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Applying formula (5.13) we have 72

1000+ -500-1.


72 1200

= .06, or 6%.

Example 5.8 Find the effective rate of interest earned during a calendar

year by an insurance company with the following data:

Assets, beginning of year ...........$10,000,000

Premium income ..................1,000,000

Gross investment income .............. 530,000

PoUcy benefits ..................... 420,000

Investinent expenses ..................20,000

Other expenses .................... 180,000

It is conventional practice to offset investinent expenses against gross invesUnent

income. Thus, we have the following:

A = 10,000,000

= 10,000,000 + 1,000,000 + 530,000 - 420,000 - 20,000 - 180,000

= 10,910,000 / = 530,000 - 20,000

= 510,000.

Therefore, using formula (5.14)

. 2(510,000)

10,000.000 + 10,910,000 - 510,000

= .05, or 5% .

force 6, on the fund balance B,, plus (2) the rate of contribution (positive or negative) to the fund at exact time /. The derivation of formula (5.20) is left as an exercise.

Formulas such as (5.18), (5.19), and (5.20) are instructive conceptually. However, they are not widely used in practical applications.

Example 5.7 At the beginning of the year an investment fund was estabUshed with an initial deposit of $1000. A new deposit of $500 was made at the end of four months. Withdrawals of $200 and $100 were made at the end of six months and eight months, respectively. amount in the fund at the end of the year is $1272. Find the approximate effective rate of interest earned by the fund during the year, using formula (5.13).

The interest earned / is computed to be

1272 - (1000 + 500 - 200 - 100) = 72.


The methods for computing the yield rate earned by an investment fund outlined in Section 5.5 are sensitive to the amounts of money invested during various subperiods when the investment experience is volatile during the year. For example, if "large" amounts happen to be invested when the earnings on the fund are "high" and "small" amounts when the earnings are "low," the overall yield rate will be quite favorable. The reverse situation will, of course, produce die opposite result.

We will demonstrate this phenomenon with an extreme illustration. Assume an investor has an investment fund in which a $1000 investment is worth only $500 at the end of six months, but is worth $10(X) again at the end of the year. If no principal is deposited or withdrawn during the year, then the yield rate for the entire year is obviously zero.

Now consider what happens if the investor doubles the outstanding investment at the end of six months. The original $1( is worth only $500 at the end of six months, so the investor deposits another $500 at that time. The

J$ pew balance of $10( is then worth $20 at the end of the year. The equation

-4= of value for this transaction is

: 1000(1 + /) + 500(1 + = 2000.

This equation can be solved as a quadratic in (1 + o, which produces the yield rate i = .4069, or 40.69%.

Next consider what happens if the investor halves the outstanding investment at the end of six months. The original $1000 is worth only $500 at the end of Six months, so the investor withdraws $250 at that time. The new balance of 50 is then worth $500 at the end of the year. The equation of value for this saction is

1000(1 + 0 - 250(1 + if = 500.

ain, solving die quadratic in (1 + if produces the yield rate / = -.2892, -28.92%.

( It would seem that the underlying yield rate for the fund based on its actual ytment performance should be zero. However, in the first illustration above -computed yield rate is considerably greater than zero because the investor sited principal just as the investment experience was about to become very yfirable. In the second illustration just the opposite happened. The investor drew principal which caused the yield rate to become significantly negative. Since the amount invested clearly affects the computed yield rate, rates uted by the methods in Section 5.5 are sometimes called dollar-weighted

Time: /(> = 0 /

Contribution; Cl

Fund value: b b

Yield rate: z.

b,, b;

Figure 5.4 Time diagram for time-weighted rates of interest

Let the amount of the net contribution to the fund (positive or negafive) at time be denoted by C for A: = 1, 2,. . . , m - 1. The prime on the symbol is being used to distinguish this notation from that used earlier in the chapter, i.e. C\ is actually equal to C,.

Let the fund values immediately before each contribution to the fund be denoted by for A: = 1, 2, . . . , m - 1. Also the fund value at the beginning of the year is denoted by Bq = Bq, while the fitnd value at the end of year is

denoted by B = B. Finally, let the yield rates over the m subintervals be denoted by j/ for = 1,2,..., .

The yield rates over the m subintervals by the time-weighted method are

given by

1 +4 =

Bk-l + Cjt-l

Jk =

- 1 for A: = 1, 2, .. . , m.



In words, 1 plus the yield rate for each subinterval is equal to the fimd balance at the end of the subinterval divided by the fund balance at the beginning of the subinterval.

The overall yield rate for the entire year is then given by

1 + / = (1 +;i)(l +J2) (1 +;J i = (1 +yi)(i +;2)- (i+;J-l.


It is important to note that yield rates computed by the time-weighted method are not consistent with an assumption of compound interest. Nevertheless, time-weighted calculations do provide better indicators of underlying investment performance than dollar-weighted calculations. However, dollar-weighted calculations provide a valid measure of the actual investment results achieved.

