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

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

###r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.06/12 = 0.005###

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

###\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 450,000 =& C_{\text{monthly}} \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \\ C_{\text{monthly}} =& 450,000 \div \left(\frac{1}{0.06/12}\left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \right) \\ =& 450,000 \div \left(\frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{360}} \right) \right) \\ =& 450,000 \div 166.7916144 \\ =& 2,697.977363 \\ \end{aligned} ###

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

We will use the annuity equation and the present value of a single cash flow equation. Keep in mind that the annuity equation gives a value that is one period before the first cash flow at t=3, so the value of the annuity will be at t=2 and needs discounting by 2 periods to get to t=0.

###\begin{aligned} V_{0} &= \dfrac{C_{3} \times \dfrac{1}{r_\text{eff annual}} \left(1 - \dfrac{1}{(1+r_\text{eff annual})^{10}} \right)}{(1+r_\text{eff annual})^2} + \dfrac{C_{5.5}}{(1+r_\text{eff annual})^{5.5}} \\ &= \dfrac{60 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right)}{(1+0.1)^2} + \dfrac{400}{(1+0.1)^{5.5}} \\ &= \dfrac{60 \times 6.144567106}{(1+0.1)^2} + \dfrac{400}{(1+0.1)^{5.5}} \\ &= 304.689278 + 236.810101 \\ &= 541.4993789 \\ \end{aligned} ###

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

###\begin{aligned} r_\text{credit card, eff yrly} &= \left(1 + \frac{r_\text{credit card, apr comp monthly}}{12} \right)^{12} - 1 \\ &= \left(1 + \frac{0.18}{12} \right)^{12} - 1 \\ &= 0.195618171 \\ \end{aligned}###

###\begin{aligned} r_\text{bond, eff yrly} &= \left(1 + \frac{r_\text{bond, apr comp 6 monthly}}{2} \right)^{2} - 1 \\ &= \left(1 + \frac{0.06}{2} \right)^{2} - 1 \\ &= 0.0609 \\ \end{aligned}###

###\begin{aligned} r_\text{stock, eff yrly} &= \left(1 + \frac{r_\text{stock, apr comp yearly}}{1} \right)^{1} - 1 \\ &= r_\text{stock, apr comp yearly} \\ &= 0.1 \\ \end{aligned}###

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

###\begin{aligned} r_\text{eff semi-annual} &= \frac{r_\text{apr comp semi-annually}}{2} \\ &= \frac{0.12}{2} \\ &= 0.06 \\ \end{aligned}###

###\begin{aligned} r_\text{eff yearly} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{2} - 1 \\ &= \left(1+\frac{0.12}{2}\right)^{2} - 1 \\ &= 0.1236 \\ \end{aligned}###

###\begin{aligned} r_\text{eff daily} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{2/360}-1 \\ &= \left(1+\frac{0.12}{2}\right)^{2/360}-1 \\ &= 0.000323769\\ \end{aligned}###

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

###\begin{aligned} r_\text{eff semi-annual} &= \frac{r_\text{apr comp semi-annually}}{2} \\ &= \frac{0.05}{2} \\ &= 0.025 \\ \end{aligned}###

###\begin{aligned} r_\text{eff yearly} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{2} - 1 \\ &= \left(1+\frac{0.05}{2}\right)^{2} - 1 \\ &= 0.050625 \\ \end{aligned}###

###\begin{aligned} r_\text{eff daily} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{2/360}-1 \\ &= \left(1+\frac{0.05}{2}\right)^{2/360}-1 \\ &= 0.000137191 \\ \end{aligned}###

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

Since the loan is interest-only, the perpetuity without growth formula can be used.

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ###

We already know the price ##P_0## and interest rate ##r_\text{eff monthly}##. We're interested in finding the monthly cash flow ##C_{\text{monthly}}##, so make it the subject.

###\begin{aligned} C_{\text{monthly}} &= P_0 \times r_\text{eff monthly} \\ &= P_0 \times \frac{r_\text{apr compounding monthly}}{12} \\ &= 450,000 \times \frac{0.06}{12} \\ &= 2,250 \\ \end{aligned}###

These interest payments are paid monthly in arrears which means they occur at the end of each month.

