Katya offers to pay you $10 at the end of every year for the next 5 years (t=1,2,3,4,5) if you pay her $50 now (t=0). You can borrow and lend from the bank at an interest rate of 10% pa, given as an effective annual rate. Ignore credit risk.
Paying $50 to Katya now in exchange for receiving $10 at the end of each year for 5 years might seem like a fair deal since ##5 \times 10 = 50##, but it's not. Katya is proposing an unfair deal that benefits her at your expense.
An amount of $50 now is worth more than 5 payments of $10 over 5 years since Katya can put your $50 payment in the bank, pay you $10 at the end of the first year, and keep the $5 ##(=0.1\times 50)## interest earned on the $50 for herself. In the second year, she starts with $45 ##(=50+5-10)## in the bank, and she receives 4.5 ##(=0.1\times 45)## interest that she can keep for herself, and so on.
To find the value of the deal to you right now, we find the present value using the annuity equation. The cash flows received are positive and the cash flows paid are negative.
###\begin{aligned} V_0 &= C_0 + \frac{C_{1,2...T}}{r} \left( 1-\frac{1}{(1+r)^T} \right) \\ &= -50 + \frac{10}{0.1} \left( 1-\frac{1}{(1+0.1)^5} \right) \\ &= -50 + 37.90786769 \\ &= -12.092132306 \\ \end{aligned}###
Therefore, accepting the deal would reduce your current wealth by $12.09, and increase Katya's current wealth by $12.09.
There are many ways to write the ordinary annuity formula.
Which of the following is NOT equal to the ordinary annuity formula?
All answers are mathematically equivalent except (e).
This annuity formula ##\dfrac{C_1}{r}\left(1-\dfrac{1}{(1+r)^3} \right)## is equivalent to which of the following formulas? Note the 3.
In the below formulas, ##C_t## is a cash flow at time t. All of the cash flows are equal, but paid at different times.
The annuity formula with T cash flows sums the present value of each, where the first is at time 1 and the last at time T:
###\dfrac{C}{r}\left(1-\dfrac{1}{(1+r)^T} \right) = \dfrac{C_1}{(1+r)^1} +\dfrac{C_2}{(1+r)^2} + ... + \dfrac{C_T}{(1+r)^T} ###The annuity formula with 3 cash flows sums the present value of each, where the first is at time 1 and the last at time 3:
###\dfrac{C}{r}\left(1-\dfrac{1}{(1+r)^3} \right) = \dfrac{C_1}{(1+r)^1} +\dfrac{C_2}{(1+r)^2} + \dfrac{C_3}{(1+r)^3} ###The following cash flows are expected:
- 10 yearly payments of $60, with the first payment in 3 years from now (first payment at t=3 and last at t=12).
- 1 payment of $400 in 5 years and 6 months (t=5.5) from now.
What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate?
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} ###You are promised 20 payments of $100, where the first payment is immediate (t=0) and the last is at the end of the 19th year (t=19). The effective annual discount rate is ##r##.
Which of the following equations does NOT give the correct present value of these 20 payments?
No explanation provided.
Telsa Motors advertises that its Model S electric car saves $570 per month in fuel costs. Assume that Tesla cars last for 10 years, fuel and electricity costs remain the same, and savings are made at the end of each month with the first saving of $570 in one month from now.
The effective annual interest rate is 15.8%, and the effective monthly interest rate is 1.23%. What is the present value of the savings?
The annuity formula is perfectly suited to this problem since the payments are all equal. Since the first cash flow is exactly one month away, the annuity equation will give a present value one month before that which is at time zero. Perfect.
###\begin{aligned} V_0 &= \frac{C_{1}}{r} \left( 1-\frac{1}{(1+r)^T} \right) \\ V_0 &= \frac{C_\text{1 monthly}}{r_\text{eff monthly}} \left( 1-\frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ &= \frac{570}{0.0123} \left( 1-\frac{1}{(1+0.0123)^{10 \times 12}} \right) \\ &= 35,654.3278 \\ \end{aligned}###The present value of an annuity of 3 annual payments of $5,000 in arrears (at the end of each year) is $12,434.26 when interest rates are 10% pa compounding annually.