Example 5.9 On January 1 an investment account is worth $100,000. On May 1 the value has increased to $112,000 and $30,000 of new principal is deposited. On November 1 the value has declined to $125,000 and $42,000 is withdrawn. On January 1 of the following year the investment account is again worth $100,000. Compute the yield rate by: (1) the dollar-weighted method, and (2) the time-weighted method. Figure 5.5 illustrates the transactions involved in diis example.


Contribution Fund Value


5-1 +30,000 112,000

11-1 -42,000 125,000


Figure 5.5 Time diagram for Example 5.9

rates of interest. It is important to observe that compound interest calculations as developed in previous chapters are done on this basis.

Now assume that the investment decisions for the fimd are being made by an investment manager, while the decisions to deposit or withdrau principal are made by the owner of the fund. Although the dollar-weighted calculations in the above two illustrations provide an accurate measure of the actual return realized by the owner of the fitnd, they do not provide a good measure of the "true" performance of the investment manager, which was zero.

Such a measure is provided by an alternative basis for calculating fund yields called time-weighted rates of interest. In this method we consider successive subintervals of the year each time a deposit or withdrawal is made. Thus, in the illustrations given above, the yield rate for the first six months of the year is /j = -50% and for the second six months is yj = 100%- We can combine these for the entire year to obtain

!+/ = (! +;,)(1 +J2) = (1 - .5)(1 + 1) = 1.

Thus, / = 0 regardless of when principal is deposited or withdrawn.

We can generalize this approach as follows. Assume that m - \ principal

deposits or withdrawals are made during the year at times fj,\-

This will divide the year into m subintervals as illustrated in Figure 5.4.

= A + C + l

which gives

100,000 = 100,000 + (30,000 - 42,000) + /

/ = 12,000.

We will use formula (5.13) to find the dollar-weighted rate of interest


i =.

100,000 -f- I 30,000 - 1 42,000 3 6

= 2.000 = 1062, or 10.62%. 113,000

A more refined answer could be obtained by using compound interest instead of formula (5.13), but solving the equation of value would require a difficult iteration. The extra effort is not worth the trouble for purposes of this example. It is important to note diat the intermediate fund balances ($112,000 and $125,000 in diis example) do not have any effect on the answer when using the dollar-weighted method.

2. Using formulas (5.21) and (5.22) the time-weighted rate of interest is found to be





100,000 83,000

= (1.12) (.880282) (1.204819) - 1

= .1879, or 18.79% . Thus, the time-weighted rate of interest is dramatically higher than the dollar-weighted rate of interest. The reason for this becomes evident upon analyzing die three subintervals. Investment experience was very fevorable during the first four mondis and die last two mondis of die year. However, it was quite adverse during the intervening six months. Since new principal was deposited just before the experience was about to turn sour and principal was withdrawn just before the experience was about to become fevorable again, the dollar-weighted calculation was quite adversely affected.

In summary, die dollar-weighted yield, 10.62%, is a measure of die actual financial results achieved by die investor. The time-weighted yield, 18.79%, is a measure of the actual performance of the investinent fund independent of the amount that happens to be invested.

The reader should not think that differences of this magnitude between the two methods are typical. It took the combination of bad timing in making principal

1. The total amount of interest earned can be found from formula (5.9)

deposits and withdrawals, together widi highly volatile investment performance (such as could occur in a fund heavily invested in common stocks), to create such a dramatic effect. The differences between the two methods would be much smaller in the event diat eidier the invesUnent performance was more stable, or diat principal deposits and withdrawals were smaller in relation to the fund balance, or both. In fact, in stable invesUnent funds invested at stable interest rates, die difference between the two mediods would generally be insignificant.


Consider the commonly-encountered situation in which an investment fund is being maintained for a number of different entities, i.e. individuals or companies. An example would be a pension fimd in which each plan participant has an individual account. However, the investment fund is commingled, i.e. each account does not have its own separate group of segregated assets, but rather a pro rata share of the entire fund.

An issue arises in connection with the crediting of interest to the various accounts. Two distinctiy different approaches to allocating interest to the various accounts are in common use; namely, die portfolio method and tiie investment year method.

Under the portfolio method an average rate based on the earnings of the entire fund is computed and credited to each account. This method is quite straightforward and simple, to implement. It is a method of long-standing usage in a variety of different situations.

However, problems arise in using the portfolio method during periods of uctuating interest rates. For example, consider a situation in which interest rates have risen significantly in the recent past. The portfolio method might . )roduce an average rate of 8%, while new deposits might be able to earn 10% on their own. The portfolio rate is lower because the fund includes a collection gf lower yielding investments made in the past. In this situation diere is a lificant disincentive for anyone to make new deposits to the und, and diere i also an increased incentive for withdrawals.

The investment year method was developed to handle this problem by ognizing the date of investment, as well as the current date, in crediting Interest. It is a newer method that came into vogue during the 1960s and 1970s en there was a long period of rising interest rates. The rate on new deposits ier the investment year method (10% in the above example) is often called je new money rate.

The investment year method is obviously more complicated than the portfolio method to apply in practice. However, many financial institutions, ch as banks and insurance companies, felt it necessary to utilize the

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