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

The cash flows occur every month, so the discount rate needs to be an effective monthly rate. This also means that time (T), the unknown variable, will be measured in months.

Since the money needed to buy the car will be ready in the future, we need the future value of the annuity, not the present value. An easy way to do this is to find the present value of the annuity of $100 monthly payments, and then grow that amount out to the future. This is a simple step which doesn't require memorisation of the 'future value of an annuity equation'. Then we can solve for the only unknown variable which is 'T'.

###V_0 = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} } \right) ### ###V_T = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} } \right) \left(1+\dfrac{r_\text{apr comp monthly}}{12} \right) ^{T}### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{0.06}{12}\right)^{T}} } \right) \left(1+\dfrac{0.06}{12} \right) ^{T} ### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \vcenter{ \left(1+\dfrac{0.06}{12} \right) ^{T} - \dfrac{\left(1+\dfrac{0.06}{12} \right) ^{T}}{\left(1+\dfrac{0.06}{12}\right)^{T}} } \right) ### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \left(1+\dfrac{0.06}{12}\right)^{T} -1 \right) ### ###\dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1 = \left(1+\dfrac{0.06}{12}\right)^{T} ### ###\ln \left( \left( 1+\frac{0.06}{12}) \right) ^{T} \right) = \ln \left( \dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1 \right) ### ###T \times \ln \left( 1+\frac{0.06}{12} \right) = \ln \left( \vcenter{ \dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1} \right) ###

###\begin{aligned} T &= \dfrac{\ln \left( \dfrac{2,500}{100} \times \dfrac{0.06}{12} +1 \right)} {\ln \left( 1+\dfrac{0.06}{12} \right)} \\ &= \dfrac{\ln \left( 1.125 \right) } {\ln \left( 1.005 \right)} \\ &= 23.6154497 \text{ months}\\ \end{aligned}###

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

The cash flows occur every month, so the discount rate needs to be an effective monthly rate and the time must be measured in months.

First we have to find the amount borrowed when the payments are $1,500 per month for 30 years.

###\begin{aligned} V_0 &= C_{\text{monthly}} \times \dfrac{1}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right) \\ &= 1,500 \times \dfrac{1}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{0.09}{12}\right)^{30 \times 12}} \right) \\ &= 1,500 \times 124.2818657 \\ &= 186,422.7985 \\ \end{aligned}###

When the present value of the 'T' months of $2,000 payments are equal to the amount borrowed, then the loan will be paid off. So the only job left is to solve for T.

###V_0 = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right) ### ###186,422.7985 = \dfrac{2,000}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{0.09}{12}\right)^{T}} \right) ### ###\frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right) = 1 - \frac{1}{\left(1+\frac{0.09}{12}\right)^{T}} ### ###\frac{1}{\left(1+\frac{0.09}{12}\right)^{T}} = 1 - \frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right) ### ###\left(1+\frac{0.09}{12}\right)^{-T} = 0.300914506 ### ### \ln \left( \left(1+\frac{0.09}{12}) \right) ^{-T} \right) = \ln \left(0.300914506 \right) ### ### -T \times \ln \left(1+\frac{0.09}{12} \right) = \ln \left(0.300914506 \right) ###

###\begin{aligned} T &= -\dfrac{\ln \left(0.300914506 \right)} {\ln \left( 1+\dfrac{0.09}{12} \right)} \\ &= 160.7235953 \text{ months}\\\\ &= 13.39363294 \text{ years}\\ \end{aligned}###

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

The winnings now less the present value of the future withdrawals should equal the present value of the amount left at the end, which is supposed be zero. Once this equation is set up, we can substitute figures and solve for the monthly withdrawal ##C_\text{0,withdrawal}##. The symbol ##r_\text{apr}## is the annualised precentage rate compounding per month.

Note that the monthly withdrawal occurs at t=0, not at t=1, so the annuity equation gives a value one period before which is at t=-1. To correct for this we grow the annuity by one period. This equation is sometimes called the 'annuity due'.