If the same amount of $12,434.26 is put in the bank at the same interest rate of 10% pa compounded annually and the same cash flow of $5,000 is withdrawn at the end of every year, how much money will be in the bank in 3 years, just after that third $5,000 payment is withdrawn?
There will be nothing left in the bank account since the three $5,000 payments in one year, two years and three years are exactly equivalent to $12,434.26 now when interest rates are 10% pa. So the annuity of 3 payments of $5,000, which are negative cash flows, will exactly offset the $12,434.26 asset value at the very start such that the net present value (NPV) is zero:
###\begin{aligned} NPV &= V_0 - C_1 \times \dfrac{1}{r} \left(1- \dfrac{1}{(1+r)^T} \right) \\ &= 12,434.26 - 5,000 \times \dfrac{1}{0.1} \left(1- \dfrac{1}{(1+0.1)^3} \right) \\ &= 12,434.26 - 5,000 \times 2.48685199098 \\ &= 12,434.26 - 12,434.26 \\ &= 0 \\ \end{aligned}###Another way to show that there will be nothing left is to:
- Grow the $12,434.26 now by one year since it will accrue interest in the bank at 10% pa: ###V_{1\text{, just before payment}}=12,434.26(1+0.1)^1 = 13,677.686###
- Then subtract the $5,000 payment at the end of year one: ###V_{1\text{, just after payment}} =13,677.686 - 5,000 = 8,677.686###
- Then grow what remains out to the second year: ###V_{2\text{, just before payment}}=8,677.686*(1+0.1)^1 = 9,545.4546###
- Then subtract the $5,000 payment at the end of the second year: ###V_{2\text{, just after payment}}=9,545.4546 - 5,000 = 4,545.4546###
- Then grow what remains out to the third year: ###V_{3\text{, just before payment}}=4,545.4546*(1+0.1)^1 = 5,000###
- This is just enough to afford the final payment of $5,000 at the end of the third year! ###V_{3\text{, just after payment}}=5,000 - 5,000 = 0###
In summary:
###\begin{aligned} V_{3\text{, just after payment}} &= ((V_0(1+r)^1 - C_1)(1+r)^1 - C_2)(1+r)^1 - C_3 \\ &= ((12,434.26(1+0.1)^1 - 5,000)(1+0.1)^1 - 5,000)(1+0.1)^1 - 5,000 \\ &= 0 \\ \end{aligned}###Your friend overheard that you need some cash and asks if you would like to borrow some money. She can lend you $5,000 now (t=0), and in return she wants you to pay her back $1,000 in two years (t=2) and every year after that for the next 5 years, so there will be 6 payments of $1,000 from t=2 to t=7 inclusive.
What is the net present value (NPV) of borrowing from your friend?
Assume that banks loan funds at interest rates of 10% pa, given as an effective annual rate.
The annuity formula can be applied to find the present value of the 6 equal payments from t=2 to 7. But care must be taken since the present value of an annuity is one period before the first cash flow (at t=2), so the whole annuity value will be at t=1 so it needs to be discounted back one extra period to get a present value.
###\begin{aligned} V_0 &= C_0 - \dfrac{ C_2.\dfrac{1}{r}\left( 1-\dfrac{1}{(1+r)^6} \right) }{(1+r)^1} \\ &= 5,000 - \dfrac{ 1,000 \times \dfrac{1}{0.1}\left( 1-\dfrac{1}{(1+0.1)^6} \right) }{(1+0.1)^1} \\ &= 5,000 - 3,959.3279 \\ &= 1,040.6721 \\ \end{aligned}###Question 58 NPV, inflation, real and nominal returns and cash flows, Annuity
A project to build a toll bridge will take two years to complete, costing three payments of $100 million at the start of each year for the next three years, that is at t=0, 1 and 2.
