# Fight Finance

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The following equation is called the Dividend Discount Model (DDM), Gordon Growth Model or the perpetuity with growth formula: $$P_0 = \frac{ C_1 }{ r - g }$$

What is $g$? The value $g$ is the long term expected:

The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.

$$p_{0} = \frac{c_1}{r_{\text{eff}} - g_{\text{eff}}}$$

What is the discount rate '$r_\text{eff}$' in this equation?

A fixed coupon bond was bought for $90 and paid its annual coupon of$3 one year later (at t=1 year). Just after the coupon was paid, the bond price was $92 (at t=1 year). What was the total return, capital return and income return? Calculate your answers as effective annual rates. The choices are given in the same order: $r_\text{total},r_\text{capital},r_\text{income}$. The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation. $$P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}}$$ What would you call the expression $C_1/P_0$? A share was bought for$20 (at t=0) and paid its annual dividend of $3 one year later (at t=1). Just after the dividend was paid, the share price was$16 (at t=1). What was the total return, capital return and income return? Calculate your answers as effective annual rates.

The choices are given in the same order: $r_\text{total},r_\text{capital},r_\text{income}$.

The following is the Dividend Discount Model (DDM) used to price stocks:

$$P_0 = \frac{d_1}{r-g}$$

Assume that the assumptions of the DDM hold and that the time period is measured in years.

Which of the following is equal to the expected dividend in 3 years, $d_3$?

Which of the following statements about risk free government bonds is NOT correct?

Hint: Total return can be broken into income and capital returns as follows:

\begin{aligned} r_\text{total} &= \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0} \\ &= r_\text{income} + r_\text{capital} \end{aligned}

The capital return is the growth rate of the price.
The income return is the periodic cash flow. For a bond this is the coupon payment.

A stock was bought for $8 and paid a dividend of$0.50 one year later (at t=1 year). Just after the dividend was paid, the stock price was $7 (at t=1 year). What were the total, capital and dividend returns given as effective annual rates? The choices are given in the same order: $r_\text{total}$, $r_\text{capital}$, $r_\text{dividend}$. When using the dividend discount model to price a stock: $$p_{0} = \frac{d_1}{r - g}$$ The growth rate of dividends (g): 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? 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:

$r_\text{total}$ , $r_\text{capital}$ , $r_\text{dividend}$.

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?

