ANNEX A. Methodology for calculating marginal effective tax rates on household savings
Introduction
This annex outlines in detail the underlying methodology for calculating the marginal effective tax rates (METRs) presented in Chapter 3. The methodology follows broadly the approach of the OECD (1994) Taxation and Household Savings study, which itself drew on the methods used by King and Fullerton (1984).
The analysis considers a saver who is contemplating investing an additional currency unit in one of a range of assets. The investment is a marginal investment, both in terms of being an incremental purchase of the asset, and in terms of generating net returns just sufficient to make the purchase worthwhile (as compared to the next best savings opportunity). The approach assumes a fixed pretax real rate of return and calculates the minimum posttax real rate of return that will, at the margin, make the savings worthwhile. The METR can then be calculated as the difference between the pre and posttax rates of return divided by the pretax rate of return.
The pretax rate of return is determined by explicitly modelling the stream of returns and taxes associated with a marginal investment over time. The modelling incorporates the impact of a wide range of taxes in the one indicator, including taxes on cash income (at the personal level as well as the level of the savings intermediary), taxes on realised capital gains (with or without indexation), taxes on asset purchases and/or sales, transaction taxes and wealth taxes. The tax gain as a result of the deductibility of savings from taxable income as well as of interest payments which have to be paid if the investment is financed with borrowed funds are modelled as well.
METRs are calculated for the following types of savings vehicles:

Bank deposits

Corporate and Government bonds

Equities (purchase of corporate shares)

Investment fund assets (marketable, collective investment vehicles)

