Fantastic Mr. Finance
Exploring Future Value of Fixed Investments
Assignment 12
By Amber Candela
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Future Values
Future values can be used to find the value of a fixed investment in a certain number of years. A fixed investment is when you put a lump sum of money in an account and let it sit, gathering interest. If you put $100 in an account with an interest of 10%, at the end of the first year you would have earned $10 so your new account total would be $110. The following year you would earn 10% on your new amount and so on until you removed the money.
While it would be nice to put a lump sum of money in an account, most people put money into an account and add to it on a regular basis. You can add monthly, every six months or once a year. This is called an annuity, or an account in which you add money on a regular basis. Say you initially invest $100 at the start of the year with 10% interest and add $100 at the start of every year. At the end of the first year you would have $110 dollars just like the first example, then the next day you add $100. This gives you $210 in the account, at the end of the second year you would earn $21 in interest yielding a total of $231. This would continue until you closed the account or stopped putting money into it.
Future Value of a Fixed Investment
To find the amount of future value on a fixed amount, we consider placing $P into an account and calculating the value at the end of each interest period.
We use a function F(t) to be the future value of an investment of "P" dollars after a time period of "t" at an interest rate "r" per period. This gives us:
F(0) = P → initial investment
F(1) = P + r*P = P(1+r) → Amount of money in account after year 1
F(2) = P*(1+r)+r*P*(1+r) = P*(1+r)² → Amount of money in account after year 2
This leads to the general formula of:
F(t)=P*(1+r)^t
When the fixed interest is yearly.
If the fixed interest is not yearly, you would multiply "t" by the number of times the interest is added and divide "r" by that same number since the interest is being split over that many times. For example
Semi-Annual: F(t) = P*(1+r/2)^2*t
Quarterly: F(t) = P*(1+r/4)4*t
Monthly: F(t) = P*(1+r/12)12*t
Daily: F(t) = P*(1+r/365)365*t
Continuous F(t) = e^r*t
Notice the continuous formula looks quite different! Continuous interest is when you would calculate the interest every hour of every day. In order to derive the formula, you can use the limit concept in math. for the sake of time I will give you the formula!
Let's Use Excel to Explore!
Example 1:
Find the future Values of $1000 at 8% annual interest calculated annually for 5 years.
The formula in D2 = $B$1*($B$2+1)^C2
Drag down the number of years, to find your value at the end of that year.
In order to explore more click on the Excel file below
Hint: In order to make a cell permanently in your formula, you need to put dollar signs around it.
What if our interest is compounded semi-annually?
In order to find the future value semi-annually, you can use a recursive formula in Excel.
In cell D2 you want the formula =$B$1*(1+$B$2/2)
In cell D3 you want the formula =D2*(1+$B$2/2) and then drag down.
You need to leave spaces in between the years because you are adding interest on your account twice, once at the beginning of the year and once in the middle of the year, this means you need to add interest 10 times to find out you value at the end of year 10.
What if our interest is compounded continuously?
In order to get "e" in excel you must type in "EXP".
Your formula in D1 = =$B$1*EXP($B$2*C2)
Your Turn!
1. Find the future value of a fixed investment of $1,000 for 10 years paying 8% compound interest annually
2. Find the future value of a fixed investment of $2,000 for 12 years at 9% semiannual interest.
3. Find the future value of a fixed investment of $12,000 for 35 years paying 7.5% continuous interest.
4. Your grandfather many times removed left $500 in the First Bank of Boston 350 years ago – and forgot about it. The bank now wishes to pay off this account to the living heirs and has found and verified 1,231 heirs. How much does each heir get if an annual interest of 3.5% was paid the first 200 years, 3.75% for the next 100 years, and 7.75% for the next 50 years. (A situation similar to this actually occurred).