We have modelled annuities using geometric series. We can also use financial tables to assist us in calculating the final value of an investment or to find the size of the regular repayments needed to pay off a certain loan. Tables may also be used to calculate the equivalent single sum that could be invested to grow to the same value as an annuity.
Terminology
First, some language. The final value of an investment is also known as the future value, $FV$FV, and the principal we invest can be called the present value, $PV$PV. For this reason, the compound interest formula we know as:
$A=P(1+r)^n$A=P(1+r)n
may also be written as:
$FV=PV(1+r)^n$FV=PV(1+r)n
In terms of investments, the total value of regular deposits and the interest earned at the end of its lifespan is the future value of the annuity. The equivalent single sum that could be invested to grow to that same amount is known as the present value of an annuity.
Financial tables for loans and savings
Banks offer different interest rates depending on the size of the repayments and the length of the loan. The table below does the heavy lifting for us, providing a multiplication factor for each $\$1000$$1000 borrowed so that we can work out the monthly repayments on a loan.
From Table $1$1, you can see that for a $20$20-year loan at an annual interest rate of $6%$6%, you would need a monthly repayment of $\$7.16$$7.16 for each $\$1000$$1000 that is borrowed in order to pay off the loan in $20$20 years.
Table 1: Monthly repayments on a $\$1000$$1000 loan
| Term of loan of $\$1000$$1000 (years) | |||||
| Annual interest rate | $10$10 | $15$15 | $20$20 | $25$25 | $30$30 |
| $3%$3% | $9.66$9.66 | $6.91$6.91 | $5.55$5.55 | $4.74$4.74 | $4.22$4.22 |
| $4%$4% | $10.12$10.12 | $7.40$7.40 | $6.06$6.06 | $5.28$5.28 | $4.77$4.77 |
| $5%$5% | $10.61$10.61 | $7.91$7.91 | $6.60$6.60 | $5.85$5.85 | $5.37$5.37 |
| $6%$6% | $11.10$11.10 | $8.44$8.44 | $7.16$7.16 | $6.44$6.44 | $6.00$6.00 |
| $7%$7% | $11.61$11.61 | $8.99$8.99 | $7.75$7.75 | $7.07$7.07 | $6.65$6.65 |
| $8%$8% | $12.13$12.13 | $9.56$9.56 | $8.36$8.36 | $7.72$7.72 | $7.34$7.34 |
If we borrow $\$15000$$15000 at $6%$6% per annum over $20$20 years, we would be borrowing $15$15 groups of $\$1000$$1000 and the monthly repayments would be:
$15\times\$7.16=\$107.40$15×$7.16=$107.40
The total repayments are given by:
$\$7.16\times\text{12 months }\times\text{20 years }\times\text{15 lots }=\$25776$$7.16×12 months ×20 years ×15 lots =$25776
So over $20$20 years, we'd actually end up paying $\$10776$$10776 interest on top of the $\$15000$$15000 repayment of the loan.
Tables may be constructed with values given for each $\$1$$1 or $\$1000$$1000.
To calculate the total amount of paid for a loan, we need to multiply the repayment amount, the length of the loan and how often the repayment is to be made (e.g. weekly, monthly) all together.
$\text{Total repayments }=\text{Individual repayment }\times\text{Number of time periods each year }\times\text{Number of years }$Total repayments =Individual repayment ×Number of time periods each year ×Number of years
We can also use financial tables to determine how long it will take us to save a certain amount. The numbers in them will be different to those in a table tailored to a loan repayment, but they work in the same way.
Table $2$2 shows us how much we need to save each month to reach $\$1000$$1000 after a certain number of years.
