Question
TEME is a manufacturer of toy construction equipment. If it pays out all of its earnings as dividends, it will have earnings of 0.3 million per quarter in perpetuity. Suppose that the discount rate, expressed as an effective annual rate (EAR), is 16%. TEME pays dividends quarterly.
Suppose that TEME is considering a one-time expansion into toy xylophones. It is estimated that this will cost 1M. Assume that this cost will be incurred at the end of the year, one year from now. As a result of expansion, earnings in subsequent quarters (i.e. starting in 1 year and 1 quarter from now) would be 0.05 million higher than without the expansion. Calculate the value of TEME if it undertakes the investment.
*Make sure to input the answer without any currency symbols, commas, and remove M as Million value.
e.g. 6,000,000 = 6M, so the answer should be written as 6
Suppose that TEME is considering a one-time expansion into toy xylophones. It is estimated that this will cost 1M. Assume that this cost will be incurred at the end of the year, one year from now. As a result of expansion, earnings in subsequent quarters (i.e. starting in 1 year and 1 quarter from now) would be 0.05 million higher than without the expansion. Calculate the value of TEME if it undertakes the investment.
*Make sure to input the answer without any currency symbols, commas, and remove M as Million value.
e.g. 6,000,000 = 6M, so the answer should be written as 6
Answers
GPT-4o mini
To calculate the value of TEME if it undertakes the investment in toy xylophones, we need to consider both the current value of the company without the expansion and the additional value that the expansion brings after the initial investment.
### Step 1: Calculate the current value of dividends
**Earnings per quarter:** 0.3 million
**Annualized earnings:** \(0.3 \text{ million} \times 4 = 1.2 \text{ million}\)
**Discount rate (qt):** Given that the effective annual rate (EAR) is 16%, we need to convert this to a quarterly rate.
The quarterly discount rate can be calculated as follows:
\[
(1 + EAR)^{(1/4)} - 1 = (1 + 0.16)^{(1/4)} - 1
\]
Calculating this:
\[
(1.16)^{0.25} \approx 1.0372 - 1 \approx 0.0372 \quad (\text{quarterly rate})
\]
Thus, \(r \approx 0.0372\).
Using this quarterly rate, the present value of the dividends can be calculated using the perpetuity formula:
\[
PV = \frac{D}{r} = \frac{0.3 \text{ million}}{0.0372}
\]
Calculating this gives:
\[
PV \approx 8.0645 \text{ million}
\]
### Step 2: Calculate the value after expansion
**Increased earnings per quarter:** \(0.3 \text{ million} + 0.05 \text{ million} = 0.35 \text{ million}\)
**New annualized earnings:** \(0.35 \text{ million} \times 4 = 1.4 \text{ million}\)
The present value of the future earnings post-expansion, starting from 1 year and 1 quarter (or 5 quarters) from now, must be calculated. First, we find the present value of the new earnings at the time just after expansion (which happens at period 0 of the new earnings).
Using the same perpetuity formula:
\[
PV_{\text{new}} = \frac{0.35 \text{ million}}{0.0372} \approx 9.3968 \text{ million}
\]
Now, this value is at the end of period 5 (4 quarters for 1 year, plus the 1 additional quarter until earnings start). We need to calculate its present value back to today (time 0).
Using the present value formula:
\[
PV_{\text{return back to today}} = \frac{PV_{\text{new}}}{(1 + r)^5}
\]
Calculating:
\[
PV_{\text{return back to today}} = \frac{9.3968 \text{ million}}{(1.0372)^5} \approx \frac{9.3968}{1.1913} \approx 7.8673 \text{ million}
\]
### Step 3: Calculate total value after the investment
Now we need to account for the investment cost of 1 million incurred at the end of the year (which is at time period 4).
So, the total value after the investment becomes:
\[
\text{Final value} = PV_{\text{current}} + PV_{\text{return back to today}} - \text{Investment}
\]
Calculating that:
\[
\text{Final value} = 8.0645 \text{ million} + 7.8673 \text{ million} - 1 \text{ million} \approx 14.9318 \text{ million}
\]
### Final Answer
Thus, without any currency symbols or commas, the answer is:
14.9318
### Step 1: Calculate the current value of dividends
**Earnings per quarter:** 0.3 million
**Annualized earnings:** \(0.3 \text{ million} \times 4 = 1.2 \text{ million}\)
**Discount rate (qt):** Given that the effective annual rate (EAR) is 16%, we need to convert this to a quarterly rate.
The quarterly discount rate can be calculated as follows:
\[
(1 + EAR)^{(1/4)} - 1 = (1 + 0.16)^{(1/4)} - 1
\]
Calculating this:
\[
(1.16)^{0.25} \approx 1.0372 - 1 \approx 0.0372 \quad (\text{quarterly rate})
\]
Thus, \(r \approx 0.0372\).
Using this quarterly rate, the present value of the dividends can be calculated using the perpetuity formula:
\[
PV = \frac{D}{r} = \frac{0.3 \text{ million}}{0.0372}
\]
Calculating this gives:
\[
PV \approx 8.0645 \text{ million}
\]
### Step 2: Calculate the value after expansion
**Increased earnings per quarter:** \(0.3 \text{ million} + 0.05 \text{ million} = 0.35 \text{ million}\)
**New annualized earnings:** \(0.35 \text{ million} \times 4 = 1.4 \text{ million}\)
The present value of the future earnings post-expansion, starting from 1 year and 1 quarter (or 5 quarters) from now, must be calculated. First, we find the present value of the new earnings at the time just after expansion (which happens at period 0 of the new earnings).
Using the same perpetuity formula:
\[
PV_{\text{new}} = \frac{0.35 \text{ million}}{0.0372} \approx 9.3968 \text{ million}
\]
Now, this value is at the end of period 5 (4 quarters for 1 year, plus the 1 additional quarter until earnings start). We need to calculate its present value back to today (time 0).
Using the present value formula:
\[
PV_{\text{return back to today}} = \frac{PV_{\text{new}}}{(1 + r)^5}
\]
Calculating:
\[
PV_{\text{return back to today}} = \frac{9.3968 \text{ million}}{(1.0372)^5} \approx \frac{9.3968}{1.1913} \approx 7.8673 \text{ million}
\]
### Step 3: Calculate total value after the investment
Now we need to account for the investment cost of 1 million incurred at the end of the year (which is at time period 4).
So, the total value after the investment becomes:
\[
\text{Final value} = PV_{\text{current}} + PV_{\text{return back to today}} - \text{Investment}
\]
Calculating that:
\[
\text{Final value} = 8.0645 \text{ million} + 7.8673 \text{ million} - 1 \text{ million} \approx 14.9318 \text{ million}
\]
### Final Answer
Thus, without any currency symbols or commas, the answer is:
14.9318