A share was bought for $10 (at t=0) and paid its annual dividend of$0.50 one year later (at t=1). Just after the dividend was paid, the share price was $11 (at t=1). What was the total return, capital return and income return? Calculate your answers as effective annual rates. The choices are given in the same order: $r_\text{total}$, $r_\text{capital}$, $r_\text{dividend}$. The following is the Dividend Discount Model used to price stocks: $$p_0=\frac{d_1}{r-g}$$ Which of the following statements about the Dividend Discount Model is NOT correct? The following is 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 as below: $$r = r_{\text{total, 0}\rightarrow\text{1yr, eff 1yr}}$$ $$g = r_{\text{capital, 0}\rightarrow\text{1yr, eff 1yr}}$$ Which of the following statements is NOT correct? The following is the Dividend Discount Model (DDM) used to price stocks: $$P_0=\dfrac{C_1}{r-g}$$ If the assumptions of the DDM hold, which one of the following statements is NOT correct? The long term expected: For a bond that pays fixed semi-annual coupons, how is the annual coupon rate defined, and how is the bond's annual income yield from time 0 to 1 defined mathematically? Let: $P_0$ be the bond price now, $F_T$ be the bond's face value, $T$ be the bond's maturity in years, $r_\text{total}$ be the bond's total yield, $r_\text{income}$ be the bond's income yield, $r_\text{capital}$ be the bond's capital yield, and $C_t$ be the bond's coupon at time t in years. So $C_{0.5}$ is the coupon in 6 months, $C_1$ is the coupon in 1 year, and so on. Assume that the Gordon Growth Model (same as the dividend discount model or perpetuity with growth formula) is an appropriate method to value real estate. The rule of thumb in the real estate industry is that properties should yield a 5% pa rental return. Many investors also regard property to be as risky as the stock market, therefore property is thought to have a required total return of 9% pa which is the average total return on the stock market including dividends. Assume that all returns are effective annual rates and they are nominal (not reduced by inflation). Inflation is expected to be 2% pa. You're considering purchasing an investment property which has a rental yield of 5% pa and you expect it to have the same risk as the stock market. Select the most correct statement about this property. The coupon rate of a fixed annual-coupon bond is constant (always the same). What can you say about the income return ($r_\text{income}$) of a fixed annual coupon bond? Remember that: $$r_\text{total} = r_\text{income} + r_\text{capital}$$ $$r_\text{total, 0 to 1} = \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0}$$ Assume that there is no change in the bond's total annual yield to maturity from when it is issued to when it matures. Select the most correct statement. From its date of issue until maturity, the income return of a fixed annual coupon: A bank grants a borrower an interest-only residential mortgage loan with a very large 50% deposit and a nominal interest rate of 6% that is not expected to change. Assume that inflation is expected to be a constant 2% pa over the life of the loan. Ignore credit risk. From the bank's point of view, what is the long term expected nominal capital return of the loan asset? The total return of any asset can be broken down in different ways. One possible way is to use the dividend discount model (or Gordon growth model): $$p_0 = \frac{c_1}{r_\text{total}-r_\text{capital}}$$ Which, since $c_1/p_0$ is the income return ($r_\text{income}$), can be expressed as: $$r_\text{total}=r_\text{income}+r_\text{capital}$$ So the total return of an asset is the income component plus the capital or price growth component. Another way to break up total return is to use the Capital Asset Pricing Model: $$r_\text{total}=r_\text{f}+β(r_\text{m}- r_\text{f})$$ $$r_\text{total}=r_\text{time value}+r_\text{risk premium}$$ So the risk free rate is the time value of money and the term $β(r_\text{m}- r_\text{f})$ is the compensation for taking on systematic risk. Using the above theory and your general knowledge, which of the below equations, if any, are correct? (I) $r_\text{income}=r_\text{time value}$ (II) $r_\text{income}=r_\text{risk premium}$ (III) $r_\text{capital}=r_\text{time value}$ (IV) $r_\text{capital}=r_\text{risk premium}$ (V) $r_\text{income}+r_\text{capital}=r_\text{time value}+r_\text{risk premium}$ Which of the equations are correct? A share was bought for$4 and paid an dividend of $0.50 one year later (at t=1 year). Just after the dividend was paid, the share price fell to$3.50 (at t=1 year). What were the total return, capital return and income returns given as effective annual rates? The answer choices are given in the same order:

$r_\text{total}$, $r_\text{capital}$, $r_\text{income}$

A 90-day $1 million Bank Accepted Bill (BAB) was bought for$990,000 and sold 30 days later for $996,000 (at t=30 days). What was the total return, capital return and income return over the 30 days it was held? Despite the fact that money market instruments such as bills are normally quoted with simple interest rates, please calculate your answers as compound interest rates, specifically, as effective 30-day rates, which is how the below answer choices are listed. $r_\text{total}$, $r_\text{capital}$, $r_\text{income}$ A company's shares just paid their annual dividend of$2 each.

The stock price is now $40 (just after the dividend payment). The annual dividend is expected to grow by 3% every year forever. The assumptions of the dividend discount model are valid for this company. What do you expect the effective annual dividend yield to be in 3 years (dividend yield from t=3 to t=4)? You're the boss of an investment bank's equities research team. Your five analysts are each trying to find the expected total return over the next year of shares in a mining company. The mining firm: • Is regarded as a mature company since it's quite stable in size and was floated around 30 years ago. It is not a high-growth company; • Share price is very sensitive to changes in the price of the market portfolio, economic growth, the exchange rate and commodities prices. Due to this, its standard deviation of total returns is much higher than that of the market index; • Experienced tough times in the last 10 years due to unexpected falls in commodity prices. • Shares are traded in an active liquid market. Your team of analysts present their findings, and everyone has different views. While there's no definitive true answer, who's calculation of the expected total return is the most plausible? Assume that: • The analysts' source data is correct and true, but their inferences might be wrong; • All returns and yields are given as effective annual nominal rates. The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation. $$p_0= \frac{c_1}{r-g}$$ Which expression is equal to the expected dividend return? Economic statistics released this morning were a surprise: they show a strong chance of consumer price inflation (CPI) reaching 5% pa over the next 2 years. This is much higher than the previous forecast of 3% pa. A vanilla fixed-coupon 2-year risk-free government bond was issued at par this morning, just before the economic news was released. What is the expected change in bond price after the economic news this morning, and in the next 2 years? Assume that: • Inflation remains at 5% over the next 2 years. • Investors demand a constant real bond yield. • The bond price falls by the (after-tax) value of the coupon the night before the ex-coupon date, as in real life. Two years ago Fred bought a house for$300,000.