Private pensions

Individual taxfavoured savings accounts

Equityfinanced owneroccupied and rented residential property

Debtfinanced owneroccupied and rented residential property
The overall methodological approach is first detailed, before METR equations for each savings vehicle are explicitly derived.
Overall approach
Consider a saver contemplating investing one currency unit in a particular savings vehicle. The present value of the returns and costs (taxes) of investing in that savings vehicle over time can be expressed as:
(1)
where is the stream of returns and the stream of taxes on those returns (which will vary depending on the particular savings vehicle and the way in which it is taxed). The returns and costs are discounted at a rate .
The length of time for which an asset is held is often crucial in determining tax liabilities. The approach taken here (and in the OECD, 1994 study) is to assign a probability to each possible holding period. The expected return is then calculated in each case, on the basis of risk neutrality. It is assumed that the probability of sale in each period is fixed, so that there is, for example, a 10 per cent chance of sale in the first period; if the asset is not sold, there is a 10 per cent chance of sale in the second period, and so on.
The time of sale of asset ‘t’ may be thought of as a random variable, which follows an exponential probability density function with rate parameter :
(2)
This setup implies that the investment will earn a return in period 0 with full certainty while the probability that the investment yields a return in the following periods decreases over time until n reaches infinity, when the probability that the investment earns a return is zero.1
This endogenous asset holding period approach has a number of advantages. First, it allows focusing on different holding periods while avoiding timerelated variables in the analytical solution of the derived effective tax rates. Second, where tax rates vary with the holding period, the method implicitly weights the different tax rates (avoiding the need to calculate different METRs for each holding period). It therefore gives a reasonable summary of the overall tax treatment where incentive effects vary with the holding period.
Incorporating the endogenous holding period into equation 1 gives:
(3)
Setting V = 0 and solving for will yield an expression for the investor’s aftertax nominal rate of return on the particular savings vehicle. The aftertax real rate of return of investing in a particular savings vehicle, , given an inflation rate of π, is then:
(4)
The investment is assumed to earn a fixed real return, . Consequently, the METR, , is:
(5)
Explicit equations for each savings vehicle
In this section we derive explicit METR equations for each of the savings vehicles considered in the study. A basic form of the equations is first derived for each savings vehicle (representing a “typical country” approach). A number of potential extensions are then presented to illustrate how a wide range of different taxes and tax designs can be incorporated into the METR modelling for each savings vehicle. Where countries rules vary from the possibilities presented here, the modelling is altered accordingly.
Bank deposits
Basic case
The net present value of an investment of one currency unit in a bank account equals:
where:
= real interest rate
π = inflation rate
= tax on purchase of the asset
= tax on sale of the asset
= tax on income / return that the asset generates in every period
= tax on wealth (levied on the principal); to some extent, it is assumed that the aftertax interest is consumed)
= expected holding period of the asset
= discount rate/the aftertax nominal return the household realises on a marginal investment
= present value of the investment
A transaction tax, , has to be paid as a result of the one currency unit of savings in a bank deposit. The cost of the savings is therefore equal to one plus the transaction tax. The investment earns a nominal return, , which is taxed at the rate . A yearly wealth tax on the value of the deposit, , has to be paid as well. This recurrent aftertax return is discounted at the rate . The probability that the household keeps the one currency unit as a bank deposit for periods is assumed to decrease exponentially as increases, as implied by the integral of the term . The inverse of is the expected holding period of the savings. The one currency unit is withdrawn from the bank account after n years, after which a tax on the sale of the asset has to be paid.
Setting V = 0 allows us to solve for :
The marginal effective tax rate equals:
Extension 1: x% of return is taxexempt
If x% of the return is taxexempt, then:
Extension 2: a savings subsidy is paid by government for each currency unit saved
In some countries, the government may provide a (tax or other type of) subsidy “su” for each currency unit of savings. This may function as a negative transaction tax, reducing the cost of the savings (in case the subsidy is given upfront when the savings are made):
The net present value of an investment of 1 currency unit in a bank account equals:
Setting V = 0 and solving for gives:
Corporate and Government bonds
Issued at par
Corporate and Government bonds issued at par will be taxed in the same manner as bank interest.
Issued at a discount
Instead of offering the nominal interest rate, the bond only pays the interest rate . As a result, the price that will have to be paid for the bond will not be 1 but instead (where ). The bondholder will recover the full face value of the bond when it is sold and a capital gains tax will have to be paid on the gain.
First note that the discount equals the net present value of the interest foregone:
The net present value of an investment in a bond issued at a discount where of the return is in the form of a capital gain equals:
Setting V = 0 and solving for gives:
where:
Equities (purchase of corporate shares)
Basic case
The net present value of an investment of one currency unit in a share of a business equals (first assuming no tax on purchase and sale):
where (in addition to the previously defined parameters):
= fraction of return that is distributed as dividends
= tax on dividends
= capital gains tax (capital gains are taxed upon realization)
The household buys a share at a price of one currency unit; the investment yields a nominal return, , which is partly distributed (and taxed) as dividends, and partly retained and reinvested. It is assumed that the reinvested earnings yield a nominal return of as well. At time t, the value of the share is . The first term within the square brackets reflects the present value of the aftertax dividends. The household sells the share in period ; a capital gains tax has to be paid on the increase in value of the share; the second term within the square brackets reflects the aftertax capital gain. The household recovers the original one currency unit of savings taxfree (third term). A yearly wealth tax is also paid on the value of the share (fourth term).
Setting V = 0 and solving for gives:
The marginal effective tax rate then equals:
Extension 1: tax on purchase of the share
Incorporating a tax on the purchase of the share changes the first term to . Setting V = 0 and solving for gives:
Extension 2: (partly or fully) indexing the capital gains for inflation
If the capital gains tax is levied only on the real capital gains (and not the nominal gains), the aftertax nominal rate of return amounts to (in the absence of sales taxes and if x and y are 0):
where is the degree to which capital gains are indexed for inflation.
The indexation appears twice. First, the capital gains tax, which is levied on the nominal interest rate, is now discounted at a real instead of nominal rate. Second, the aftertax nominal return increases with the term .
Investment fund assets (marketable, collective investment vehicles, where 100% of the return is retained and reinvested)
Basic case
The net present value of an investment of one currency unit in a share of an investment fund equals:
where = the tax on the investment fund’s earnings.
In the first period, the household buys a share of the investment fund at price of one currency unit. The investment fund invests these funds in saving opportunities which earn a nominal return, . The tax is due on the investment fund’s earnings. Consequently, the fund will reinvest in every period the return . After n periods, the household sells its share in the investment fund. It realises the capital gains and recovers the original investment. The increase in value of the asset is taxed under the capital gains tax at rate . Additionally, in every period the tax authorities levy a wealth tax, , on the value of the asset in that particular period.
Setting V = 0 and solving for gives:
The marginal effective tax rate then equals:
Extension 1: purchase tax + sales (exit) tax on the value of the share at the moment of sale
The net present value of an investment of one currency unit in a share of an investment fund becomes:
The household’s aftertax nominal rate of return then amounts to:
Private pensions
Basic case
The net present value of a household investment of 1 currency unit in a pension fund amounts to:
where:
= the tax on the pension fund’s earnings
= income tax rate at which pension savings can be deducted from taxable personal income
= income tax rate at which the pension is taxed (note that this excludes employee social security contributions)
The household has one currency unit which it saves for a pension through a pension fund. Even though the pension fund receives savings of one currency unit, the cost to the household is only because pension savings can be deducted from taxable personal income at rate , which will typically be the household’s marginal income tax rate at the time of saving. The pension fund invests the funds in savings opportunities that earn a nominal return, . A tax, , is due on the fund’s earnings. The fund then reinvests in every period the return, . After n years, the total return, which equals the original contributions plus the return on the investment, is distributed and entirely taxed at the rate , typically the household’s personal income tax rate. In addition, wealth taxes are due yearly on the value of the pension savings in that particular year.
Setting V = 0 and solving for gives:
The marginal effective tax rate then equals:
Extension 1: purchase tax on the contributions paid to the pension fund
The household may have to pay a purchase tax on the contributions paid to the pension fund, which may be either deductible (in which case d=1) or not (d=0) from personal income tax. The cost of the investment then amounts to: .
The household’s nominal aftertax rate of return amounts to:
Individual taxfavoured savings accounts
Basic case
The net present value of an investment of one currency unit in a taxfavoured savings account (assuming that 100% of the return is paid out each year) equals:
where g% of the savings are taxdeductible at a rate ; x% of the return is taxexempt; and (1x)% is taxed at rate . There is also a wealth tax, .
Setting V = 0 and solving for gives:
The marginal effective tax rate amounts to:
Extension 1: purchase tax, sale tax, and tax on principal
There may be a tax on purchase , which may be deductible (if d=1) or not (if d=0), and a tax on sale, , on the principal investment. Part of the principal (w%) when recovered (i.e. when the asset is sold) may also be taxed at a rate, . The net present value of an investment of one currency unit in a taxfavoured savings account would then equal (assuming that 100% of the return is paid out each year):
The household’s aftertax nominal rate of return then amounts to:
Equityfinanced owneroccupied and rented residential property
Basic case
The net present value of a one currency unit equityfinanced investment in owneroccupied housing equals:
Equityfinanced investment in housing costs the household one currency unit plus a transaction tax (property transfer tax), . The investment yields a pretax real return (imputed or actual rental income), , and a return to pay for the depreciation of the house, . The value of the house is increasing in the inflation rate, π, and decreasing in . Earnings are discounted at the nominal rate . x% of the return is taxexempt; (1x)% of the return is taxed under an income tax, . The imputed rental income (irv) may be taxed instead at a rate . Additionally, the value of the house is taxed under a local property tax, , and may be subject to a wealth tax, . After n years, the household sells the house and recovers the value of one currency unit.
The household’s aftertax nominal rate of return is:
The marginal effective tax rate then equals:
Extension 1: capital gains, and depreciation provisions
Instead of deriving the return entirely each year (from imputed or actual rental income) part of the return, , is now earned each year and the rest of the return, , is earned in the form of capital gains. The house depreciates in value due to wear and tear and the owner receives a return to pay for the economic depreciation of the asset; this return may be taxed, and the taxpayer may be able to claim tax depreciation allowances, . The value of the house, which is used for tax purposes, is updated yearly (and increases with the inflation rate, the economic return and decreases with the economic depreciation rate, ). The tax depreciation allowances equal:

