Answer: Good choice. You earned $10. Poor choice. You lost $10.

Let ##C_t## be consumption at time t and ##V_t## be wealth at time t.

Common sense method:

We initially have ##V_0## wealth in the bank. Then we consume or spend ##C_0## a moment later, still at time zero. The amount left in the bank accrues interest so it grows over the next year by the interest rate. To find this future value we multiply by ##(1+r)^1##. At time one, everything that's left in the bank is consumed ##(C_1)## with nothing left over at the end.

###(V_0 -C_0)(1+r)^1 - C_1 = 0 ###

Formula Building Steps
Time	Event	Formula
0	Starting wealth	##V_0##
0	Consume	##V_0 - C_0##
1	Lend to bank for one year	##(V_0 - C_0)(1+r)^1##
1	Consume all so there's nothing left	##(V_0 - C_0)(1+r)^1 - C_1 = 0##

 

The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##. So we can solve simultaneously and substitute numbers (k represents thousands),

###(V_0 -C_0)(1+r)^1 - C_1 = 0 ### ###(V_0 -C_0)(1+r)^1 - C_\color{red}{0} = 0 ### ###(100k -C_0)(1+0.1)^1 - C_0 = 0 ### ###100k(1+0.1)^1 -C_0(1+0.1)^1 - C_0 = 0 ### ###C_0\left(1+(1+0.1)^1\right) = 100k(1+0.1)^1 ### ###\begin{aligned} C_0 &= \frac{100k(1+0.1)^1}{1+(1+0.1)^1} \\ &= \frac{100,000 \times 1.1}{2.1} \\ &= 52,380.9524 \\ \end{aligned}### ###C_1 = C_0 = 52,380.9524###

Present value method:

This method is easier to formulate. Since all wealth will be consumed, the present value of the positive wealth and negative consumption must equal zero.

###V_0 -C_0 - \frac{C_1}{(1+r)^1} +\frac{V_1}{(1+r)^1} = 0 ###

The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##. Also ##V_1 = 0## since there's no wealth left over at the end.

Solving simultaneously and substituting numbers (k represents thousands),

###100k -C_0 - \frac{C_0}{(1+0.1)^1} +\frac{0}{(1+0.1)^1} = 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1}\right) = 100k ### ###\begin{aligned} C_0 &= \frac{100k}{\left(1+ \frac{1}{(1+0.1)^1}\right)} \\ &= 52,380.9524 = C_1 \\ \end{aligned}###

Future value method:

Similarly to the present value method, this method is easy to formulate. Since all wealth will be consumed, the future value of the positive wealth and negative consumption must equal zero.

###V_0(1+r)^1 -C_0(1+r)^1 - C_1 = 0 ###

The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##.

Solving simultaneously and substituting numbers (k represents thousands),

###100k(1+0.1)^1 -C_0(1+0.1)^1 - C_0 = 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1}\right) = 100k ### ###C_0 = 52,380.9524 = C_1 ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Let ##C_t## be consumption at time t and ##V_t## be wealth at time t.

Common sense method:

We have ##V_0## in the bank then we consume ##C_0## of that at time zero. The amount left in the bank accrues interest over the year so we grow it by ##(1+r)^1##. Again at time one we consume ##C_1##, and the amount remaining in the bank accrues more interest. At time two we consume ##C_2## and the amount left after this is ##V_2##.

###\left( (V_0 -C_0)(1+r)^1 - C_1 \right)(1+r)^1-C_2 = V_2 ###

Formula Building Steps
Time	Event	Formula
0	Starting wealth	##V_0##
0	Consume	##V_0 - C_0##
1	Lend to bank for one year	##(V_0 - C_0)(1+r)^1##
1	Consume more	##(V_0 - C_0)(1+r)^1 - C_1##
2	Lend to bank for another year	##((V_0 - C_0)(1+r)^1 - C_1)(1+r)^1##
2	Consume again but leave some wealth aside	##((V_0 - C_0)(1+r)^1 - C_1)(1+r)^1 - C_2 = V_2##

 

The question stated that consumption at t=0,1 and 2 are equal, so ##C_0 = C_1 = C_2##.

