Any form of investment has an objective of increasing the amount of money or value of the asset which the investor has invested. Return is the parameter which can be used to measure the performance of the financial asset and is also helpful in evaluation of the asset. Return is generally which attracts the investors towards a financial asset over others, assuming all other conditions to be same.
A rate of return is the gain received from an investment over a period of time expressed as a percentage. Though the returns for a simple investor can be easy and require only unitary skills, the formulas can turn to be complex for the following reason:
- Selection of proper inputs to calculate return
- Treatment of additional client contributions and withdrawals to and from the investment account
- Adjusting the return to reflect the timing of these contributions and withdrawals
- Differentiating between the return produced by the investment manager and the return experienced by the investor
- Computing the returns spanning multiple valuation periods
- Averaging periodic rates of return
Single Period Rate of Return
The rate of return concentrates a lot of information into a single statistic. Individual data points about the beginning and ending market values, income earned, cash contributions and withdrawals, and trades for all of the holdings in the portfolio are compressed into a single number. It is faster for an investor to analyse proportions than absolute numbers. Returns are comparable even if the underlying figures are not. Returns calculated for different periods are comparable; that is, an investor can compare this year’s return to last year’s. Return is the value reconciling the beginning investment value to the ending value over the time period we are measuring.
MVE = MVB * (1 + Decimal Return)
MVE = market value at the end of the period
MVB = market value at the beginning of the period
Return in per cent = (gain or loss) / (investment made) * 100
Generally the returns are calculated ignoring the time value of money whereby the investment made is assumed to have been made only prior to the sale of the asset. The rate of return can be rewritten as,
(gain or loss)/(investment made)*100 = ((current value/investment made)–1)*100
The equation leads to,
(MVE – MVB)/MVE, or, (MVE/MVB – MVB/MVB), or, (MVE/MVB – 1)
The numerator is the unrealised gain or loss; in fact it should be given as market value plus the accrued income. The denominator is always the investment made or the money which is put to risk.
For instance, the investor invests on 1 January, Rs. 100, and it rises to Rs. 110 at the end of January and then to Rs. 120 at the end of February. The gains are thus calculated as:
1 – January: 100
1 – February: 110 return = (110-100)/100 = 10%
1 – March: 120 return = (120-110)/110 = 9.09%
This procedure where the market value of the investment has been used to calculate, the gain on the investment has been recognised, though it is not actually realised as there has been no sale done. For calculating returns which include the unrealised gain, the returns are measured at the end of each measurement period, the dates of which are called valuation dates. A return between two valuation dates is known as a single period, holding period, or periodic return.
To sum up, when there are no transactions into and out of an investment account and no income is earned the ending value is simply divided with the beginning value. In case of portfolio, the total market value is derived by summing up the individual asset value in the portfolio.
The unit value is simply calculated by dividing this value with the total number of shares outstanding. This is done using a methodology called trade date accounting. Here the securities are included in the portfolio valuation on the day the manager agrees to buy or sell the securities, as opposed to waiting for the day the trades settle with the broker.
The market value of each security is the amount we would expect to receive if the investment were sold on valuation date. There are a number of factors like liquidity, estimating the exact current value and future cash flows which make this process hard.
The individual security market value include a measure of income earned or accrued income. This is the income which the investor has earned but is yet to receive. For example, in bonds, there are generally semi-annual coupons, thus between the time of coupon payments, the investor has earned the interest, though that would be received only on the coupon date. Returns which include both the change in the market value and the income earned is called the total return. Similar adjustments are made when considering the portfolio as a unit.
Thus the single period return calculation formula can be written as:
[((ending market value + ending accrued income)/(beginning market value + beginning accrued income)) – 1] * 100
Here the cost of investment has not been included in the calculation. Also for each subsequent period, the ending market value for the previous period is used as the beginning value for the next period. Here it is assumed that for each valuation period the amount of money invested is the unrealised gain and not the initial investment made.
Return on investment
The previous section gave information about the single period return calculation. In real life scenarios, the investors make additional investments and also withdraw from the portfolio. In case of contributions/withdrawals the calculation for the gain/loss is:
Gain/loss = (current value – original investment – net cash inflows + net cash outflows)
If the above cash flows are adjusted into the formula for the return, the result is the return on investment.
ROI in percent = ((EMV+NOF)-(BMV+NIF))/(BMV+NIF) * 100
In all the above formulas the timing of the cash flow and also the duration for which the investment has been made is not considered.
Time value of money
To project the future value of money, the formula used is
FV = PV * (1 + R)N
Here the present amount of money is multiplied with the interest rate applicable. This is possible because of the fact that interest rates are known whereas the returns are not. Thus the future value of an investment is equal to the present value plus the interest and other gains earned over the period.
Compounding is the reinvestment of income to earn more income in subsequent periods. Here the implicit assumption is that any cash flow is reinvested at the same rate of interest and thus the future value is calculated. As it can be seen from simple calculations the power of compounding makes the returns to be large in nature. As time passes, the total returns increase more than what the simple interest would have given and also the interest-on-interest increases. Interest-on-interest is the interest which is earned on an initial interest which has now been assumed to have been reinvested at the same rate of interest.
Interest rates are quoted in annual basis and the same can be adjusted for lesser time period compounding:
MVE=MVB*(1+rate of interest/compounding frequency)(periods * compounding frequency)
When the numerator and the denominator values are adjusted for the time, and the contributions and withdrawals are taken as such that the time component have been added using the time value of money formula, the denominator is known as the average capital employed or the average interest balance.
