###### Key Properties of the Normal Distribut ...

A random variable is said to have the normal distribution (Gaussian curve) if... **Read More**

The money-weighted return considers the money invested and gives the investor information on the actual investment return. Calculating money-weighted return is similar to calculating an investment’s internal rate of return (IRR).

The money-weighted rate of return (MWRR) is like the portfolio’s internal rate of return (IRR). It’s the rate at which the present value of cash flows equals zero. In simple terms, it’s a way to measure how well a portfolio is performing.

$$\sum_{t=0}^{T}{\frac{CF_t}{\left(1+IRR\right)^t}=0}$$

Where:

\(T\) = Number of periods.

\(CF_t\) = Cash flow at time *t*.

\(IRR\) = Internal rate of return (or money-weighted rate of return).

The money-weighted rate of return (MWRR) looks at a fund’s starting and ending values and all the cash flows in between. In an investment portfolio, cash inflows are a part of it. These inflows could be from deposits or investments made during a certain period. The MWRR considers these inflows and calculates the overall rate of return for the portfolio:

- The beginning value.
- Dividends/interest reinvested.
- Contributions made.

Cash outflows, on the other hand, refer to:

- Withdrawals made.
- Dividends or interest received.
- The final value of the fund.

An investor makes the following investments in a portfolio over a two-year period:

- At the beginning of year one, the investor invests $10,000.
- At the end of the first year, after the portfolio’s value increases to $12,000, the investor adds $5,000, making the total portfolio value $17,000.
- At the end of the second year, the portfolio value further increases to $25,000.

The money-weighted rate of return for the investor’s portfolio is *closest* to:

**Solution**

We need to calculate the internal rate of return (IRR) considering the following cash flows:

- \( CF_0 = -\$10,000 \) (Initial investment)
- \( CF_1 = -\$5,000 \) (Additional investment at the end of year one)
- \( CF_2 = +\$25,000 \) (Final portfolio value at the end of year two)

To find the money-weighted rate of return, solve the equation for IRR:

$$ \frac{CF_0}{(1+IRR)^0} + \frac{CF_1}{(1+IRR)^1} + \frac{CF_2}{(1+IRR)^2} = \frac{-10,000}{1} + \frac{-5,000}{(1+IRR)} + \frac{25,000}{(1+IRR)^2} = 0 $$

Using BA II Plus Calculator, \(IRR\approx 35.08\%\).

Calvin Hair purchased a share of Superior Car Rental Company for $85 at the beginning of the first year. He bought an additional unit for $87 at the end of the first year. At the end of the second year, he sold both shares at $90. During both years, Hair received a dividend of $4 per share, which was not reinvested.

Calculate the money-weighted return.

**Solution**

To calculate the money-weighted return in this example, we need to consider the timing and amounts of cash flows and their respective investment periods.

**Step 1: **Calculate the total investment at the beginning (*t*=0):

$$ \text{Initial investment}= -$85 $$

**Step 2: **Calculate the total investment at *t *= 1:

$$\begin{align} \text{Initial investment + Additional investment} &= $87 – $4 \text{( Dividend received at}\\&\text{ the end of the first year, which is not reinvested)}\\& = -$83 \end{align}$$

**Step 3: **Calculate the final portfolio value at *t *= 2:

$$ \begin{align}\text{Number of shares sold × Selling price}& = 2 \text{ shares} × $90 = $180 + 8 \text{( Dividend received for}\\ &\text{both shares)}\\& =$188\end{align} $$

As such, we have:

\(CF_0=-85\).

\(CF_1=-83\).

\(CF_2=188\).

Using the BA II Plus calculator, you will get \(IRR=7.71\%\), which is equivalent to the money-weighted rate of return.

The money-weighted rate of return (MWRR) considers all cash flows, such as withdrawals or contributions. If an investment spans multiple periods, MWRR gives more importance to the fund’s performance when the account is at its largest. This can be a problem for fund managers because it might make their performance seem worse due to factors they can’t control.

The time-weighted rate of return (TWRR) calculates the compound growth of an investment. Unlike the money-weighted rate, it doesn’t care about withdrawals or contributions. TWRR is like finding the average return of different time periods within your investment.

**Steps of Calculating Time-weighted Rate of Return**

**Step 1: **Value the portfolio immediately before any significant cash inflow or outflow of funds. Divide the evaluation period into subperiods based on dates of significant additions or withdrawals of funds.