Table 2: Monthly instalments required to save $\$1000$$1000
| Time to save $\$1000$$1000 (years) | |||||
| Annual interest rate | $5$5 | $10$10 | $15$15 | $20$20 | $25$25 |
| $1%$1% | $16.26$16.26 | $7.93$7.93 | $5.15$5.15 | $3.77$3.77 | $2.94$2.94 |
| $2%$2% | $15.86$15.86 | $7.53$7.53 | $4.77$4.77 | $3.39$3.39 | $2.57$2.57 |
| $3%$3% | $15.47$15.47 | $7.16$7.16 | $4.41$4.41 | $3.05$3.05 | $2.24$2.24 |
| $4%$4% | $15.08$15.08 | $6.79$6.79 | $4.06$4.06 | $2.73$2.73 | $1.95$1.95 |
| $5%$5% | $14.70$14.70 | $6.44$6.44 | $3.74$3.74 | $2.43$2.43 | $1.68$1.68 |
| $6%$6% | $14.33$14.33 | $6.10$6.10 | $3.44$3.44 | $2.16$2.16 | $1.44$1.44 |
For example, if we save each month for $10$10 years at $2%$2% per annum compounded monthly, we would need to save $\$7.53$$7.53 each month to get to $\$1000$$1000 by the end of $10$10 years.
We can calculate the monthly instalments required to save $\$30000$$30000 in $10$10 years if the savings account earns $2%$2% interest per annum, compounded monthly.
We want to save a total of $\$\left(30\times1000\right)$$(30×1000):
| $\text{Instalment }$Instalment | $=$= | $30\times\$7.53$30×$7.53 |
| $=$= | $\$225.90$$225.90 |
Practice questions
Question 1
Petra has a $15$15-year loan for $\$10000$$10000 at $4%$4% p.a. monthly reducible interest. Using Table $1$1, calculate each of the following:
The monthly instalments required to repay the loan.
The total amount she will repay
The interest she will be charged over the term of the loan.
The total interest expressed as a percentage of the principal loan.
Question 2
Jacob wants to save $\$30000$$30000 over $10$10 years in a savings account that earns $3%$3% interest p.a., compounded monthly. Using Table $2$2, calculate:
The monthly instalments required to save the target amount.
The total amount she will invest herself.
The amount earned through interest over the saving period.
Question 3
Trey and Greta each borrow $\$5000$$5000. Trey's loan is at $5%$5% p.a. reducing interest for $15$15 years. Greta's loan is at $6%$6% p.a. interest reducing interest for $10$10 years.
Using Table $1$1, calculate the difference between the total repayments for each loan. Who pays the least amount?
Question 4
Bart made regular monthly contributions to an annuity, which was worth $\$10000$$10000 after $15$15 years. His annuity earned $3%$3% interest per annum, compounded monthly.
Using Table $2$2, calculate the size of his monthly instalments.
How much did Bart contribute in total over $15$15 years?
Annie made a single large payment when Bart made his first monthly contribution. Her investment earned the same interest rate, with the same compounding periods as Bart, and was also worth $\$10000$$10000 after $15$15 years. Find the value of the single payment that Annie made, correct to the nearest dollar. You may use the following rearrangement of the present value formula: $PV=\frac{FV}{\left(1+r\right)^n}$PV=FV(1+r)n
Compare the total amounts that each person paid into the savings fund.
Effective interest rates
The table below compares the effect of changing the number of compounding periods when $\$1000$$1000 is invested for one year at a nominal rate of $5%$5% p.a. The final amount is calculated using the compound interest formula $A=P\times(1+\frac{r}{n})^{nt}$A=P×(1+rn)nt
|
Number of compounding periods |
$1$1 |
$4$4 |
$365$365 |
|---|---|---|---|
| Calculation | $A=1000\times(1+\frac{0.05}{1})^1$A=1000×(1+0.051)1 $=1000\times(1.05)$=1000×(1.05) |
$A=1000\times(1+\frac{0.05}{4})^4$A=1000×(1+0.054)4 $=1000\times(1.05095)$=1000×(1.05095) |
$A=1000\times(1+\frac{0.05}{365})^{365}$A=1000×(1+0.05365)365 $=1000\times(1.05127)$=1000×(1.05127) |
| Final amount | $\$1050$$1050 | $\$1050.95$$1050.95 | $\$1051.27$$1051.27 |
| Amount of interest | $\$50$$50 | $\$50.95$$50.95 | $\$51.27$$51.27 |
| Effective annual interest rate | $\frac{50}{1000}=0.05=5%$501000=0.05=5% | $\frac{50.95}{1000}=0.05095=5.095%$50.951000=0.05095=5.095% | $\frac{51.27}{1000}=0.05127=5.127%$51.271000=0.05127=5.127% |
From the table we can see that the amount of interest earned increases when the number of compounding periods increases.
The effective annual interest rate has been calculated using the formula:
$\text{effective interest rate}=\frac{\text{amount of interest in first year}}{\text{loan amount}}\times100%$effective interest rate=amount of interest in first yearloan amount×100%
The published rate of $5%$5% per annum is called the nominal interest rate.
Note: If we only compound once per year then this nominal interest rate is the same as the effective interest rate.
$i_{effective}=(1+\frac{i}{n})^n-1$ieffective=(1+in)n−1
where $i_{effective}$ieffective is the effective interest rate per annum, expressed as a decimal
$i$i is the nominal (or published) interest rate per annum, expressed as a decimal
$n$n is the number of compounding periods per annum
Being able to calculate the effective interest rate can come in handy when we are choosing and comparing investments or loans. They allow us to more easily work out how much interest the investment or loan will actually earn and quickly compare rates that have different compounding periods.
When investing money, we want to have the highest possible effective interest rate.
When borrowing money, we want to have the lowest possible effective interest rate.
Worked example
example 1
A bank advertises a nominal interest rate for an investment of $5.6%$5.6% per annum, compounded quarterly.
Calculate the effective interest rate.
Think: This is an effective interest problem so we can use the formula. Note that it's on the formula sheet! We are solving for $i_{effective}$ieffective where $i=0.056$i=0.056 and $n=4$n=4.
Do: Write $i_{effective}=(1+\frac{i}{n})^n-1$ieffective=(1+in)n−1
Therefore $i_{effective}=(1+\frac{0.056}{4})^4-1$ieffective=(1+0.0564)4−1
Using calculator $i_{effective}=1.057187-1$ieffective=1.057187−1
Therefore $i_{effective}=5.72%$ieffective=5.72% p.a.
Practice questions
Question 5
James invested $\$3000$$3000 at $4.6%$4.6% p.a. compounded daily.
Find the amount of interest earned in a year. You may assume that there are $365$365 days in a year (ignoring leap years).
Write your answer to the nearest cent.
Find the effective annual interest rate as a percentage to two decimal places.
Question 6
An investment earns interest at a rate of $7.2%$7.2% compounding semiannually.
What is the effective rate correct to two decimal places?
Using a table of future value interest factors
A table of this type doesn't look particularly different to those previously shown. However, they provide less specific detail to a given scenario, meaning it is important that we understand how they work.
Here is a table of future value interest factors. As it contains no mention of a dollar amount, we conclude that the factor applies to each single dollar involved in our calculations.
Most importantly, we must remember that the future value is the present value multiplied by the correct interest factor selected from the table.
| Interest rate | |||||
|---|---|---|---|---|---|
| Time period | $1%$1% | $2%$2% | $3%$3% | $4%$4% | $5%$5% |
| $1$1 | $1.0000$1.0000 | $1.0000$1.0000 | $1.0000$1.0000 | $1.0000$1.0000 | $1.0000$1.0000 |
| $2$2 | $2.0100$2.0100 | $2.0200$2.0200 | $2.0300$2.0300 | $2.0400$2.0400 | $2.0500$2.0500 |
| $3$3 | $3.0301$3.0301 | $3.0604$3.0604 | $3.0909$3.0909 | $3.1216$3.1216 | $3.1525$3.1525 |
| $4$4 | $4.0604$4.0604 | $4.1216$4.1216 | $4.1836$4.1836 | $4.2465$4.2465 | $4.3101$4.3101 |
| $5$5 | $5.1010$5.1010 | $5.2040$5.2040 | $5.3091$5.3091 | $5.4163$5.4163 | $5.5256$5.5256 |
| $6$6 | $6.1520$6.1520 | $6.3081$6.3081 | $6.4684$6.4684 | $6.6330$6.6330 | $6.8019$6.8019 |
| $7$7 | $7.2135$7.2135 | $7.4343$7.4343 | $7.6625$7.6625 | $7.8983$7.8983 | $8.1420$8.1420 |
| $8$8 | $8.2857$8.2857 | $8.5830$8.5830 | $8.8923$8.8923 | $9.2142$9.2142 | $9.5491$9.5491 |
And so, the future value of an annuity that involves investing $\$1200$$1200 at the start of the $3$3rd year at $5%$5% compounding annually is:
$3.1525\times\$1200=\$3783$3.1525×$1200=$3783
We can also work backwards. What is the value of an annuity that would provide a future value of $\$407100$$407100 at the start of the $7$7th year at $5%$5% per annum compound interest? Well, this means the future value is given, not the present value. So we need to construct an equation of sorts, with the future $FV$FV equalling $PV$PV multiplied by the correct future value interest factor.
$407100=PV\times8.1420$407100=PV×8.1420
$PV=\frac{407100}{8.1420}=\$50000$PV=4071008.1420=$50000
Practice questions
Question 7
Find the interest generated on an annuity in which $\$2700$$2700 is invested every 3 months for $2$2 years at $9%$9% p.a. with interest compounded quarterly. Give your answer correct to the nearest cent.
| Table of future value interest factors | |||||
| Interest rate per 3 months | |||||
| Periods | $2.25%$2.25% | $2.75%$2.75% | $3.25%$3.25% | $3.75%$3.75% | $4.25%$4.25% |
| $8$8 | $8.6592$8.6592 | $8.8138$8.8138 | $8.9716$8.9716 | $9.1326$9.1326 | $9.2967$9.2967 |
| $9$9 | $9.8540$9.8540 | $10.0562$10.0562 | $10.2632$10.2632 | $10.4750$10.4750 | $10.6918$10.6918 |
| $10$10 | $11.0757$11.0757 | $11.3328$11.3328 | $11.5967$11.5967 | $11.8678$11.8678 | $12.1462$12.1462 |
| $11$11 | $12.3249$12.3249 | $12.6444$12.6444 | $12.9736$12.9736 | $13.3129$13.3129 | $13.6624$13.6624 |
| $12$12 | $13.6022$13.6022 | $13.9921$13.9921 | $14.3953$14.3953 | $14.8121$14.8121 | $15.2431$15.2431 |
| $13$13 | $14.9083$14.9083 | $15.3769$15.3769 | $15.8631$15.8631 | $16.3676$16.3676 | $16.8909$16.8909 |
| Table of future value interest factors | |||||
| Interest rate per 3 months | |||||
| Periods | $2.25%$2.25% | $2.75%$2.75% | $3.25%$3.25% | $3.75%$3.75% | $4.25%$4.25% |
| $8$8 | $8.6592$8.6592 | $8.8138$8.8138 | $8.9716$8.9716 | $9.1326$9.1326 | $9.2967$9.2967 |
| $9$9 | $9.8540$9.8540 | $10.0562$10.0562 | $10.2632$10.2632 | $10.4750$10.4750 | $10.6918$10.6918 |
| $10$10 | $11.0757$11.0757 | $11.3328$11.3328 | $11.5967$11.5967 | $11.8678$11.8678 | $12.1462$12.1462 |
| $11$11 | $12.3249$12.3249 | $12.6444$12.6444 | $12.9736$12.9736 | $13.3129$13.3129 | $13.6624$13.6624 |
| $12$12 | $13.6022$13.6022 | $13.9921$13.9921 | $14.3953$14.3953 | $14.8121$14.8121 | $15.2431$15.2431 |
| $13$13 | $14.9083$14.9083 | $15.3769$15.3769 | $15.8631$15.8631 | $16.3676$16.3676 | $16.8909$16.8909 |
Question 8
Find the contribution Beth needs to deposit into her savings account every 3 months, which pays $9%$9% p.a. with interest compounded quarterly, in order to obtain an amount in $6$6 years with a present value of $\$5516.71$$5516.71. Give your answer correct to the nearest dollar
| Table of present value interest factors | |||||
| Interest rate per 3 months | |||||
| Periods | $0.25%$0.25% | $0.75%$0.75% | $1.25%$1.25% | $1.75%$1.75% | $2.25%$2.25% |
| $20$20 | $19.4845$19.4845 | $18.5080$18.5080 | $17.5993$17.5993 | $16.7529$16.7529 | $15.9637$15.9637 |
| $21$21 | $20.4334$20.4334 | $19.3628$19.3628 | $18.3697$18.3697 | $17.4475$17.4475 | $16.5904$16.5904 |
| $22$22 | $21.3800$21.3800 | $20.2112$20.2112 | $19.1306$19.1306 | $18.1303$18.1303 | $17.2034$17.2034 |
| $23$23 | $22.3241$22.3241 | $21.0533$21.0533 | $19.8820$19.8820 | $18.8012$18.8012 | $17.8028$17.8028 |
| $24$24 | $23.2660$23.2660 | $21.8891$21.8891 | $20.6242$20.6242 | $19.4607$19.4607 | $18.3890$18.3890 |
| $25$25 | $24.2055$24.2055 | $22.7188$22.7188 | $21.3573$21.3573 | $20.1088$20.1088 | $18.9624$18.9624 |
Using a table of present value interest factors
The main purpose of this kind of table is to calculate the equivalent single sum that can be invested that will grow to the same amount as an annuity where regular deposits are being made. The present value of an annuity is less than its future value since it is the pre-investment amount that has yet to accrue any interest. It would also be less than the sum of all regular deposits into the annuity. This might be harder to visualise, as both the present value and annuity will eventually be worth the same total amount. However, since the single investment would be in the account for the entire period of the investment, while each subsequent annuity deposit is earning interest for shorter and shorter periods of time, the single investment must earn more interest and thus have a relatively smaller principal when compared to the deposits made into the annuity.
