An Investment Model

Fund Rate of Return

Consider the calculation of rate of return of a fund. Let us assume a continuous gain fund model with midperiod flows. The period used here is the month. The value of the fund at a time t during the period, 0=t=1, can be written

                        v(t) = v₀e^rt + fe^r(t-1/2)

                        v(0) = v₀                                      v(1) = v₁ = v₀e^r+ fe^r/2

where v₀ is the intial value, v₁ the final value, f the midperiod flow r the continuous rate of return, and t-1/2 is limited to t=1/2, that is, t-1/2 is 0 for t<1/2.  The last equation can be solved for r knowing v₀, v₁, and f, with the substitution

                        u = e^r/2                                         0 = v₀u² + fu - v₁

                        u = v₁ / f                                                        v₀ = 0

                        u = ( -f ± Ö(f²+4v₀v₁)) / (2v₀)                      v₀ ¹ 0

                        r = 2 ln(u)

If the flow is 0, u = Ö(v₁/v₀).  The rate has no real solution if f²+ 4v₀v₁< 0, such as if f is zero and v₀v₁ < 0, or if f and v₀ are zero but v₁is non-zero.

The continuous rate of return can be converted to the simple period rate of return as follows.   These equations can be used to convert any others between simple and continuous rates.

                        e^r = (1+r_s)

                        r_s = e^r - 1                                      r = ln(1+r_s)

The mid-month flows and month-end valuations from the fund history table are used to compute a rate of return time series for each fund segment and the total fund. Index or market rate of return time series are retrieved from the index history table. 

Balanced Index Rate of Return

A balanced market index is computed by maintaining a constant asset balance.  As the performance of each segment varies, this requires continuous adjustment.   For a balanced index, where the kth index segment has a r_sk rate of return on an asset allocation of ak portion of index, the rate of return, r_bs, over a period is

                        r_b = S_k r·ak                                    r_bs = P_k (1+r_sk)^ak -  1

 A balanced index rate of return time series is constructed for a fund having an equity segment, a fixed segment and a cash segment, and an asset allocation between them. 

Unit Values

Unit value time series, u(n), are then calculated from these rate of return time series.

                        u(n) = exp( S_i=1..n r(i) )                u(n) = P_i=1..n (1+r_s(i))

This shows the value of 100 dollars invested in the fund since inception.

Period Rates of Return

Once unit value time series are available, the rate of return over any time period may be calculated from the ratio of the last unit value to the first unit value over the selected time period. 

                        r(i,j) = ln(u(j)/u(i))                        r_s(i,j) = u(j)/u(i) - 1

This is the way quarterly and selected period rate of returns are calculated, including those over the market cycles.  The fiscal year to date and since inception periods change from fund to fund, while the market cycle periods are the same for any asset allocation. 

Annualization

Certain period rates are annualized.  The conversion of a rate from n = j - i periods to a rate over m periods is

                        ra = r · m/n                                      ra_s = (1+r_s)^m/n - 1

where m is 12 months per annum.  Only the three and five year returns are annualized. 

Average Allocation

The average value of the fund during the period can be calculated to be

                        a = ò_0..1v(t)dt = v(t)/r = (v1 - (v0+f))/r

                           = (v1+v0)/2                                 r = 0

The average allocation during the period is ratio of the average allocated value to the average total value, and the average allocation over several periods is simply the average of the allocations during each period.  The quarterly average asset allocations for market timing analysis are computed by averaging the monthly averages this way. 

The actual balanced index rate of return is calculated using average allocations for each month and the returns for each segment for that month.  As a result, it is not sensitive to cash flows occuring within the period of a month.   

Variance and Regression Coefficients, a, b, r²

Next consider the mean and variance of the risk adjusted rates of return.  If the fund, index and cash equivalent rate of return during the period j are r_fj, r_ij, and r_cj, then the means, variances, and covariance of the risk adjusted rates of return over n periods, m_f, m_i, s_f², s_i², and s_if², can be computed from the following formulas. 

                        m_f =  (1/n) S_j=1..n (r_fj - r_cj)           s_f² = (1/n) S_j=1..n (r_fj - r_cj)² - >m_f²

                        m_i =  (1/n) S_j=1..n (r_ij - r_cj)           s_i² = (1/n) S_j=1..n (r_ij - r_cj)² - >m_i²

                        s_if² = (1/n) S_j=1..n (r_fj - r_cj)(r_ij - r_cj) - m_f·m_i

The non-risk adjusted versions are computed simply by setting the cash equivalent rates of return to zero.  Using the risk adjusted values, the linear regression coefficients, a, b, and r² can be calculated as

                        a = m_f - b m_i

                        b = s_if² / s_i²

                              r² = s_if⁴ / (s_i² s_f²)

The fund risk calculations are calculated this way using the monthly rates of return.  The other calculations are performed as indicated in the table.   

Investment Objectives

The investment objective rate of return is defined to be an index plus an annualized offset.  If the index is null, the offset alone is used, while if the offset is zero, the index alone is used.  The monthly offset is computed and it is added to the index rate of return time series for an objective rate of return time series.  The final year objective is stored in the objectives table. 

A few special indicies have been fabricated including the one year median performance and three year top quartile performance of the universe table for several asset allocations.   A few other special balanced indicies have been fabricated for the use of some accounts.  Past reconstruction of these indicies have been made. 

Valuations

The value of a fund segment having the same initial value and cash flows as the fund segment can be computed by successive evaluation of each month-end value, v(t) at t = 1, using the selected index rate of return. 

The valuations of the fund, the investment objective, consumer prices, the balanced index, and each segment of the fund and the balance index are computed in this manner. 

Percentile Ranking

The percentile performance of the fund relative to the universe is computed by cubic spline interpolation of the funds rate of return in the universes rate of return for funds having the same asset allocation objective.  Linear interpolation with limits of 0 and 100 percent is resorted to when the fund performance lies beyond the 5th and 95th percentiles.  Linear or zero order interpolation is used when the function fails to have a derivative. 

An asset allocation is determined by its equity segment, the fixed income and cash equivalent segments being lumped together for the universe calculations. 

Assumptions and Cautions

Mid-month flows and a continuous gain model is assumed.  All statistics include reinvestment of income.  The balance market index maintains a constant asset allocation. 

The valuations and cash flows are accurate only to the last digit.  Some valuations are estimated.  The rate of return of a segment having no allocation is zero.  This can result in lower rates of return when a period covers a time when no allocation was available.  While monthly rates of return are calculated, some investments may pay returns less frequently and this can perturb the month to month rates.  Calculation of rates of return are impossible when the value is zero or when the value passes through zero.  Calculation of the rates of return near zero may be suspect if the actual value represents large cash balances and large cash borrowings as the rate of return on net balance is calculated. 

As balanced market indicies do not include the cost of allocation adjustment or transaction costs, the results can be optimistic.  As the market timing analysis is based on monthly data, cash flows within a month are neglected.  This can underestimate the effects of timing during periods of substantial volatility. 

The regression is based on monthly data.  A regression on one or two samples will provide perfect correlation, r² = 1.  The coefficients can change significantly in the first quarters of a fund.  Confidence intervals can be computed for these values.