Straight line tax depreciation allowances:
where is the income tax rate; is the number of years over which the house has to be depreciated.

Declining balance tax depreciation allowances:
where is the tax depreciation rate.
The net present value of an equityfinanced investment in owneroccupied housing equals:
where takes the value 1 if the capital gains are indexed for inflation and 0 if they are not.
Setting V=0 and solving for gives:
Debtfinanced owneroccupied and rented residential property
Basic case
Instead of buying residential property with own funds (equity), households may decide to borrow money to finance the investment and invest their own funds in an alternative savings opportunity. In some countries, mortgage interest payments are deductible from taxable personal income. As a result, the aftertax borrowing cost may be lower than the return which the household can realise on an alternative savings opportunity. This will reduce the financing, FC, below the equityfinanced cost of investing in residential property (i.e. (1 + t_{p} )).
The net present value of a debtfinanced investment in owneroccupied housing then equals:
The financing cost amounts to:
The household borrows currency units. Interest payments have to be paid, but may be deductible from the household’s personal income, thereby reducing the cost to . Moreover, the debt may be deducted from the household’s taxable wealth, which reduces the yearly cost by . After n years, the originally borrowed funds must be paid back. The discount rate is the opportunity return, , which is assumed to be the aftertax return on savings in a bank deposit.
The financing costs can be written as , where denotes the financing gains (or losses) and depends on the difference between the household’s opportunity return on an alternative savings opportunity and the borrowing costs. The opportunity return is assumed to be the aftertax return on savings in a bank deposit. If this is not tax favoured (i.e is taxed at ), then F will likely be equal to zero. If it is taxfavoured, or if an alternative taxfavoured savings vehicle was chosen as the opportunity return, F will be positive (and the effective tax rate would be lower).
The household’s aftertax nominal rate of return then amounts to:
Extension 1: capital gains, and depreciation provisions
Instead of deriving the return entirely each year (from imputed or actual rental income) part of the return, , is now earned each year and the rest of the return, , is earned in the form of capital gains. The household now borrows . The house depreciates in value due to wear and tear and the owner receives a return to pay for the economic depreciation of the asset; this return may be taxed, and the taxpayer may be able to claim tax depreciation allowances, . The value of the house, which is used for tax purposes, is updated yearly (and increases with the inflation rate, the economic return and decreases with the economic depreciation rate, ). The tax depreciation allowances are as detailed for an equity financed investment.
The net present value of a debtfinanced investment in owneroccupied housing equals:
The financing cost amounts to:
where
Setting V = 0 and solving for gives:
References
King, M. and D. Fullerton (1984), Taxation of income from capital: a comparative study of the United States, United Kingdom, Sweden and West Germany, Chicago University Press, Chicago.
OECD (1994), “Taxation and Household Saving”, OECD Publishing, Paris.
Note
← 1. The expected holding period of the asset (i.e. the expected period when the asset will be sold) is the inverse of the rate parameter (i.e. ), and is calculated as the expected value of the exponential distribution: