1. What is the present value of an investment worth $500 eight years from now if it pays 6.2% interest compounded monthly? How much interest would you earn over the term of this investment? What is the annual percentage yield for this investment?
N=96 I%=6.2 PV=? PMT=0 FV=500 P/Y=12
Solve for PV.
The present value is $304.87 and the interest you would earn is $195.13.
APY = (1+ i)ppy - 1 = (1 + 0.062 / 12)12 - 1 = 6.379% (to three decimal places).
Period Initial Balance Interest New Balance ------ --------------- -------- ----------- 0 304.87 1 304.87 1.58 306.45 2 306.45 1.58 308.03
2. If you invested $120 a month into a savings account paying 3.4% compounded monthly, how much would you have in ten years and how much interest would you have earned? What is the annual percentage yield for this investment?
N=120 I%=3.4 PV=0 PMT=-120 FV=? P/Y=12
Solve for FV.
The future value is $17,122.12 and you would earn $2,722.12 in interest.
APY = (1+ i)ppy - 1 = (1 + .034 / 12)12 - 1 = 3.453% (to three decimal places).
Period Initial Balance Interest Payment New Balance ------ --------------- -------- ------- ----------- 0 0.00 1 0.00 0.00 120.00 120.00 2 120.00 0.34 120.00 240.34
3. How much could you borrow on a four year loan at 6.9% compounded monthly if you could afford to pay $250 a month? How much interest would you pay? What is the annual percentage rate for this loan?
N=48 I%=6.9 PV=? PMT=-250 FV=0 P/Y=12
Solve for PV.
You could borrow $10,460.30 and you would pay $1,539.70 in interest.
APR = (1+ i)ppy - 1 = (1 + .069 / 12)12 - 1 = 7.122% (to three decimal places).
Period Initial Balance Interest Payment New Balance ------ --------------- -------- ------- ----------- 0 10,460.30 1 10,460.30 60.15 250.00 10,270.45 2 10,270.45 59.06 250.00 10,079.51
4. If you deposited $1,000 in a savings account paying 5% compounded semiannually, how much would you have in 60 years and how much interest would you have earned? What is the annual percentage yield?
N=120 I%=5 PV=-1000 PMT=0 FV=? P/Y=2
Solve for FV.
The future value is $19,358.15 and you would earn $18,358.15 in interest.
APY = (1+ i)ppy - 1 = (1.025)2 - 1 = 5.0625% (to four decimal places).
Period Initial Balance Interest New Balance ------ --------------- -------- ----------- 0 1,000.00 1 1,000.00 25.00 1,025.00 2 1,025.00 25.63 1,050.63
5. How much would you have to invest each quarter at 4.1% compounded quarterly if your goal was to have $50,000 in ten years? How much interest would you earn? What is your annual percentage yield?
N=40 I%=4.1 PV=0 PMT=? FV=50000 P/Y=4
Solve for PMT.
You should invest $1,017.52 every quarter. You would earn $9,299.20 in interest.
APY = (1+ i)ppy - 1 = (1.01025)4 - 1 = 4.163% (to three decimal places).
Period Initial Balance Interest Payment New Balance ------ --------------- -------- -------- ----------- 0 0.00 1 0.00 0.00 1,017.52 1,017.52 2 1,017.52 10.43 1,017.52 2,045.47
6. What would be your monthly payments on a $200,000 25-year mortgage at 7.2% compounded monthly? How much interest would you pay? What is the annual percentage rate for this loan? Generate a table showing how the balance due changes over the first two periods.
N=300 I%=7.2 PV=200000 PMT=? FV=0 P/Y=12
Solve for PMT.
Your monthly payment would be $1,439.18 and you would pay $231,754.00 in interest.
APR = (1+ i)ppy - 1 = (1.006)12 - 1 = 7.442% (to three decimal places).
Period Initial Balance Interest Payment New Balance ------ --------------- -------- -------- ----------- 0 200,000.00 1 200,000.00 1,200.00 1,439.18 199,760.82 2 199,760.82 1,198.56 1,439.18 199,520.20
7. Supposed you saved $400 a month for 30 years in a retirement fund earning 5% compounded monthly. When you retire, you elect to take your retirement as a guaranteed 20-year annuity. That is, you (or your heir, in the event of your death) will receive a monthly retirement benefit every month for 20 years at which time the account will be empty. What will be the size of the monthly retirement benefit? How much interest will this account have earned over its entire 50-year span?
In order to determine the monthly retirement benefit, we must first calculate the amount in the account when you retire.
N=360 I%=5 PV=0 PMT=-400 FV=? P/Y=12
Solve for FV.
The future value is $332,903.45. This represents the present value of the 20-year annuity you'll receive upon retirement. Now calculate the monthly retirement benefit.
N=240 I%=5 PV=-332903.45 PMT=? FV=0 P/Y=12
Solve for PMT.
The monthly retirement benefit will be $2,197.02.
To find the total interest earned, you find the interest for the first 30 years (during which you are investing) and the interest for the last 20 years (during which you are receiving payments) and add them together:
Total Interest = [332903.45 - 360*400] + [240*2197.02 - 332903.45]
= 188903.45 + 194381.35
= $383,284.80
The graph below illustrates the balance in the retirement account over its 50-year span. For the first 30 years, the balance increases because $400 is being put into the account each month. For the last 20 years, the balance decreases because $2,197.02 is being withdrawn each month.
8. A couple finances a new house with a 25-year, $150,000 mortgage at 5.67% compounded monthly. How much is the monthly payment on this mortgage? After four years, the interest rate on the mortgage goes up to 8.05%. How much is the new monthly payment?
First find the initial monthly payment (effective for the first 4 years of the mortgage).
N=300 I%=5.67 PV=150000 PMT=? FV=0 P/Y=12
Solve for PMT.
The monthly payment for the first four years is $936.42.
Next, find the amount due at the end of the first four years. This is the future value of an annuity whose present value is 150,000 and whose monthly payments are $936.42.
N=48 I%=5.67 PV=150000 PMT=-936.42 FV=? P/Y=12
Solve for FV.
The balance due on the loan after four years will be $137,765.53. Conceptually, this is the present value of a second loan for the remaining 21 years at 8.05% compounded monthly.
N=252 I%=8.05 PV=137765.53 PMT=? FV=0 P/Y=12
Solve for PMT.
The new monthly payment is $1,134.62.
9. Suppose a man invested $5,000 ten years ago. The value of that investment today is $10,000. If interest is compounded quarterly, what nominal annual rate of interest is this investment earning?
N=40 I%=? PV=-5000 PMT=0 FV=10,000 P/Y=4
The nominal annual rate is 6.992% compounded monthly.
10. If a couple put $450 a month into an account paying 1.3% compounded monthly, how long would it take to save up $180,000?
N=? I%=1.3 PV=0 PMT=-450 FV=180,000 P/Y=12
It would take 333 months (27 years and 9 months)