### V_\text{0, winnings} - V_\text{-1, annuity of withdrawals starting now} \left(1+\frac{r_\text{apr}}{12}\right)^1 = \frac{V_T}{\left(1+\frac{r_\text{apr}}{12}\right)^T} ### ### V_\text{0, winnings} - \dfrac{C_\text{0,withdrawal} }{\left( \frac{r_\text{apr}}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\frac{r_\text{apr}}{12}\right)^{T}} \right) \left(1+\frac{r_\text{apr}}{12}\right)^1 = \frac{V_T}{\left(1+\frac{r_\text{apr}}{12}\right)^T} ### ### 1,000,000 - \dfrac{C_\text{0,withdrawal} }{\left( \frac{0.06}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\frac{0.06}{12}\right)^{60 \times 12}} \right) \left(1+\frac{0.06}{12}\right)^1 = \frac{0}{\left(1+\frac{0.06}{12}\right)^{60 \times 12}} ### ### C_\text{0,withdrawal} \times \dfrac{1}{\left( \frac{0.06}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\frac{0.06}{12}\right)^{60 \times 12}} \right) \left(1+\frac{0.06}{12}\right)^1 = 1,000,000 ### ### C_\text{0,withdrawal} \times 195.4584457 = 1,000,000 ###

###\begin{aligned} C_\text{0,withdrawal} &= \frac{1,000,000}{195.4584457} \\ &= 5,116.176978 \\ \end{aligned}###

As an alternative to growing the annuity forward by one period to find the 'annuity due' as above, we could instead exclude the very first withdrawal from the annuity, and simply add it on separately since it's already a present (t=0) value. Using this method we would have:

###\begin{aligned} V_\text{0, winnings} - V_\text{0, annuity of withdrawals starting in one month} + C_\text{0,withdrawal} &= \frac{V_T}{\left(1+\frac{r_\text{apr}}{12}\right)^T} \\ V_\text{0, winnings} - \dfrac{C_\text{1,withdrawal} }{\left( \frac{r_\text{apr}}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\frac{r_\text{apr}}{12}\right)^{60 \times 12 -1}} \right) + C_\text{0,withdrawal} &= \frac{V_T}{\left(1+\frac{r_\text{apr}}{12}\right)^{60 \times 12}} \\ \end{aligned}###

Remembering that the withdrawals are all equal, so ##C_\text{0,withdrawal} = C_\text{1,withdrawal}##, this method should give the same answer.

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

Right now (at t=0), borrow $109.09 from the bank and lend $100 of it to your neighbour now. This leaves us with a positive cash flow of $9.09 at t=0.

In on year (t=1), take the $120 from your neighbour and use it to pay back the bank the $120 owed ##\left(109.09\times(1+0.1)^1\right)##. This leaves us with a cash flow of zero at t=1.

To work out the amounts, use an arbitrage table.

Arbitrage Table of Cash Flows
Instrument	Time 0	Time 1
Buy debt from (lend to) neighbour now, and wait for repayment in one year.	-100	120
Sell debt to (borrow from) bank at 10% pa and pay it back in one year.	109.09 Step 3	-120 Step 2
Total	9.09 Step 4	0 Step 1

The steps used to calculate the table's values are given here.

Step 1: All future cash flows need to total zero, that way only the initial (t=0) cash flow will be non-zero.

Step 2: The bank loan cash flow at time 1 must equal -120 so that total cash flows are zero. Since we're paying this $120 at the end, we must be borrowing using this bank loan.

Step 3: Since we're paying back 120 in one year, we must be borrowing the present value of that which is 109.09, calculated as follows: ###V_0 = -\dfrac{C_1}{(1+r)^1} = -\dfrac{-120}{(1+0.1)^1} = 109.09###

Step 4: Adding up the total cash flows at time zero, -100+109.09 = 9.09 which is the NPV of the arbitrage.

Arbitrage tables are great since they show how to a create a positive arbitrage cash flow of the NPV right now ($9.09) with no risk and no capital required.