**Step 2: **Compute the holding period return on the portfolio for each period.

**Step 3: **Compound or link the holding period returns to the annual rate of return, which is the time-weighted rate of return.

$$TWRR\ =\ {(1+{HPR}_1\times(1+{ HPR}_2)\times(1+{HPR}_3)\ldots\times(1+{HPR}_{n-1})\times(1+{HPR}_n)}\ – 1$$

If the evaluation period is more than one year, compute the geometric mean of the annual returns to get the time-weighted return for the investment period.

$$\begin{align}{\bar{R}}_{Gi}&=\sqrt[N]{\left(1+{\rm HPR}_1\ \right)\times\left(1+{\rm HPR}_1\ \right)\ldots\times\left(1+{\rm HPR}_n\right)}-1\\&=\left[\left(1+{\rm HPR}_1\ \right)\times\left(1+{\rm HPR}_1\ \right)\ldots\times\left(1+{\rm HPR}_n\right)\right]^\frac{1}{N}-1\end{align}$$

An investor purchases a share of stock at t = 0 for $200. At the end of the year (at t = 1), the investor purchases an additional share of the same stock, this time for $220. She then sells both shares at the end of the second year for $230 each. She also received annual dividends of $3 per share at the end of each year. Calculate the annual time-weighted rate of return on her investment.

**Solution**

First, we break down the two years into two one-year periods.

**Holding period 1:**

Beginning value = 200.

Dividends paid = 3.

Ending value = 220.

**Holding period 2:**

Beginning value = 440 (2 shares × 220)

Dividends paid = 6 (2 shares × 3)

Ending value = 460 (2 shares × 230)

Secondly, we calculate the HPR for each period:

$$\begin{align}{HPR}_1&=\frac{(220-200+3)}{200}=11.5\%\\{HPR}_2&=\frac{\left(460-440+6\right)}{440}=5.9\% \end{align}$$

Lastly, we need to find the geometric mean of the HPRs since we are dealing with a period of more than a year.

$$\begin{align}TWRR&=\left[\left(1+{\rm HPR}_1\ \right)\times\left(1+{\rm HPR}_1\ \right)\ldots\times\left(1+{\rm HPR}_n\right)\right]^\frac{1}{n}-1\\&=\left(1.115\times1.059\right)^{0.5}-1=8.7\%\end{align}$$

The beginning value of a portfolio as of January 1, 2020, was $1,000,000. On February 10, the portfolio’s value was $1,100,000, including an additional contribution of the $50,000 injected into the portfolio on this date. The portfolio’s ending value at the beginning of April was $1,350,000.

The time-weighted rate of return is *closest to:*

**Solution**

The time-weighted return is calculated as follows:

$$\begin{align}{HPR}_1&=\frac{V_1-V_0}{V_0}=\frac{\left(1,100,000-50,000\right)-1,000,000}{1,000,000}=5\%\\{HPR}_2&=\frac{V_2-V_1}{V_1}=\frac{1,350,000-1,100,000}{1,100,000}=22.73\%\\\Rightarrow TWRR &=\left(1+{HPR}_1\right)\times\left(1+{HPR}_1\right)-1\\&=1.05\times1.2273-1=28.87\%\end{align}$$

## Question

A chartered analyst buys a share of stock at time t = 0 for $50. At t = 1, he purchases an extra share of the same stock for $53. The share gives a dividend of $0.50 per share for the first year and $0.60 per share for the second year. He sells the shares at the end of the second year for $55 per share. Calculate the annual time-weighted rate of return.

A. 5.90%.

B.12.24%.

C. 7.00%.

The correct answer is

A.We have two one-year holding periods:

$$\begin{array}{cc}HP_1&HP_2\\P_0=50&P_0=106\\D=0.5&D=1.2\\P_1=53&P_1=110\\ \end{array}$$

We now calculate the holding period returns:

$$\begin{align}{HPR}_1&=\frac{(53-50+0.5)}{50}=7\%\\{HPR}_2&=\frac{\left(110-106+1.2\right)}{106}=4.9\%\\ \Rightarrow TWRR &=1.07\times1.049-1=12.24\%\end{align}$$

Therefore,

$$\text{Annual TWRR}={(1+0.1224)}^{0.5}-1=5.9\%$$