We can compare present values of different investment options to determine which is the more financially appealing option. Just like when comparing future values, the larger the present value, the larger the overall return will be.
| Table of present value interest factors | ||||||
|---|---|---|---|---|---|---|
| Interest rate $r$r | ||||||
| Time period $n$n | $0.6%$0.6% | $0.65%$0.65% | $0.7%$0.7% | $0.75%$0.75% | $0.8%$0.8% | $0.85%$0.85% |
| $45$45 | $39.33406$39.33406 | $38.90738$38.90738 | $38.48712$38.48712 | $38.07318$38.07318 | $37.66545$37.66545 | $37.26383$37.26383 |
| $46$46 | $40.09350$40.09350 | $39.64965$39.64965 | $39.21263$39.21263 | $38.78231$38.78231 | $38.35859$38.35859 | $37.94133$37.94133 |
| $47$47 | $40.84841$40.84841 | $40.38714$40.38714 | $39.93310$39.93310 | $39.48617$39.48617 | $39.04622$39.04622 | $38.61311$38.61311 |
| $48$48 | $41.59882$41.59882 | $41.11986$41.11986 | $40.64856$40.64856 | $40.18478$40.18478 | $39.72839$39.72839 | $39.27924$39.27924 |
| $49$49 | $42.34475$42.34475 | $41.84785$41.84785 | $41.35905$41.35905 | $40.87820$40.87820 | $40.40515$40.40515 | $39.93975$39.93975 |
| $50$50 | $43.08623$43.08623 | $42.57113$42.57113 | $42.06459$42.06459 | $41.56645$41.56645 | $41.07653$41.07653 | $40.59470$40.59470 |
Jacky decides to invest $\$300$$300 each month into a savings account for the next four years. The account will pay $7.2%$7.2% interest per annum compounding monthly. What is the present value of her annuity?
To use the table above, we must first convert the rate and time period into months, as specified by the compounding period in the question. Four years gives $n=48$n=48 months and $7.2%$7.2% p.a. is equivalent to $0.6%$0.6% or $0.006$0.006 per month. This gives us the present value interest factor $41.59882$41.59882. And so, the present value of Jacky's annuity is $\$300\times41.59882=\$12479.65$$300×41.59882=$12479.65.
The other application of a present value interest factor table is to use it to find out the size of the repayments required for a loan. The principal value of the loan amount is the present value of an annuity which is the sum of all the repayments that will be made. After working out the size of the repayments, we could then calculate the total amount of interest paid over the course of the loan.
Jake borrows $\$15000$$15000 for a car loan that is charged $9%$9% per annum over four years, compounding monthly. This gives $n=48$n=48 again and $r=0.0075$r=0.0075. in this case, we know the present value: it is $\$15000$$15000. What we don't know is the regular repayment, which we can call $M$M.
$\$15000$$15000 $=M\times40.18478$=M×40.18478, giving us $M=$M= $\$373.28$$373.28.
Practice questions
Question 9
Question 10
To be inserted after new questions created using present value tables