreciably in 2012 and witnessed a seven-year astronomical rise until 2019, with minimal deviations (see Figure 1). Figure 1 : Trend of Savings in Ghana (1981Ghana ( -2019

THEORETICAL REVIEW The Lifecycle Model of Modigliani
Most of the theories on household saving have been replaced for irrelevance and others improved with the passage of time.
One of such theories that have undergone metamorphosis is the Life-Cycle Model of Modigliani (1949;Ando & Modigliani,1963;Modigliani & Brumberg, 1990). This theory sees saving as an inevitable and unpredictable variable that enhances productivity and economic growth. The early classical economic theories on individual saving behaviour before the 1930s linked savings and investment linearly to deposit interest rate (see Caminati, 1981 andSmyth, 1993). However, a print controversy rose between Keynes' (1936) general theory of savings and investment (cited in Richert-Kazmierska, 2019) and the classical economic thought over saving-investment equality. Keynes's position on saving is dichotomous; at one level, Keynes sees saving as being equal to investment on their levels. In a different instance, Keynes thinks that only when the economy exhibits equilibrium characteristics will the equality of saving-investment be accepted, replacing the early classical economic thought. This double approach to saving-investment behaviour triggered debates among economists and has led to the reliance on the life-cycle theory of consumption and savings by Modigliani (1949;cited in Ando & Modigliani, 1963). The life-cycle theory of savings argues that the willingness of a person to save surges only with an improvement in income. The life-cycle theory affirms the linear saving-individual age relationship. Thus, the demographic structure of the individual is a dominant determinant of a potential depositor's behaviour. These powerful raging arguments on what determines saving propensities in an economy have partly mooted the idea to undertake a study verifying the grounds on which these entrenched positions are taken. These academic interactions have not stopped at only the theoretical concerns but have extended to include several cited empirical studies on the various roles played by certain macroeconomic variables to either boost or discourage a person's willingness to save (e.g., Griffin, 1972 andJuster &Wachtel, 1972;Larbi, 2013).
The compelling fact is even that, a reliance on macroeconomic variables to forecast household saving behaviour will mean policy decision-makers will have to consider the implications of their actions for household saving-consumption reactions.
Since the saving-consumption hypothesis is transitionary, policies should reflect every stage of the transition. The lifecycle theory underpins this study in several facets; it response to questions on how households maintain the same level of consumption as far as human lives are concern, reminiscent of certain factors, and in this study; financial development, interest rates and inflation, which are the exogenous variables presumed to work for or against household saving. In a sense, Keynes (1936) first saw saving as a good that households seek to expend their scarce financial resources on; deferring consumption at the initial stages of life. And in doing so, saving is regarded as a function of national policies on the variables that are studied in this research paper. The facts about the lifecycle theory in this paper are stylized. First, we look at how saving becomes a function of financial market (financial system) development. Financial sector development describes conditions under which clients do business fairly and at comparatively low cost. Entering into a contractual relationship to transact financial business should not threaten potential profits, part of which is slated for saving. This encourages potential customers of financial institutions to improve saving by increasing old accounts till and opening new ones. Consumption must be prudently discouraged to positively affect saving since the life-cycle model professes the maintenance of a uniform level of consumption throughout lifetime. How is this done in order not to venture or invite recessionary tendencies? Deposit interest rates are increased optimally. Inflation, particularly the hyper type, works against the lifecycle model since it deteriorates the real purchasing power. This enjoins national governments to implement policies that contain inflation to boost both saving and consumption.