After completion, the toll bridge will yield a constant $50 million at the end of each year for the next 10 years. So the first payment will be at t=3 and the last at t=12. After the last payment at t=12, the bridge will be given to the government.
The required return of the project is 21% pa given as an effective annual nominal rate.
All cash flows are real and the expected inflation rate is 10% pa given as an effective annual rate. Ignore taxes.
The Net Present Value is:
Since the cash flows are real but our discount rate is nominal, we need to convert the nominal discount rate to a real rate. Using the exact Fisher equation,
###\begin{aligned} 1+r_\text{real} &= \frac{1+r_\text{nomimal}}{1+r_\text{inflation}} \\ r_\text{real} &= \frac{1+r_\text{nomimal}}{1+r_\text{inflation}} -1 \\ &= \frac{1+0.21}{1+0.1} -1 \\ &= 0.1 \\ \end{aligned} ###
Now just discount the cash flows using two annuity equations.
###\begin{aligned} V_0 &= -C_\text{0, 1, 2}.\frac{1}{r}\left(1 - \frac{1}{(1+r)^{3}} \right).(1+r)^{1} + C_\text{3, 4, ..., 12}.\frac{1}{r}\left(1 - \frac{1}{(1+r)^{10}} \right).\frac{1}{(1+r)^{2}} \\ &= -100m \times \frac{1}{0.1} \times \left(1 - \frac{1}{(1+0.1)^{3}} \right) \times (1+0.1)^{1} + 50m \times \frac{1}{0.1} \times \left(1 - \frac{1}{(1+0.1)^{10}} \right) \times \frac{1}{(1+0.1)^{2}} \\ &= -100m \times 2.48685199 \times 1.1 + 50m \times 6.14456711 \times 0.82644628 \\ &= -273.553719m + 253.9077316m \\ &= -19.64598737m \\ &= -19,645,987.37 \\ \end{aligned} ###
Some countries' interest rates are so low that they're zero.
If interest rates are 0% pa and are expected to stay at that level for the foreseeable future, what is the most that you would be prepared to pay a bank now if it offered to pay you $10 at the end of every year for the next 5 years?
In other words, what is the present value of five $10 payments at time 1, 2, 3, 4 and 5 if interest rates are 0% pa?
When the yield is zero, there is no time value of money. Therefore we can just sum cash flows like an accountant.
###\begin{aligned} V_0 &= T \times C \\ &= 5 \times 10 = 50 \\ \end{aligned}###
Interestingly, the normal way to value an annuity with the annuity equation will not work since there will be a divide by zero problem which is mathematically impossible:
###\begin{aligned} V_0 &= C_\text{1} \times \frac{1}{r_\text{eff yrly}} \left( 1 - \frac{1}{(1+r_\text{eff yrly})^{T}} \right) \\ &= 1 \times \color{red}{\frac{1}{0}} \left( 1 - \frac{1}{(1+0)^{5}} \right) \\ \end{aligned}###
Since 1/0 is mathematically undefined, that is a dead-end.
But present-valuing the individual payments separately will still work.
###\begin{aligned} P_0 &= \frac{C_\text{1 yr}}{(1+r_\text{eff yrly})^1} + \frac{C_\text{2 yr}}{(1+r_\text{eff yrly})^2} + \frac{C_\text{3 yr}}{(1+r_\text{eff yrly})^3} + \frac{C_\text{4 yr}}{(1+r_\text{eff yrly})^4} +\frac{C_\text{5 yr}}{(1+r_\text{eff yrly})^5} \\ &= \frac{10}{(1+0)^1} + \frac{10}{(1+0)^2} + \frac{10}{(1+0)^3} + \frac{10}{(1+0)^4} +\frac{10}{(1+0)^5} \\ &= 10+10+10+10+10 \\ &= 5 \times 10 \\ &= 50 \\ \end{aligned}###