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Chapter 5: Problem 24

Let \(A\) be the fund balance on January \(1, B\) the balance on June \(30,\) and \(C\) the balance on December 31 a) If there are no deposits or withdrawals, show that yield rates computed by the dollar-weighted method and the time-weighted method are both equal to \((C-A) / A\) b) If there was a single deposit of \(D\) immediately after the June 30 balance was calculated, find expressions for the dollar-weighted and time-weighted yield rates. c) Rework ( \(b\) ) if the deposit occurred immediately before the June 30 balance was calculated. d) Verbally interpret the fact that the dollar-weighted yield rates in \((b)\) and \((c)\) are equal. e) Show that the time-weighted yield rate in \((b)\) is greater than in \((c)\)

Short Answer

Expert verified
For a fund with no deposits or withdrawals, both the dollar-weighted and time-weighted yield rates are equal to: \[Yield Rate = \frac { (C-A) } { A }\] With a deposit of \(D\) after June 30, the dollar-weighted yield rate is: \[ r_{DW} = \frac { (C-B-D) } { A(B/A) + D(B/A) } \] and the time-weighted yield rate is: \[ r_{TW} = \left ( \left ( \frac { B } { A } \right ) \left ( \frac { C } { B+D } \right ) \right ) - 1 \] With the deposit before June 30, the dollar-weighted yield rate remains the same while the time-weighted yield rate changes to: \[ r_{TW} = \left ( \left ( \frac { B+D } { A } \right ) \left ( \frac { C } { B } \right ) \right ) - 1 \] The dollar-weighted yield rates being the same shows that this rate is independent of deposit timing. However, the time-weighted yield rate is greater in scenario b than in c, indicating that earlier deposits lead to a higher yield rate.

Step by step solution

01

Calculating yield rates with no deposits or withdrawals

For Part a, with no deposits or withdrawals, both the dollar-weighted and the time-weighted yield rates can be calculated using the following formula: \[Yield Rate = \frac { (C-A) } { A }\] When applying this formula, \(A\) is the balance at the beginning of the year, \(C\) is the end balance, and the difference \((C - A)\) represents the total earnings over the year. Dividing these earnings by the initial balance \(A\) gives the yield rate as a decimal fraction.
02

Calculating yield rates with a deposit

For Part b, if there was a deposit of \(D\) immediately after the June 30 balance was calculated, the dollar-weighted yield rate \((r_{DW})\) and the time-weighted yield rate \((r_{TW})\) can be calculated using the following formulas: \[ r_{DW} = \frac { (C-B-D) } { A(B/A) + D(B/A) } \] \[ r_{TW} = \left ( \left ( \frac { B } { A } \right ) \left ( \frac { C } { B+D } \right ) \right ) - 1 \] According to dollar-weighted method, the yield rate is calculated based on the amount of money in the fund and the time it was there. The time-weighted method circumvents this by breaking the measurement period into sub-periods and calculates the geometric mean of sub-period returns.
03

Calculating yield rates with a deposit before the balance

For Part c which reworks b with the deposit happening before June 30, the formulas change slightly due to this timing difference: \[ r_{DW} = \frac { (C-D-B) } { D(B/D) +B(B/D) } \] \[ r_{TW} = \left ( \left ( \frac { B+D } { A } \right ) \left ( \frac { C } { B } \right ) \right ) - 1 \]
04

Interpreting dollar-weighted yield rates

Part d asks for a verbal interpretation. The fact that the dollar-weighted yield rate is the same in both cases (b and c) signifies that this rate is independent of timing deposits throughout the measurement period. As long as the total amount of funds and their time in the account remain constant, the yield rate will not change regardless of when deposits occurred.
05

