In addition to calculating \(\)the future value (\(FV\)), a TVM solver can be used to calculate the present value (\(PV\)), the regular payment (\(PMT\)), the number of payments (\(N\)), and the annual interest rate (\(I\%\)). It can also calculate \(C/Y\) and \(P/Y\), but those values are typically known when an investment or loan is initiated. The next examples deal with each of the four cases.
Example 2
Amari is planning to take four years off of work to travel. He expects that he will need \($60~000\) each year, starting in one year's time. The interest rate on his bank account is \(3\%\), compounded annually. How much money does he need to have in his bank account now, in order to afford this trip?
Solution
Method 1: Using the TVM Solver
For this question, we need to find the present value (\(PV\)) of an annuity.
Enter the other quantities into the TVM Solver as follows:
- \(\texttt{N=4}\)
- \(\texttt{I%=3}\)
- \(\texttt{PV=}\)
- \(\texttt{PMT=60000}\)
- \(\texttt{FV=0}\)
- \(\texttt{P/Y=1}\)
- \(\texttt{C/Y=1}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
Notes:
- The value for \(PMT\) is positive, since Amari will be receiving money.
- \(FV=0\), since if Amari has saved the perfect amount of money, he will have none left after the end of four years.
- The payments take place at the end of each year.
After pressing "Enter" or "Solve" we see the following display:
- \(\texttt{N=4}\)
- \(\texttt{I%=3}\)
- \(\texttt{PV=-223025.9}\)
- \(\texttt{PMT=60000}\)
- \(\texttt{FV=0}\)
- \(\texttt{P/Y=1}\)
- \(\texttt{C/Y=1}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
Note:
- The value for \(PV\) is negative, because it is the amount Amari needs to pay into his bank account.
Therefore, there needs to be \($223~025.90\) in Amari's bank account now, in order to afford this trip.
Method 2: Use the Formula
Since the payments take place at the end of each year, and interest is also compounded annually, this is an ordinary simple annuity.
The formula for the present value of an ordinary simple annuity is: \(PV=\dfrac{R(1-(1+i)^{-n})}{i}\).
Substitute \(R=60~000\), \(i=0.03\), \(n=4\) into the formula:
\(\begin{align*} PV &= \dfrac{60~000(1-(1.03)^{-4})}{0.03} \\ &= 223~025.904 \ldots \end{align*}\)
As we found earlier, Amari needs to have \($223~025.90\) in his bank account now.
Example 3
Denise is hoping to purchase a new, high-end laptop for \($5000\). In order to do this, she is depositing \($100\) a month (at the end of the month) into a bank account with an annual interest rate of \(2\%\), compounded monthly. How long will it take for Denise to save enough money to buy the laptop?
Solution
\(\)For this question, we need to calculate number of payments, \(N\). This is challenging to calculate algebraically, so use the TVM Solver.
Enter the known quantities into the TVM Solver:
- \(\texttt{N=}\)
- \(\texttt{I%=2}\)
- \(\texttt{PV=0}\)
- \(\texttt{PMT=-100}\)
- \(\texttt{FV=5000}\)
- \(\texttt{P/Y=12}\)
- \(\texttt{C/Y=12}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
Note:
- \(PMT\) is negative, since Denise will be paying the bank
After pressing "Enter" or "Solve" we see the following display:
- \(\texttt{N=48.07}\)
- \(\texttt{I%=2}\)
- \(\texttt{PV=0}\)
- \(\texttt{PMT=-100}\)
- \(\texttt{FV=5000}\)
- \(\texttt{P/Y=12}\)
- \(\texttt{C/Y=12}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
\(N=48.07\) represents just over \(48\) months, or just over \(4\) years.
This means that Denise will be able to buy her laptop in \(4\) years, if she pays a little bit extra in the final month. Or, it will take her \(4\) years and \(1\) month, but in the last month her payment will be less than \($100\).
Example 4
Neville's parents are going to start saving for his university education. They have opened a bank account with a yearly interest rate of \(2.4\%\), compounded monthly, and are hoping that they will have saved \($8 0~000\) in \(18\) years. How much money do they need to save at the end of each month to meet this goal?
Solution
Method 1: Using the TVM Solver
This time we need to solve for the regular payment. Enter all of the other values into the TVM Solver, and then solve for \(PMT\).
- \(\texttt{N=216}\)
- \(\texttt{I%=2.4}\)
- \(\texttt{PV=0}\)
- \(\texttt{PMT=-296.48}\)
- \(\texttt{FV=80000}\)
- \(\texttt{P/Y=12}\)
- \(\texttt{C/Y=12}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
Notes:
- There are 12 payments per year for 18 years, so the number of payments is \(12(18)=216\).
- \(PMT\) is negative, since Neville's parents will be paying the bank.
Therefore, Neville's parents need to save \($296.48\) at the end of each month in order to have a total of \($80~000\) after \(18\) years.
Method 2: Using the Formula
This is an ordinary simple annuity (since the payments take place at the end of each month, and interest is also compounded monthly).
Since we are given the future value, we can use the formula \(FV=\dfrac{R((1+i)^n-1)}{i}\) or the rearranged version, \(R=\dfrac{iFV}{(1+i)^n-1}\).
Substitute \(i=0.002\) (the monthly interest rate), \(n=216\) (the number of monthly payments), and \(FV=80~000\):
\(\begin{align*} R&=\dfrac{0.002(80~000)}{(1.002)^{216}-1} \\ &= 296.477 \ldots \end{align*}\)
The payment is \($296.48\), the same answer that we obtained using the TVM Solver.
Example 5
Elena is considering purchasing a \($20~000\) car. Instead of buying it outright, she decides to lease it for four years. The lease payment is \($300\) at the end of each month, and when she returns the car to the dealership at the end of the four years it will have a value of \($10~000\). What annual interest rate, compounded monthly, is the dealership charging Elena for the lease?
Solution
We need to find \(I\%\). This is another calculation that is difficult to complete algebraically. Instead, use a TVM Solver.
- \(\texttt{N=48}\)
- \(\texttt{I%=7.17}\)
- \(\texttt{PV=20000}\)
- \(\texttt{PMT=-300}\)
- \(\texttt{FV=-10000}\)
- \(\texttt{P/Y=12}\)
- \(\texttt{C/Y=12}\)
- \(\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}\)
Notes:
- There are 12 payments a year for 4 years.
- \(PV\) is positive, since the dealership is giving Elena the equivalent of a loan.
- \(PMT\) is negative, since Elena is paying the dealership.
- \(FV=-10~000\), since Elena is returning the car to the dealership (i.e., paying them back) while it still has some value.
Therefore, the dealership is charging Elena an annual interest rate of \(7.17\%\).
Advantages of a TVM Solver
You may have already noticed that using a TVM solver or financial calculator can be faster than using the annuity formulas. Here are some other advantages to using a TVM solver:
- It allows us to calculate both \(n\) and \(i\), which are quite complicated to solve for algebraically. (In fact, the calculations for both require algebraic techniques that you likely have not seen yet.) For now, we will only be calculating these values using a TVM solver.
- It allows us to consider cases where the compounding period is not the same as the payment period, which is quite common in real financial situations. We can take care of this on the TVM Solver by entering different values for \(C/Y\) and \(P/Y\). Otherwise, we would have to use a much more complicated version of the annuity formula. We will see this in later examples.
- We can also specify on the TVM solver whether the payments are made at the beginning or the end of the payment period, while the formulae we have been using are exclusively for payments made at the end of the payment period (i.e. for ordinary annuities). We will also see this in later examples.