# Section 4C

### exercises

...

#### Stocks

Price-to-Earning Ratio. Exercise 42, p. 234

General Mills close at $52.65 per share with a P/E ratio of 16.14 (a) How much were earnings per share? Solution. Recall that $$\text{P/E ratio} = \frac{\text{share price}}{\text{earnings per share }}$$ Thus in this case $$16.14= \frac{52.65}{\text{earnings per share }}$$ and $${\text{earnings per share }} = \frac{52.65}{16.14} \approx \ 3.26$$ #### Price-to-Earning Ratio. Exercise 42, p. 234 ... continued (b) Does the stock seem to be overpriced, underpriced, or about right given that the historical P/E ratio is 12 - 14? Solution. The P/E ratio of 16.14 is higher than the historical numbers 12-14. Recall that $$\text{P/E ratio} = \frac{\text{share price}}{\text{earnings per share }}$$ With given earning per share, the price must be lower in order to have the P/E ratio within the historical interval. The stock is overpriced. For example, if the P/E ratio was 13, the share price would be $$\text{share price} = \text{P/E ratio} \times \text{earnings per share } = 13 \times 3.26 = 42.38$$ #### Stocks Price-to-Earning Ratio. Exercise 44, p. 234 Google closed at$393.50 with P/E ratio of 28.78

(a) How much were earnings per share?

Solution. Recall that
$$\text{P/E ratio} = \frac{\text{share price}}{\text{earnings per share }}$$
Thus in this case $$28.78= \frac{393.50}{\text{earnings per share }}$$
and
$${\text{earnings per share }} = \frac{393.50}{28.78} \approx \ 13.67$$

#### Price-to-Earning Ratio. Exercise 44, p. 234 ... continued

(b) Does the stock seem to be overpriced, underpriced, or about right given that the historical P/E ratio is 12 - 14?

Solution. The P/E ratio of 28.78 is higher than the historical numbers 12-14. Recall that
$$\text{P/E ratio} = \frac{\text{share price}}{\text{earnings per share }}$$
With given earning per share, the price must be lower in order to have the P/E ratio within the historical interval. The stock is overpriced. For example, if the P/E ratio was 13, the share price would be
$$\text{share price} = \text{P/E ratio} \times \text{earnings per share } = 13 \times 13.67 = 177.71$$

#### Bond yields

Exercise 48, p. 234

Compute the current yield of a $$\ 1000$$ Treasury bond with coupon rate of 2.5% that has a market value of $$\ 1050$$.

Solution. Recall that
$$\text{current yield} = \frac{\text{annual interest payment}}{\text{current price of bond}}$$
While the current price of the bond is $1050, the annual interest payment = face value $$\times$$ coupon rate = $$1000 \times 0.025 = 25$$ Thus $$\text{current yield} = \frac{25}{1050} \approx 0.0238 = 2.38\%.$$ #### Bond yields. Exercise 50, p. 234 Compute the current yield of a $$\ 10,000$$ Treasury bond with coupon rate of 3% that has a market value of $$\ 9,500$$. Solution. Recall that $$\text{current yield} = \frac{\text{annual interest payment}}{\text{current price of bond}}$$ While the current price of the bond is$ 9500,
the annual interest payment = face value $$\times$$ coupon rate = $$10000 \times 0.03 = 300$$
Thus
$$\text{current yield} = \frac{300}{9500} \approx 0.0316 = 3.16\%.$$

#### Bond interest

Exercise 52, p. 234

Compute the annual interest that you would earn with a $1000 Treasury bond with a current yield of 1.5% that is quoted at 98 points Solution. Recall that $$\text{current yield} = \frac{\text{annual interest }}{\text{current price }}$$ Thus $$\text{annual interest } = \text{current yield} \times \text{current price }$$ While the current yield is 1.5% = 0.015, current price = 98% of the face value = 0.98 $$\times$$1000 =$980
Thus $$\text{annual interest } = 0.015 \times 980 = \ 14.7$$

#### Bond interest

Exercise 54, p. 234

Compute the annual interest that you would earn with a $10,000 Treasury bond with a current yield of 3.6% that is quoted at 102.5 points Recall that $$\text{current yield} = \frac{\text{annual interest }}{\text{current price }}$$ Thus $$\text{annual interest } = \text{current yield} \times \text{current price }$$ While the current yield is 3.6% = 0.036, current price = 102.5% of the face value = 1.025 $$\times$$10000 =$10250
Thus $$\text{annual interest } = 0.036 \times 10250 = \ 369$$

#### Further applications. Savings plans

Exercise 58, p. 235

Polly deposits $$\ 50$$ per month in an account with an APR of 6%.
Quint deposits deposits $$\ 40$$ per month in an account with an APR of 6.5%.