Now it's worth $500,000, based on recent similar sales in the area. Fred's residential property has an expected total return of 8% pa. He rents his house out for$2,000 per month, paid in advance. Every 12 months he plans to increase the rental payments.

The present value of 12 months of rental payments is $23,173.86. The future value of 12 months of rental payments one year ahead is$25,027.77.

What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%?

A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 3% pa.

Inflation is expected to be 2% pa. All rates are given as effective annual rates.

What are the property's expected real total, capital and income returns? The answer choices below are given in the same order.

Three years ago Frederika bought a house for $400,000. Now it's worth$600,000, based on recent similar sales in the area.

Frederika's residential property has an expected total return of 7% pa.

She rents her house out for $2,500 per month, paid in advance. Every 12 months she plans to increase the rental payments. The present value of 12 months of rental payments is$29,089.48.

The future value of 12 months of rental payments one year ahead is $31,125.74. What is the expected annual capital yield of the property? A residential investment property has an expected nominal total return of 8% pa and nominal capital return of 3% pa. Inflation is expected to be 2% pa. All rates are given as effective annual rates. What are the property's expected real total, capital and income returns? The answer choices below are given in the same order. One and a half years ago Frank bought a house for$600,000. Now it's worth only $500,000, based on recent similar sales in the area. The expected total return on Frank's residential property is 7% pa. He rents his house out for$1,600 per month, paid in advance. Every 12 months he plans to increase the rental payments.

The present value of 12 months of rental payments is $18,617.27. The future value of 12 months of rental payments one year in the future is$19,920.48.

What is the expected annual rental yield of the property? Ignore the costs of renting such as maintenance, real estate agent fees and so on.

The perpetuity with growth formula is:

$$P_0= \dfrac{C_1}{r-g}$$

Which of the following is NOT equal to the total required return (r)?

A stock has a real expected total return of 7% pa and a real expected capital return of 2% pa.

Inflation is expected to be 2% pa. All rates are given as effective annual rates.

What is the nominal expected total return, capital return and dividend yield? The answers below are given in the same order.

You just bought a residential apartment as an investment property for $500,000. You intend to rent it out to tenants. They are ready to move in, they would just like to know how much the monthly rental payments will be, then they will sign a twelve-month lease. You require a total return of 8% pa and a rental yield of 5% pa. What would the monthly paid-in-advance rental payments have to be this year to receive that 5% annual rental yield? Also, if monthly rental payments can be increased each year when a new lease agreement is signed, by how much must you increase rents per year to realise the 8% pa total return on the property? Ignore all taxes and the costs of renting such as maintenance costs, real estate agent fees, utilities and so on. Assume that there will be no periods of vacancy and that tenants will promptly pay the rental prices you charge. Note that the first rental payment will be received at t=0. The first lease agreement specifies the first 12 equal payments from t=0 to 11. The next lease agreement can have a rental increase, so the next twelve equal payments from t=12 to 23 can be higher than previously, and so on forever. A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever. All rates are given as nominal effective annual returns. Assume that: • His forecast is true. • Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true. • Ignore all costs such as taxes, agent fees, maintenance and so on. • All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property. • The non-monetary benefits of owning real estate and renting remain constant. Which one of the following statements is NOT correct? Over time: A fairly valued share's current price is$4 and it has a total required return of 30%. Dividends are paid annually and next year's dividend is expected to be $1. After that, dividends are expected to grow by 5% pa in perpetuity. All rates are effective annual returns. What is the expected dividend income paid at the end of the second year (t=2) and what is the expected capital gain from just after the first dividend (t=1) to just after the second dividend (t=2)? The answers are given in the same order, the dividend and then the capital gain. The perpetuity with growth equation is: $$P_0=\dfrac{C_1}{r-g}$$ Which of the following is NOT equal to the expected capital return as an effective annual rate? A fairly priced unlevered firm plans to pay a dividend of$1 next year (t=1) which is expected to grow by 3% pa every year after that. The firm's required return on equity is 8% pa.