Solving simultaneously and substituting numbers (k represents thousands),

###\left( (V_0 -C_0)(1+r)^1 - C_1 \right)(1+r)^1-C_2 = V_2 ### ###\left( (100k -C_0)(1+0.1)^1 - C_0 \right)(1+0.1)^1-C_0 = 50k ### ###\begin{aligned} C_0 &= \dfrac{100k(1+0.1)^2 - 50k }{1+(1+0.1)^1 + (1+0.1)^2} \\ &= 21,450.1511 = C_1 = C_2 \\ \end{aligned}###

Present value method:

###V_0 -C_0 - \frac{C_1}{(1+r)^1} -\frac{C_2}{(1+r)^2}- \frac{V_2}{(1+r)^2}= 0 ###

Also, consumption at t=0, 1 and 2 are all equal, so

###C_0 = C_1 = C_2 ###

Solving simultaneously and substituting numbers (k represents thousands),

###100k -C_0 - \frac{C_0}{(1+r)^1} -\frac{C_0}{(1+r)^2} - \frac{50k}{(1+r)^2}= 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1} + \frac{1}{(1+0.1)^2}\right) = 100k - \frac{50k}{(1+0.1)^2} ### ###\begin{aligned} C_0 &= \frac{100k - \dfrac{50k}{(1+0.1)^2}}{\left(1+ \dfrac{1}{(1+0.1)^1} + \dfrac{1}{(1+0.1)^2}\right)} \\ &= 21,450.1511 = C_1 = C_2 \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the bond pays coupons semi-annually, we can assume that the yield is given as an annualised percentage rate (APR) compounding every 6 months. Therefore we'll divide the 10% APR compounding semi-annually by 2 to get the yield as an effective 6 month rate:

###\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 0.05}{2} \times \frac{1}{0.1/2}\left(1 - \frac{1}{(1+0.1/2)^{3\times2}} \right) + \frac{100}{(1+0.1/2)^{3 \times 2}} \\ =& 2.5 \times 5.075692067 + 74.62153966 \\ =& 12.68923017 + 74.62153966 \\ =& 87.31076983 \\ \end{aligned} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

This is a par bond since the coupon rate is equal to the yield. Therefore the price is equal to the face value, $100.

Using the fixed interest bond pricing formula gives the same answer, but takes a lot longer:

###\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 0.005}{2} \times \frac{1}{0.005/2}\left(1 - \frac{1}{(1+0.005/2)^{2\times2}} \right) + \frac{100}{(1+0.005/2)^{2 \times 2}} \\ =& 0.25 \times 3.975124455 + 74.62153966 \\ =& 0.993781114 + 99.00621889 \\ =& 100 \\ \end{aligned} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The NPV of buying any fairly priced asset is zero. Therefore the NPV of buying a fairly priced bond is also zero. Whether the bond is a premium or discount bond is irrelevant, it's unrelated to the NPV of buying it.

The fair price of a bond is the present value (PV) of its expected future cash flows, which is the present value of coupons and face value:

###\begin{aligned} P_\text{0, bond} &= PV(\text{coupons}) + PV(\text{face value}) \\ &= \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^T} \right) + \frac{F_T}{(1+r)^T} \\ \end{aligned}###

The net present value (NPV) of buying an asset is the present value of costs less gains.

###\begin{aligned} NPV &= -PV(\text{costs}) + PV(\text{gains}) \\ \end{aligned}###

The cost of a bond is its price, and the gains from a bond are the coupons and face value. Since the price of a fairly priced bond equals the present value of the coupons and face value, then the net present value of buying a fairly priced bond must be zero.

Mathematically, we can re-arrange the bond price formula to be in the same form as the NPV formula, which shows that the NPV must be zero:

###P_\text{0, bond} = PV(\text{coupons}) + PV(\text{face value}) ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{costs})} + \underbrace{PV(\text{coupons}) + PV(\text{face value})}_{PV(\text{gains})} ###

Note that premium bonds can also be fairly priced. The NPV of buying a fairly priced premium bond is zero. The term 'premium' does not indicate that the bond's price is above (or below) the fair price, it indicates that the bond's price is above its face value which is usually the $100 or $1,000 that's paid at maturity. Premium bonds have a higher price than their face value because the coupon rate is more than the total required return (the yield). Therefore investors are willing to pay a high price for the bond, higher than the face value, making the bond a premium bond. The highest price investors will pay for the bond will be the price that makes the NPV zero.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

A premium bond's price (##p_0##) is higher than its face value (##p_1##). Therefore: ###p_0 > p_1### ###r_\text{capital} = \frac{p_1-p_0}{p_0} < 0###

So the expected capital return of a premium bond must be negative, not positive.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

From the bond pricing formula, the required return r is in the denominator of each fraction so any increase in r causes a decrease in the price P and vice versa:

###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ###

When the required return rises, the bond price falls.

When the required return falls, the bond price rises.

This is not only true for bonds but for any asset including shares and land.

The required return of a fairly priced bond is also its IRR. Remember that the IRR is the discount rate that makes the NPV zero.

###\begin{aligned} NPV &= C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + ... + \frac{C_T}{(1+r)^T} \\ 0 &= C_0 + \frac{C_1}{(1+r_{irr})^1} + \frac{C_2}{(1+r_{irr})^2} + ... + \frac{C_T}{(1+r_{irr})^T} \\ \end{aligned} ###

Re-arranging the bond-pricing equation:

###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{cost})} + \underbrace{C_\text{1,2,3,...,T} \times \frac{1}{r_\text{IRR}}\left(1 - \frac{1}{(1+r_\text{IRR})^{T}} \right) + \frac{F_\text{T}}{(1+r_\text{IRR})^{T}}}_{PV(\text{gains})} ###

Because the NPV of buying a fairly priced bond is zero, the bond's yield is equivalent to the IRR of buying it too.

Answer: Well judged, ##g## is the expected capital return of the stock (b). You gained $10. Poor choice, (b) is the answer. You lost $10.

The pronumeral ##g## is the expected growth rate of the dividend which must also be the expected growth rate of the stock price, which is the expected capital return of the stock.

If the stock price grew by a rate less than the dividend in perpetuity, then the dividend would eventually grow bigger than the stock price which is impossible.

If the stock price grew by a rate more than the dividend in perpetuity, then the stock price would eventually grow so big compared to the dividend that the dividend yield (expected dividend in one year divided by stock price now) would be close to zero. To maintain the same total return, the capital return of the stock price must increase until it is very close (or equal) to the total return. If the total return is more than the country's GDP growth rate, then the capital return of the stock will be more than the average firm in perpetuity (forever), so the firm must take over the country. This is very unlikely.

Mathematically:

### P_0 = \frac{ C_1 }{ r - g } ### ### r - g = \frac{ C_1 }{ P_0 } ### ###g = r - \frac{ C_1 }{ P_0 } ###

Substitute for the total return, ##r = \dfrac{P_1 - P_0 + C_1}{P_0}##

###\begin{aligned} g &= \frac{ P_1 - P_0 + C_1 }{ P_0 } - \frac{ C_1 }{ P_0 } \\ &= \frac{ P_1 - P_0 }{ P_0 } \\ &= r_\text{capital} \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Rearranging this equation:

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

This is equivalent to:

###\begin{aligned} r_{\text{eff, total}} =& r_{\text{eff, capital}} + r_{\text{eff, income}} \\ \end{aligned} ###

Where ##r_{\text{eff, capital}} = g## and ##r_{\text{eff, income}} = c_1/p_0##.

It's clear that ##r_{\text{eff}}## is the expected total return of the stock which is the sum of the expected capital and income returns.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

For a stock, the expected income return is the ratio of the dividend expected to be paid in one year to the stock price now. Another name for it is the expected dividend yield.

Rearranging this equation:

###P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}} ### ###r_{\text{eff}} - g_{\text{eff}} = \frac{C_1}{P_{0}} ### ###r_{\text{eff}} = g_{\text{eff}} + \frac{C_1}{P_{0}} ###

This is equivalent to:

###\begin{aligned} r_{\text{eff, total}} =& r_{\text{eff, capital}} + r_{\text{eff, income}} \\ \end{aligned} ###

Where ##r_{\text{eff, capital}} = g## and ##r_{\text{eff, income}} = d_1/p_0##.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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 + 0.08)^{1/2}-1 \\ &= 0.039230485 \\ \end{aligned}###

The $5 dividend just paid was received by the previous share owner so we ignore it. The next dividend to be paid will be in 6 months, and it will be 1% bigger than the last. 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{5(1+0.01)^1}{0.039230485 - 0.01} \\ &= 172.7648405 \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The growth rate in the dividend (##g##) must equal the growth rate in the stock price measured between a whole number of dividend periods. But the growth rate in stock price, also known as the capital return, is actually equal to the total return ##r## in between dividend payments. This is best seen in a saw-tooth graph, where the 'dividend drop-off' price fall can be seen every time a stock pays a dividend. But here it's explained in words:

• The expected capital return measured just after a dividend is paid to just after the next dividend is paid is ##g##.
• The expected capital return measured just before a dividend is paid to just before the next dividend is paid is also ##g##.
• But, the expected capital return measured just after a dividend is paid to just before the next dividend is paid is actually ##r##. The price growth must be higher than ##g## since the stock price must accumulate the next dividend payment as well as the usual price gain over a whole dividend period. Thus the price growth between dividend payments must be ##r = d_1/P_0 + g##.

Using this logic, the growth rate in the share price from just after the current (t=0) dividend was paid to just after the next dividend is paid in one year will be ##g##.

###P_\text{1, just after div} = P_\text{0, just after div}(1+g)^1### Similarly for the next year, just after that dividend is paid (at t=2).

###\begin{aligned} P_\text{2, just after div} &= P_\text{1, just after div}(1+g)^1 \\ &= P_\text{0, just after div}(1+g)^2 \\ \end{aligned}###

But from just after the second dividend is paid at t=2 to t=2.5, that period is in between dividend payments, so the share price growth will be the total return ##r##.

###\begin{aligned} P_\text{2.5} &= P_\text{2, just after div}(1+r)^{0.5} \\ &= P_\text{0, just after div}(1+g)^2(1+r)^{0.5} \\ \end{aligned}###