Savings-financial sector development nexus
Domestic saving is not done for its sake; it is intended for the individual and national wellbeing (Bayar, 2014). At the individual level, it minimizes financial stress, helps leave financial legacy, and provides a greater sense of financial freedom. Undisputedly, national saving is partly a function of economic growth (Group Deputies, 1995). Pant and Kindness (2021) note that saving does not imply the absence of spending but a deliberate setting aside of part of one's income for an intended purpose in the foreseeable future. Indeed, the financial sector of every economy is responsible for this fundamental role of saving. Financial development focuses on the process that allows for low cost of transacting business in the financial sector to enable potential customers to deposit their surplus funds to earn interest. As one of the cardinals of this study, the researchers attempt to establish the link between saving and financial development in Ghana. Numerous studies have been undertaken to confirm the presence of a link between private Saving (PS) and Financial Sector Development (FSD). Bayar (2014) studied emerging Asian Countries and reports that PS and FD are positively linked. In this study, Bayar appears to have supported results produced by the studies of Sahoo and Dash (2013), Horioka and Yin (2010).
The relevance of financial sector restructuring to domestic saving was established by Newlove (2008) in one of the pioneer studies in Ghana. Quartey (2008) equally emphasized the need to deepen financial sector development to boost domestic savings. Recent, Altaee and Al-Jafari (2019) agree with the results of these strands of studies in their revelation that financial development does not just propel domestic savings in Turkey in the short-run but has a long-lasting significant positive effect on the savings culture of the Turkish people. In Nigeria, Ewetan, Ike, and Urhie (2015) provided yet another evidence of a long-run relationship between financial sector development and domestic saving. On the contrary, Horioka and Yin (2010;Ewetan et al., 2015) studied household saving behaviour in OECD countries and its relationship with FD (computed by private credit to GDP) using Modigliani's (1949) life-cycle theory. He finds that FD is key to increasing savings in the environment of moderate interest rate, stable prices, and regularized income.

Saving-interest rates relationship
As benchmark, the Central bank uses the Policy Rate (PR) to signal the cost of credit (Kwakye, 2010). This is reflected in the financial system through inter-bank transactions on one side and bank-customer transaction relationship on the other. The old saving-interest rate model is a linear relationship (see Caminati, 1981;Smyth, 1993). One pioneer article in Uruguay on real interest rate-saving relationship is that of de Melo and Tybout (1986;cited in Balassa, 1989). Their results exhibited a positive relationship, albeit weak. The regression estimates showed a t-value of about 1.5. Another study within the same period by McDonald (1983)  The elites in the insurance sector have also waded into the empirics of interest rate-saving behaviour among households. One such authority is Patricia Paleyo-Romero. In a study of household saving behaviour in sixteen European Union countries, Paleyo-Romero (2020) used balanced panel data spanning 2008 to 2018, to analyse a cross-sectional saving behaviour of households. The study brought forth two interesting revelations; low (negative) interest regime has both a contractionary effect on saving (substitution effect) and an expansionary effect (income effect). While the substitution effect lowers the reward on private saving, the zeal to maintain financial income causes an expansion in private deposits (Paleyo-Romero, 2020). Stakeholders in the financial sector appreciate the fact that interest rates are among levers used to maintain balance in every economy, as such, central banks adjust the prime rates to influence individual bank lending decisions. Following these developments, Smith (2021) thinks that negative interest rates boost the economy by encouraging consumption and enabling banks to borrow and lend to the public. Smith (2021) further states that negative interest rates fight deflation and save the economy from recession. Low interest rates, Smith emphasizes, make it expensive to hold more money, thereby incentivizing spending. Reasoning in the same direction, Hughes (2016) theoretically advanced an argument that negative interest rates stimulate the economy. This is done by the central bank through increased lending to commercial banks at lower rates during recession.

The link between saving and inflation
Depending on whether it is an anticipated or unanticipated inflation, its effect on private saving could either be symmetrical or asymmetrical (Juster & Wachtel,1972). Thus, consumers are biased in their expectations because the individual does not expect equal changes in incomes as in price levels. Thus, empirical studies associate a positive link between private saving and unanticipated inflation (Juster & Wachtel, 1972). Campbell and Lovati (1979) associate price level increase with improvement in savings. This is so because households usually respond to increased prices through cut-back in borrowing and spending, thereby saving. In a study by Santosh and Lakshmi (2018), both domestic and private saving were negatively affected by inflation.
Source: researchers' own construct (2022) Figure 2 : Conceptual Framework  Table 4). Next in line was to establish the long-run relationship among variables using the Johansen (1991) Cointegration Estimation Model (see Table  4). There was need to meet a necessary underlying condition before all these tests were carried out, making sure issues of spurious or useless regression did not arise. E-views 9.0 software was used to perform the tests for time series properties .

RESULTS AND DISCUSSION
This study assessed the combined effect of financial development, interest rate, and inflation on private saving behaviour in Ghana. It was done by considering Private Saving behaviour (PS) as the variable of interest.