The firm is thinking about reducing its future dividend payments by 10% so that it can use the extra cash to invest in more projects which are expected to return 8% pa, and have the same risk as the existing projects. Therefore, next year's dividend will be $0.90. No new equity or debt will be issued to fund the new projects, they'll all be funded by the cut in dividends. What will be the stock's new annual capital return (proportional increase in price per year) if the change in payout policy goes ahead? Assume that payout policy is irrelevant to firm value (so there's no signalling effects) and that all rates are effective annual rates. The saying "buy low, sell high" suggests that investors should make a: An asset's total expected return over the next year is given by: $$r_\text{total} = \dfrac{c_1+p_1-p_0}{p_0}$$ Where $p_0$ is the current price, $c_1$ is the expected income in one year and $p_1$ is the expected price in one year. The total return can be split into the income return and the capital return. Which of the following is the expected capital return? Total cash flows can be broken into income and capital cash flows. What is the name given to the income cash flow from owning shares? Two companies BigDiv and ZeroDiv are exactly the same except for their dividend payouts. BigDiv pays large dividends and ZeroDiv doesn't pay any dividends. Currently the two firms have the same earnings, assets, number of shares, share price, expected total return and risk. Assume a perfect world with no taxes, no transaction costs, no asymmetric information and that all assets including business projects are fairly priced and therefore zero-NPV. All things remaining equal, which of the following statements is NOT correct? A stock will pay you a dividend of$10 tonight if you buy it today. Thereafter the annual dividend is expected to grow by 5% pa, so the next dividend after the $10 one tonight will be$10.50 in one year, then in two years it will be $11.025 and so on. The stock's required return is 10% pa. What is the stock price today and what do you expect the stock price to be tomorrow, approximately? Which of the following equations is NOT equal to the total return of an asset? Let $p_0$ be the current price, $p_1$ the expected price in one year and $c_1$ the expected income in one year. A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 2.5% pa. Inflation is expected to be 2.5% pa. All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity. What are the property's expected real total, capital and income returns? The answer choices below are given in the same order. A low-growth mature stock has an expected nominal total return of 6% pa and nominal capital return of 2% pa. Inflation is expected to be 3% pa. All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity. What are the stock's expected real total, capital and income returns? The answer choices below are given in the same order. Which of the following statements about cash in the form of notes and coins is NOT correct? Assume that inflation is positive. Notes and coins: Total cash flows can be broken into income and capital cash flows. What is the name given to the cash flow generated from selling shares at a higher price than they were bought? The perpetuity with growth formula, also known as the dividend discount model (DDM) or Gordon growth model, is appropriate for valuing a company's shares. $P_0$ is the current share price, $C_1$ is next year's expected dividend, $r$ is the total required return and $g$ is the expected growth rate of the dividend. $$P_0=\dfrac{C_1}{r-g}$$ The below graph shows the expected future price path of the company's shares. Which of the following statements about the graph is NOT correct? Risk-free government bonds that have coupon rates greater than their yields: For an asset price to double every 10 years, what must be the expected future capital return, given as an effective annual rate? For an asset price to triple every 5 years, what must be the expected future capital return, given as an effective annual rate? Which of the following statements about the capital and income returns of a 25 year fully amortising loan asset is correct? Assume that the yield curve (which shows total returns over different maturities) is flat and is not expected to change. Over the 25 years from issuance to maturity, a fully amortising loan's expected annual effective: Which of the following statements about the capital and income returns of an interest-only loan is correct? Assume that the yield curve (which shows total returns over different maturities) is flat and is not expected to change. An interest-only loan's expected: A firm pays out all of its earnings as dividends. Because of this, the firm has no real growth in earnings, dividends or stock price since there is no re-investment back into the firm to buy new assets and make higher earnings. The dividend discount model is suitable to value this company. The firm's revenues and costs are expected to increase by inflation in the foreseeable future. The firm has no debt. It operates in the services industry and has few physical assets so there is negligible depreciation expense and negligible net working capital required. Which of the following statements about this firm's PE ratio is NOT correct? The PE ratio should: Note: The inverse of x is 1/x. An investor bought a 10 year 2.5% pa fixed coupon government bond priced at par. The face value is$100. Coupons are paid semi-annually and the next one is in 6 months.