Comparing time-weighted yield rates

Finally in part e, by substituting values into the equations from parts b and c, it can be shown that the time-weighted yield rate is greater in scenario b than in c. This means that the earlier a deposit is made, the greater the growth of the investment. This is due to the fact that more money is in the fund longer to accrue interest.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dollar-Weighted Yield Rate
The dollar-weighted yield rate is a powerful tool to assess the return on an investment when cash flows occur at different times. It accounts for the effect of deposits and withdrawals on investment returns. Unlike simple interest calculations, the dollar-weighted method adjusts the contribution of each cash flow based on the time it remains in the account.
In essence, the dollar-weighted yield rate provides insight by weighting each deposit or withdrawal according to its duration in the investment period. This ensures that larger amounts, or those invested early on, significantly impact the overall yield.
This method is especially useful when dealing with irregular cash flows. For example, when a deposit of amount \( D \) is made immediately after the June 30 balance, the yield rate considers how the timing affects growth by including the effect of this deposit for part of the year. The formula used integrates these cash flows and gives us an adjusted yield rate, reflecting how the invested money and time contribute to the fund's performance.
Time-Weighted Yield Rate
The time-weighted yield rate is distinct in its approach as it eliminates the impact of cash flow timing when evaluating an investment's performance. Instead, it focuses purely on the investment's return over specified periods, making it invaluable for comparisons.
This method breaks the investment period into smaller segments and calculates the return for each segment. These returns are then geometrically linked to obtain an overall rate for the entire period. It thus circumvents any bias introduced by the amount or timing of the cash flows.
For example, when a deposit of \( D \) is made right after calculating the June 30 balance, the time-weighted yield rate assesses the segments before and after the deposit separately. This yields a more accurate picture of how well the investment performed, independent of when and how much cash was added or withdrawn.
Investment Analysis
Investment analysis aims to evaluate the performance and prospects of an investment. In one scenario, analyzing both dollar-weighted and time-weighted yield rates can offer comprehensive insights.
The dollar-weighted yield rate helps in understanding how the investment is performing considering the timing and size of cash inflows or outflows. This is beneficial when internal cash handling influences returns.
Conversely, the time-weighted yield rate provides clarity on the investment's performance devoid of cash flow interference. This rate is advantageous for assessing the manager's efficiency in handling the investment portfolio.
Together, these yield rates equip an investor with the knowledge to interpret different aspects of investment performance, taking into account both the investment outcomes and the impact of money management choices. Through this dual method analysis, investors can achieve a deeper understanding of their investments and make informed decisions about their future strategies.

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Most popular questions from this chapter

It is desired to accumulate a fund of \(\$ 1000\) at the end of 10 years by equal deposits at the beginning of each year. If the deposits earn interest at \(8 \%\) effective but the interest can be reinvested at only \(4 \%\) effective, show that the deposit necessary is $$\frac{1000}{25 \pi 1.04^{-12}}$$

The proceeds from a life insurance policy are left on deposit, with interest credited at the end of each year. The beneficiary makes withdrawals from the fund at the end of each yeart, for \(t=1,2, \ldots, 10 .\) At the minimum interest rate of \(3 \%\) guaranteed in the policy, the equal annual withdrawal would be \(\$ 1000\). However, the insurer credits interest at the rate of \(4 \%\) for the first four years and \(5 \%\) for the next six years. The actual amount withdrawn at the end of year \(t\) is $$W_{t}=\frac{F_{t}}{\vec{a}_{\overline{11-t}} \cdot . .03}$$ where \(F_{t}\) is the amount of the fund, including interest, prior to the withdrawal. Calculate \(W_{10}\)

An investor enters into an agreement to contribute \(\$ 7000\) immediately and \(\$ 1000\) at the end of two years in exchange for the receipt of \(\$ 4000\) at the end of one year and \(\$ 5500\) at the end of three years. Find: a) \(P(.09)\) b) \(P(.10)\)

An investment account is established on which it is estimated that \(8 \%\) can be earned over the next 20 years. If the interest cach year is subject to income \(\operatorname{tax}\) at a \(25 \%\) tax rate, find the percentage reduction in the accumulated interest at the end of 20 years.

Let the exposure associated with \(i\) (i.e. the denominator of formula (5.13) ) be denoted by \(E .\) By using formula (5.17) show that the force of interest at any point is given by $$\delta_{t}=\frac{I}{E+(1-t) I}$$ It should be noted that in terms of the time diagram, this is equivalent to bringing the entire amount of interest \(I\) onto the time diagram at time \(t\) instead of time \(1 .\) The expression \(E+(1-t)\) Is often called the "exposure associated with \(\delta_{t}\)."
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