Compare the balances, and the amounts deposited after 10 years.

#### Exercise 58, p. 235 ... continued

Solution. We firstly calculate the amounts deposited.

Polly deposits $$\ 50$$ per month during 10 years:

$$50 \times 12 \times 10 = \ 6,000$$

Quint deposits $$\ 40$$ per month during 10 years:

$$40 \times 12 \times 10 = \ 4,800$$

Polly deposits bigger amount of money then Quint.

#### Exercise 58, p. 235 ... continued

We now calculate their accumulated balances using Savings Plan Formula

$$A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)}$$

Polly: $$PMT=50, \ \ \ APR=0.06 \ \ \ n=12, \ \ \ Y=10$$:

$$A = 50 \times \frac{ \left[ \left( 1 + \frac{0.06}{12} \right)^{(12 \times 10)} -1 \right] }{\left( \frac{0.06}{12} \right)}=\ 8,193.97$$

Quint: $$PMT=40, \ \ \ APR=0.065 \ \ \ n=12, \ \ \ Y=10$$:

$$A = 40 \times \frac{ \left[ \left( 1 + \frac{0.065}{12} \right)^{(12 \times 10)} -1 \right] }{\left( \frac{0.065}{12} \right)}=\ 6,736.13$$

#### Exercise 58, p. 235 ... continued

Summary

Over the period of 10 years, Polly deposited $$\ 6,000$$ and accumulated $$\ 8,193.97$$

Over the period of 10 years, Quint deposited $$\ 4,800$$, and accumulated $$\ 6,736.13$$

It is interesting to calculate the total return in both cases

Polly's total return = $$\frac{8193.97-6000}{6000} \times 100\% \approx 36.6 \%$$

Quint's total return = $$\frac{6736.13 -4800}{4800} \times 100 \% \approx 40.3 \%$$

That is because Quint's APR was higher!

#### Further applications. Savings plans

Exercise 60, p. 235

George deposits $$\ 40$$ per month in an account with an APR of 7%.
Harvey deposits deposits $$\ 150$$ per quarter in an account with an APR of 7.5%.

Compare the balances, and the amounts deposited after 10 years.

#### Exercise 60, p. 235 ... continued

Solution. We firstly calculate the amounts deposited.

George deposits $$\ 40$$ per month during 10 years:

$$40 \times 12 \times 10 = \ 4,800$$

Harvey deposits $$\ 150$$ per quarter during 10 years:

$$150 \times 4 \times 10 = \ 6,000$$

Harvey deposits bigger amount of money then George.

#### Exercise 60, p. 235 ... continued

We now calculate their accumulated balances using Savings Plan Formula

$$A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)}$$

George: $$PMT=40, \ \ \ APR=0.07 \ \ \ n=12, \ \ \ Y=10$$:

$$A = 40 \times \frac{ \left[ \left( 1 + \frac{0.07}{12} \right)^{(12 \times 10)} -1 \right] }{\left( \frac{0.07}{12} \right)}=\ 6,923.39$$

Harvey: $$PMT=150, \ \ \ APR=0.075 \ \ \ n=4, \ \ \ Y=10$$:

$$A = 150 \times \frac{ \left[ \left( 1 + \frac{0.075}{4} \right)^{(4 \times 10)} -1 \right] }{\left( \frac{0.075}{4} \right)}=\ 8,818.79$$

#### Exercise 60, p. 235 ... continued

Summary

Over the period of 10 years, George deposited $$\ 4,800$$ and accumulated $$\ 6,923.39$$

Over the period of 10 years, Harvey deposited $$\ 6,000$$, and accumulated $$\ 8,818.7$$

It is interesting to calculate the total return in both cases

George's total return = $$\frac{6923.39-4800}{4800} \times 100\% \approx 44.2\%$$

Harvey's total return = $$\frac{8818.7 -6000}{6000} \times 100 \% \approx 46.9 \%$$

That is bcause Harvey's APR was higher.