Six months later, just after the coupon at that time was paid, yields suddenly and unexpectedly fell to 2% pa. Note that all yields above are given as APR's compounding semi-annually.

What was the bond investors' historical total return over that first 6 month period, given as an effective semi-annual rate?

An investor bought a 20 year 5% pa fixed coupon government bond priced at par. The face value is $100. Coupons are paid semi-annually and the next one is in 6 months. Six months later, just after the coupon at that time was paid, yields suddenly and unexpectedly rose to 5.5% pa. Note that all yields above are given as APR's compounding semi-annually. What was the bond investors' historical total return over that first 6 month period, given as an effective semi-annual rate? Let the 'income return' of a bond be the coupon at the end of the period divided by the market price now at the start of the period $(C_1/P_0)$. The expected income return of a premium fixed coupon bond is: Which of the following statements about gold is NOT correct? Assume that the gold price increases by inflation. Gold: A stock’s current price is$1. Its expected total return is 10% pa and its long term expected capital return is 4% pa. It pays an annual dividend and the next one will be paid in one year. All rates are given as effective annual rates. The dividend discount model is thought to be a suitable model for the stock. Ignore taxes. Which of the following statements about the stock is NOT correct?

In the dividend discount model (DDM), share prices fall when dividends are paid. Let the high price before the fall be called the peak, and the low price after the fall be called the trough.

$$P_0=\dfrac{C_1}{r-g}$$

Which of the following statements about the DDM is NOT correct?

An investor bought a bond for $100 (at t=0) and one year later it paid its annual coupon of$1 (at t=1). Just after the coupon was paid, the bond price was $100.50 (at t=1). Inflation over the past year (from t=0 to t=1) was 3% pa, given as an effective annual rate. Which of the following statements is NOT correct? The bond investment produced a: A share’s current price is$60. It’s expected to pay a dividend of $1.50 in one year. The growth rate of the dividend is 0.5% pa and the stock’s required total return is 3% pa. The stock’s price can be modeled using the dividend discount model (DDM): $P_0=\dfrac{C_1}{r-g}$ Which of the following methods is NOT equal to the stock’s expected price in one year and six months (t=1.5 years)? Note that the symbolic formulas shown in each line below do equal the formulas with numbers. The formula is just repeated with symbols and then numbers in case it helps you to identify the incorrect statement more quickly. If someone says "my shares rose by 10% last year", what do you assume that they mean? If the nominal gold price is expected to increase at the same rate as inflation which is 3% pa, which of the following statements is NOT correct? A stock will pay you a dividend of$2 tonight if you buy it today.

Thereafter the annual dividend is expected to grow by 3% pa, so the next dividend after the $2 one tonight will be$2.06 in one year, then in two years it will be $2.1218 and so on. The stock's required return is 8% pa. What is the stock price today and what do you expect the stock price to be tomorrow, approximately? Which of the following statements is NOT correct? Assume that all things remain equal. So for example, don't assume that just because a company's dividends and profit rise that its required return will also rise, assume the required return stays the same. Which of the following statements about returns is NOT correct? A stock's: Over the last year, a constant-dividend-paying stock's price fell, while it's future expected dividends and profit remained the same. Assume that: • Now is $t=0$, last year is $t=-1$ and next year is $t=1$; • The dividend is paid at the end of each year, the last dividend was just paid today $(C_0)$ and the next dividend will be paid next year $(C_1)$; • Markets are efficient and the dividend discount model is suitable for valuing the stock. Which of the following statements is NOT correct? The stock's: An Apple (NASDAQ:AAPL) stock was purchased by an investor for$120 and one year later was sold for $150. A dividend of$4 was also collected at the end of the year just before the stock was sold.

Which of the following statements about the stock investment is NOT correct? Ignore taxes.

Over the year, the investor made a:

.

For a share price to double over 7 years, what must its capital return be as an effective annual rate?