Source: Researchers' computation based on the world Bank development indicators (2020)
Domestic saving (DS) proxies gross domestic saving, measured as a percentage of gross domestic product. Averagely, DS has a figure of about 7%, which is way below the Sub-Saharan African average of 21.102 in its 2019 estimates (World Bank Development Indicator, 2020). There is a corresponding standard deviation of about 6% depicting the variability from the expected normal average saving of a country. While the Maximum and Minimum values of DS stood at 22.09% and -2.96%, respectively, the Lagrange test (Jacque-Bera) probability values for DS and FSD are normally distributed since they are more than 0.05. Those of INF and IR are not normally distributed since their respective Jacque-Bera probability values are less than the alpha value of 0.05. It is observed from Table 2 that data is normally distributed for domestic saving and financial sector development. With respect to the inflation rate, a higher standard deviation of 25.7734 against its mean of 28.257 is reported, which is not good enough. A higher standard deviation mostly turns to throw sway off results as not reflecting realities on the ground. When it occurs this way, it destabilizes the policy intentions of governments. Interest rate performed comparatively better, considering a mean of 16.759 as against 8.89 of its standard deviation. This describes the general relative interest stability in the economy, although not satisfactory. Note: ** denotes 5% significant level while *** denotes 1% significant level The correlation coefficient is used to determine the level and direction of the relationship between paired independent variables and or between paired dependent and independent variables in some instances. Whereas variables are said to correlate inversely when the sign of the coefficient is negative, there is always a direct relationship when the sign of the coefficient is positive. The absolute coefficient figure explains the magnitude of the relationship. A correl value of 0.8 or more between paired variables raises the issue of multicollinearity and should be fixed. The results in Table 3, have no collinearity challenges between paired explanatory variables. Again, the correl values are comparatively moderate, still confirming a case of the absence of multicollinearity issues (see Bryman & Cramer 1997). Excepting the case of INFL-R, where DS inversely relates with at 5 percent significant level, DS has positive relationships with FSD and IR at 5% and 1% respectively  (2022) The results in Table 4 indicate a non-stationarity at levels of all variables I (0). At first difference or Integration of order one [I (1)], all variables became stationary, satisfying a necessary condition for the estimation of cointegration and error correction models.

Determination of the optimal lag length
Another necessary condition before estimating the Johansen Cointegration Test is the determination of the optimal Lag Length. Table 5 provides results after running the test. Lag 1 under AIC is appropriate for the model analysis since AIC has the lowest value among the asterisked. What that means is that, if there are shocks or displacements in the system with respect to the regressors, the response variable will take a year to react to the shocks brought about by these explanatory variables (FD, IR & INFL-r). Unfortunately, this paper deals with only the short-run analysis since available data failed to establish any long-run relationship among variables, hence the use of the VAR Model. Table 6 provides details of the failure of the Trace Test to establish long-run characteristics.

