Financial Management AFIN253
Tutorial 2, Week 3
Compulsory question that will be collected and marked.
Question 143 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
An Australian company just issued two bonds:
- A 6-month zero coupon bond at a yield of 6% pa, and
- A 12 month zero coupon bond at a yield of 7% pa.
What is the company's forward rate from 6 to 12 months? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
Using the term structure of interest rates equation, also known as the expectations hypothesis theory of interest rates,
###\left(1+\frac{r_{\text{0}\rightarrow\text{12mth, apr 6mth}}}{2}\right)^2 = \left(1+\frac{r_{\text{0}\rightarrow\text{6mth, apr 6mth}}}{2}\right)^1 \left(1+\frac{r_{\text{6}\rightarrow\text{12mth apr 6mth}}}{2}\right)^1 ### ###\left(1+\frac{0.07}{2}\right)^2 = \left(1+\frac{0.06}{2}\right)^1 \left(1+\frac{r_{\text{6}\rightarrow\text{12mth apr 6mth}}}{2}\right)^1 ### ###\left(1+\frac{r_{\text{6}\rightarrow\text{12mth, apr 6mth}}}{2}\right)^1 = \frac{\left(1+\frac{0.07}{2}\right)^2}{\left(1+\frac{0.06}{2}\right)^1} ### ###\begin{aligned} r_{\text{6}\rightarrow\text{12mth, apr 6mth}} &= \left( \frac{\left(1+\frac{0.07}{2}\right)^2}{\left(1+\frac{0.06}{2}\right)^1} - 1 \right) \times 2 \\ &= 0.0800 \\ \end{aligned} ###
Tutorial 2, Week 3
Homework questions.
A 90-day Bank Accepted Bill (BAB) has a face value of $1,000,000. The simple interest rate is 10% pa and there are 365 days in the year. What is its price now?
###\begin{aligned} P_\text{0, bill} =& \frac{F_d}{1 + r_\text{simple} \times \frac{d}{365}} \\ =& \frac{1,000,000}{1 + 0.1 \times \frac{90}{365}} \\ =& 975,935.8289 \\ \end{aligned} ###
A four year bond has a face value of $100, a yield of 6% and a fixed coupon rate of 12%, paid semi-annually. What is its price?
###\begin{aligned} P_\text{0, bond, theoretical} =& \text{PV(annuity of coupons)} + \text{PV(face value)} \\ =& \text{coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{face}}{(1+r_\text{eff})^{T}} \\ =& \frac{100 \times 0.12}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{4\times2}} \right) + \frac{100}{(1+0.06/2)^{4 \times 2}} \\ =& 6 \times 7.01969219 + 78.94092343 \\ =& 42.11815314 + 78.94092343 \\ =& 121.0590766 \\ \end{aligned} ###
Which one of the following bonds is trading at par?
If a bond 'trades at par', it means that it's price is the same as its par value, and par value is a synonym of face value or principal. A par bonds' yield also equals its coupon rate.
Only answer (d) is correct since the bond's face value and price are equal so it is a par bond. The other bonds described in (a), (b) and (c) are all premium bonds.
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100), maturity (3 years) and yield (10%) as each other.
Which of the following statements is true?
Since bond A's coupon rate is equal to its yield, it must be a par bond. Similarly for bond B. So both are par bonds. Also note that they would have a price equal to their par value which is $100.
An 'interest only' loan can also be called a:
The price of an interest only loan (the amount paid by lender to borrower at the start) is not paid off at all until the very end. Only interest payments are made throughout the life of the loan.
Therefore the amount owing at the end of the loan will be equal to the original amount borrowed.
Since the amount at the end is called the principal, face value or par value, we say that the loan is priced at par. It's a par loan.
A firm wishes to raise $20 million now. They will issue 8% pa semi-annual coupon bonds that will mature in 5 years and have a face value of $100 each. Bond yields are 6% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue?
A common misunderstanding in this question is to divide the amount to be raised by the bond face value. This is wrong because the firm doesn't receive the face value at the start, actually it pays the face value at maturity.
To find the number of bonds that must be issued, divide the amount to be raised by the bond price since that's the cash flow that the issuing firm receives at the start.
To calculate the bond price,
###\begin{aligned} P_\text{0, bond} =& \text{PV(annuity of coupons)} + \text{PV(face value)} \\ =& \text{coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{face}}{(1+r_\text{eff})^{T}} \\ =& \frac{100 \times 0.08}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{5\times2}} \right) + \frac{100}{(1+0.06/2)^{5 \times 2}} \\ =& 34.12081135 + 74.40939149 \\ =& 108.5302028 \\ \end{aligned} ###
To find the number of bonds to issue right now:
###D_\text{0, new bonds} = P_\text{0,bond} . n_\text{bonds}######\begin{aligned} n_\text{bonds} =& \frac{D_\text{0, new bonds}}{P_\text{0,bond}} \\ =& \frac{$20m}{$108.5302028} \\ =& 0.1842805m \\ =& 184,280.5 \text{ bonds} \\ \end{aligned} ###
Note that issuing bonds is the same thing as selling bonds or lending. At the start the firm sells the bond contract in exchange for the bond price cash payment. At maturity, the firm will pay the bond face value to the lender. The lender can also be called the bond holder, investor or financier.
A share was bought for $30 (at t=0) and paid its annual dividend of $6 one year later (at t=1).
Just after the dividend was paid, the share price fell to $27 (at t=1). What were the total, capital and income returns given as effective annual rates?
The choices are given in the same order:
##r_\text{total}## , ##r_\text{capital}## , ##r_\text{dividend}##.