Performance of an investment: money-weighted returns
Timing of investor decision
Market timing is the term which relates the time an investor makes his investment to the market cycle. Based upon the time the investor invests or sells the investment return can vary.
The concept can be explained with a simple example. The period and the return table can be redrawn as:
Period | Market Value | Return |
1 | 100 | - |
2 | 120 | 20% |
3 | 110 | -8.33% |
4 | 115 | 4.55% |
As seen in the table, the returns for the periods are given in the third column. Had the investor invested in period 1 and sold the investment in period 2, the return is 20% whereas, if the investment had been sold in period 4, neglecting the time value of money, the return would have been only 15%. These decisions of investing and selling the investment are dependent upon the investor and not on the manager.
Timing of investment manager decision
The investment of the amount of cash at disposal of the manager would depend on the manager herself.
Take for example, that the first manager decides to keep a portion of the cash aside and not invest it and the second manager using same amount of money decides to invest the whole of the fund in the market. Now, if the market moves up, and the market returns are more than the returns which are earned on the non-invested money (can be assumed to be nominal in nature), the second manager stands to gain more than the first and vice-versa.
Segregating investor and manager timing decision
As it can be seen from the above discussion, if the investor and the manager are two different persons then the responsibility of the decisions would be dependent upon them. Thus there is a need to segregate the investor and the manager timing decisions. The decisions are:
· The timing of investor decisions to make an investment into the portfolio
· The decisions made by the manager to allocate assets and select securities within the portfolio
The money weighted return (MWR) is used when we need to measure the performance as experienced by the investor. MWR is a performance statistic reflecting how much money was earned during the measurement period. This amount is influenced by the timing of decisions to contribute or withdraw money from a portfolio, as well as the decisions made by the manager of the portfolio. The timing and size of the cash flows have an impact on the ending market value:
Transaction | Before Market | Effect on Performance |
Contribute | Goes Up | Positive |
Contribute | Goes Down | Negative |
Withdraw | Goes Up | Negative |
Withdraw | Goes Down | Positive |
Though the above data can be used to segregate the cash flows, in order to account for the timing of the cash flows, they have to be discounted for the appropriate amount of time.
Take for example, that the investments are made at the start of each period.
Time Period | Months Invested | Period Weight |
0 | 12 | 1.00 |
1 | 11 | 0.92 |
2 | 10 | 0.83 |
3 | 09 | 0.75 |
As shown in the table, the cash flows for different time periods can be adjusted using the period weights.
MWR is found by calculating the IRR (Internal Rate of Return) for all the cash flows and changing the time parameter to reflect the true value for individual cash flows. The formula for the calculation of the IRR is given as:
MVE = MVB (1+IRR) + CF1 (1+IRR) + CF2 (1+IRR)2 +….+ CFn (1+IRR)n
The values for the MVE and MVB and the different cash flows are required to calculate the IRR.
As IRR calculation is dependent upon the trial and error, it can be tedious to calculate. To minimise the time and calculate the rate of return approximately, the Modified Dietz Return (MDR) formula was devised.
MDR = (MVE-MVB-CF)/(MVB+((CD-Ci)/CD)*CFi) * 100, where
CF = net amount of cash flows for the period
CD = total days in the period
Ci = the day of the cash flow
CFi = the amount of the net cash flow on Ci
Performance of the investment manager – time-weighted returns
Time-weighted return (TWR) is the statistics that completely eliminate the impact of investor cash flows. The time weighted return (TWR) is a form of total return that measures the performance of a dollar invested in the fund over the complete measurement period. The TWR eliminates the timing effect that external portfolio cash flows have on performance, leaving only the effects of the market and manager decisions.
The steps to calculate the TWR are:
- Begin with the market value at the beginning of the period.
- Move forward through time toward the end of the period.
- Note the value of the portfolio immediately before a cash flow into or out of the portfolio.
- Calculate a subperiod return for the period between the valuation dates.
- Repeat 3 and 4 for each cash flow encountered.
- When there are no more cash flows, calculate a subperiod return for the last period using the end of period market value.
- Compound the subperiod returns by taking the product of (1 + the subperiod returns).
The last step is called the chain linking or geometric linking, which is similar to compounding.
TWR = [(1+R1) * (1+R2) * …. * (1-Rn) – 1] * 100, where,
Rn are sub-period returns.
Sub-period return is calculated using the formula MVE/(MVB + NCF).
Multiple-period return calculation
Cumulative return is the rate of return which the investor receives due to the growth rates for the different sub-periods.
CR = [GR1*GR2*….*GRn – 1]*100
Arithmetic mean return is the arithmetic average of the returns for the different sub-periods. The formula for the same is:
AMR = (Sum (periodic returns))/(count of returns)
Geometric mean return is the n-th root of the product of the returns of the different sub-periods.
GMR = [n√ (1 + cumulative return) – 1] * 100
Annualising returns less than a year – the geometric mean return when calculated for a 1-year period is called an average annual return, compound annual return, or annualised return. The formula for the same is given as:
{[(1 + period rate)# of periods] – 1} * 100
Annualising returns greater than a year – the formula is:
{[(# of years√ (1 + period rate)] – 1} * 100
AR = [(number of days / 365.25 √linked growth rates) – 1] * 100, where,
linked growth rate = (1+R1) * (1+R2) …. (1+RN)
Compound annual internal rate of return – this can be calculated from the general formula of the compound annual rate of growth where the periods are divided into multiples of a year.
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