Johansen Cointegration Test
As suggested by Granger (1969), if variables of interest are found to be cointegrated amongst themselves at first difference, then there is need to ascertain proper statistical inferences through the Vector Autoregression VAR Model and the Vector Error Correction Model (VECM). Unfortunately, after estimating the Johansen Cointegration Test, the system failed to establish the presence of cointegration equation among the variables. As such, the study opted for the VAR model (see Adeleye, 2018).  Bayar (2014), Horioka and Yin (2010), Ewetan et al (2015) including that of Park and Shin (2009). Placing emphasis on the relevance of financial sector development, Quartey (2008) and Al-Jafari (2019) explain that serious financial sector restructuring improves domestic saving. In terms of interest rate-private savings relationship, this paper reliably estimates an inverse one. In making calculations for the same purpose, the old saving model fails to support the results of this paper (Caminati, 1981 andSmyth, 1993). The results produced in this paper are further discounted by the estimates of De Melo and Tybout's study of 1986of , pp. 570, cited in Balassa (1989, as their findings exhibited a positive relationship although with weak t-static and prob values. The empirical sense of the results in this paper is that, as deposit interest rates shoot up, ceteris paribus, private saving worsens. Corroborating these results by inference, a study in Ghana by Osei-Fosu et al (2014) estimated long-run interest rate elasticity of household savings which showed a statistically negative value.
An interesting strong argument mounted in support of results in this paper, Paleyo-Romero (2020) proximates that low (negative) interest regime has both a contractionary effect on saving (substitution effect) and expansionary effect (income effect). She used a balanced panel data of sixteen European Union countries from 2008 to 2018 to analyse cross-sectional saving behaviours of households. She starts by explaining that stakeholders in the financial sector appreciate the fact that interest rates are among levers used to maintain balance in every economy, as such, central banks adjust the prime rates to influence individual bank lending decisions. There are other supporting results to the negative interest rates-saving behaviour. A very recent one is that of Smith (2021) who thinks that negative interest rate hypothesis boosts economic activities by lending at low rates to commercial banks to give out loans to the public to encourage consumption and investment during recession. Thus, negative interest rates are antidotes to deflation.
Although this paper finds inflation to influence private saving albeit weak t-statistic positively, it is worth discussing. The position of Juster and Wachtel (1972) on inflation is one of a support to the positive association with private saving. However, they succinctly state that, depending on whether the inflation regime is anticipated or unanticipated, the effect on private saving could either be symmetrical or asymmetrical. Another study whose results are in tandem with those of this paper is Campbell and Lovati's (1979) paper which associates price level increases with improvement in savings. This is done through cut-back in borrowing and spending, thereby having enough to save. Finally, Santosh and Lakshmi (2018) established results opposite to those revealed in this paper. They found both domestic and private saving to be negatively influenced by inflation in India. Source: Researchers' computation: World Bank Development Data (1980 The VAR Mode in Table 8 has PS as the core variable of interest. From the coefficients, it is realized that PS has a strong significant endogenous effect on itself. Thus, it has increased by about 81 percent over the past performance. This is observed from the t-statistic of about 6.9. This reaffirms Griffin's (1972) assertion that the saving behaviour of the depositor or policy maker is elusive and cannot be predicted. In terms of FD, it strongly influences itself, contributing about a 93% increase of its past realisation. Relative to its contribution to the saving behaviour of an average household in Ghana, a percentage change in FD will bring about approximately 47 percent improvement in private saving in Ghana. Thus, exhibiting strong endogeneity, signifying that if government's policies are tilted towards reducing expenses incurred in the financial system through reductions in the average cost of obtaining financial information and enforcing contracts, the floating customers' willingness to save, all things equal, will improve, and vice versa. In like manner, with a percentage fall in interest rates, all things held constant, there will be deterioration in the saving potential of aspiring depositors. Thus, the willingness of potential depositors will be negatively affected by a 46-percentage point. Interest rates are seen to negatively influence private savings.
In the case of the inflation rate, there is least to discuss because of its inability to significantly influence private savings as portrayed by the results (see Table 8). Besides, from past realization, FD has a strong endogenous on itself, influencing its status of increase by 93 percent, indicative of its t-statistic. Interest rate is also seen to have a strong influence on itself, depicting an influence strength of about 84 percent.

CONCLUSION AND RECOMMENDATION FOR POLICY
This study updated available literature on the determinants of private savings in Ghana. Efforts were made to establish the combined role of financial development, interest rates, and annual inflation rate in influencing private saving. Among findings of the study included the significant positive relationship between financial development and savings. It reemphasized the need to reduce the cost of transactions within the financial system to boost the households' propensity to improve saving and also to rake in new entrants into the saving web. The results also supported the negative lending rates hypothesis by the central bank to encourage consumption and investment in times of recession. Although evidence on inflation-saving relationship was positive, it rather showed an insignificant t-value. It is practical that lowering the deposit interest rates by the banks deter potential customers and also serve as panacea for panic withdrawals as in the case of the just passed into law of the electronic transaction tax that implicitly demotivates customers from doing business electronically. It is crucial that policymakers pay attention to macroeconomic variables that are central in influencing private saving encouraging new customer to buy into the culture of saving since it helps grow the economy. Besides, there appears to be less money in circulation in Ghana considering the global pandemic and the Russian-Ukrainian war that hit all nations. Therefore, a reduction in interest rates either explicitly or implicitly by financial sector operators could release more money into the system to rejuvenate the once vibrant economy.

LIMITATIONS AND SUGGESTIONS FOR FURTHER RESEARCH
Since the study is confined to Ghana's situation, the generalization of results is quite problematic, except where aspiring researchers wish to do a comparative analysis of peculiar occurrences in individual economies. Besides, the use of more variables would have appeared better; however, the original intention of researchers was to establish a link between domestic saving prospects of citizens and three macroeconomic variables (SD, INF & IR). Having identified these gaps, the researchers aim to suggest that an entailing study involving several African economies, or a cross-country analysis will fit as a solution to the established gap. The inclusion of additional variables into the model will, together, place the research in a preferred position to generalize results.