###\begin{aligned} r_\text{total} =& r_\text{capital} + r_\text{income} \\ =& \frac{P_1 - P_0}{P_0} + \frac{C_1}{P_0} \\ =& \frac{27 - 30}{30} + \frac{6}{30} \\ =& \frac{-3}{30} + \frac{6}{30} \\ =& -0.1 + 0.2 \\ \end{aligned}### So the capital return was -0.1 and the income return was 0.2. The total return is the sum: ### r_\text{total} = 0.1###
A share just paid its semi-annual dividend of $10. The dividend is expected to grow at 2% every 6 months forever. This 2% growth rate is an effective 6 month rate. Therefore the next dividend will be $10.20 in six months. The required return of the stock is 10% pa, given as an effective annual rate.
What is the price of the share now?
The dividend is paid every 6 months so we need to discount it using an effective 6 month rate. So first convert the effective annual rate to an effective 6 month rate.
###\begin{aligned} r_\text{eff 6mth} &= (1 + r_\text{eff annual})^{1/2}-1 \\ &= (1 + 0.1)^{1/2}-1 \\ &= 0.048808848 \\ \end{aligned}###
Applying the dividend discount model (DDM),
###\begin{aligned} P_0 &= \frac{C_\text{6mth}}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ &= \frac{C_\text{0}(1+g_\text{eff 6nth})^1}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ &= \frac{10(1+0.02)^1}{0.048808848 - 0.02} \\ &= 354.0578922 \\ \end{aligned}###
When using the dividend discount model to price a stock:
### p_{0} = \frac{d_1}{r - g} ###
The growth rate of dividends (g):
If the total return of the stock is more than the country's GDP (Gross Domestic Product) growth rate, and assuming that the average firm grows at the GDP growth rate, then capital return of the stock will be more than the average firm in perpetuity (forever). Therefore the firm must take over the country. This is very unlikely.
So the growth rate used in the dividend discount model should be less than the country's GDP growth rate.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
###p_0=\frac{d_1}{r_\text{eff}-g_\text{eff}}###
Which expression is NOT equal to the expected capital return?
Answer (d) is the dividend yield minus one (which is not very useful!). All of the other expressions will be equal to the firm's capital yield which is the same as the growth rate of the stock price and also the growth rate of the dividend, provided that the assumptions of the DDM hold.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
### p_0 = \frac{d_1}{r - g} ###
Which expression is NOT equal to the expected dividend yield?
Answer (e) is incorrect because it grows the stock price by the total return (r) instead of the capital return (g). While it is true that in between dividend payments the stock price grows by the total return (r), the dividend payments reduce the stock price. This means that the stock price grows by the total return (r) less the dividend yield (##d_{t+1}/p_t##) which equals the capital return (g) over a whole period.
This concept is best illustrated in a 'saw-tooth' graph of expected share price versus time.
Here's the Dividend Discount Model, used to price stocks:
### p_0=\frac{d_1}{r-g} ###
All rates are effective annual rates and the cash flows (##d_1##) are received every year. Note that the r and g terms in the above DDM could also be labelled: ###r = r_{\text{total, 0}\rightarrow\text{1yr, eff 1yr}}### ###g = r_{\text{capital, 0}\rightarrow\text{1yr, eff 1yr}}### Which of the following statements is NOT correct?
All statements are correct except (d). The total effective 3yr return over the 3 years must include dividends paid over all 3 years, not just the 3rd but also the 1st and 2nd. The correct expression would be answer (c) or:
###\begin{aligned} r_{\text{total, 0}\rightarrow\text{3yr, eff 3yr}} &= \dfrac{\text{(capital gain at t=3)}+\text{(future value of dividends at t=3)}}{\text{(price at t=0)}} \\ &= \dfrac{(p_3-p_0)+\left(d_1(1+r)^2+d_2(1+r)^1+d_3\right)}{p_0} \\ \end{aligned}###
This expression for total return will also equal the internal rate of return (IRR) of buying the stock, recieving the 3 dividends and selling the stock at the end of year 3. Note that this return will only be achievable if dividends can be re-invested in the stock, allowing them to grow at the stock's total rate of return r.
A stock is expected to pay the following dividends:
| Cash Flows of a Stock | ||||||
| Time (yrs) | 0 | 1 | 2 | 3 | 4 | ... |
| Dividend ($) | 0.00 | 1.00 | 1.05 | 1.10 | 1.15 | ... |
After year 4, the annual dividend will grow in perpetuity at 5% pa, so;
- the dividend at t=5 will be $1.15(1+0.05),
- the dividend at t=6 will be $1.15(1+0.05)^2, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in three and a half years (t = 3.5)?
###\begin{aligned} P_{3.5} =& \frac{C_4}{r-g} \times (1+r)^{0.5} \\ =& \frac{1.15}{0.1- 0.05} \times (1+0.1)^{0.5} \\ =& 23 \times (1+0.1)^{0.5} \\ =& 24.12260351 \\ \end{aligned} ###
Note that ## \frac{C_4}{r-g} ## is ## P_3 ##, and this amount is grown forward half a period by the total return (r), not the capital return (g). This may seem counter-intuitive since you would normally grow the dividend or the price forward by g. But in this case the stock price needs to be grown by the total return (r) because we are growing the price in between dividend payments. The stock price needs to grow by the higher total return so it is big enough to pay the dividend and fall in price, but still have realised a capital return (g).
Another way of thinking about it is that the growth rate in the stock price between t=3 and 3.5 needs to include not just the capital growth (g), but also the accrued dividend which will be paid at t=4 and which is part of the stock price until it is paid.
This concept is best illustrated by the 'saw-tooth' graph of expected share price vs time.