Can you explain "time value of money" and "compound interest" and provide examples of each?

Question

Can you explain the time value of money concept, including present value and future value?

How does compound interest work? Why is compound interest a powerful concept when it comes to borrowing, saving, or investing?

Answer 1

I will just explain the time value of money in general, descriptive terms and save the math for someone else.

Imagine: You have half a million dollars. I'd like to borrow it all from you. I'll pay it all back, every penny, but no more. And I'll pay it back in about, oh, thirty years or so. (Imagine also that you can be 100% sure that I'll pay it back.)

Does this sound like a good deal? Not really. Why not?

Well, you could do something with that sort of money. With that sort of money, you could do a lot of things for 30 years. You could buy a nice house and live in it for 30 years and save yourself from spending a lot of money on rent during that time (or save money on interest by paying off a mortgage early) even if the price of the house goes nowhere. If you already had a house, you could do some home improvement, like insulate the place better (to save on heating bills) or even just on something that you're going to enjoy for part of those 30 years (a patio in the back yard). If you were feeling entrepreneurial, you could take that money and start a business. Or you could invest that money in the stock market, and get a lot more back.... and if that's too risky for you, just start a savings account and earn interest. And finally, in 30 years, the value of the dollar will be lower because of inflation, so it won't buy as much now as it will then.

That's the time value of money. It's the opportunity cost of the best of the things that you could have done with that money during the time it was gone. When you take out a loan, your interest payments will depend in part on the time value of the money you're borrowing: the people making the loan could be investing that money somewhere else, like government bonds. (It will also depend on factors like the risk of default on the loan - this is why credit card debt is more expensive than debt like a mortgage that's backed by a big fat asset like a house which can be seized and sold if you happen to default.) This is how the Federal Reserve can affect interest rates across the economy by just buying or selling government bonds.

Answer 2

Time Value of Money - The simple calculation for this is FV = PV * (1+r)^N which reads The Future Value is equal to the Present Value times 1 plus the interest rate multiplied by itself by the number of periods that will pass. A simple way to look at this is that if interest rates were 5%/yr a dollar would be worth (1.05)^N where N is the number of years passing.

The concept of compound interest cannot be separated from the above. Compounding is accounting for the interest on the interest that has accrued in prior periods. If I lend you a dollar at 6% simple interest for 30 years, you would pay me back $1 + $1.80 or $2.80. But - 1.06^30 = 5.74 so that dollar compounded at 6% annually for 30 years is $5.74. Quite a difference.

Often, the time value of money is discussed in light of inflation. A dollar today is not the same dollar as 30 years ago or 30 years hence. In fact, inflation has eroded the value of the dollar by a factor of 3 over the past 30 years. An average item costing $100 would now cost $300. So when one invests, at the very least they try to stay ahead of inflation and seek additional return for their risk.

One quirk of compounding is the "rule of 72." This rule states that if you divide the interest rate into the number 72 the result is the number of years to double. So 10% per year will take about 7.2 years to double, 8%, 9 years, etc. It's not 100% precise, but a good "back of napkin" calculation.

When people talk about the total payments over the thirty year life of a mortgage, they often ignore the time value of money. That payment even ten years from now has far less value than the same payment today.

Answer 3

The fundamental concept of the time value of money is that money now is worth more than the same amount of money later, because of what you can do with money between now and later. If I gave you a choice between $1000 right now and $1000 in six months, if you had any sense whatsoever you would ask for the money now. That's because, in the six months, you could use the thousand dollars in ways that would improve your net worth between now and six months from now; paying down debt, making investments in your home or business, saving for retirement by investing in interest-bearing instruments like stocks, bonds, mutual funds, etc. There's absolutely no advantage and every disadvantage to waiting 6 months to receive the same amount of money that you could get now.

However, if I gave you a choice between $1000 now and $1100 in six months, that might be a harder question; you will get more money later, so the question becomes, how much can you improve your net worth in six months given $1000 now? If it's more than $100, you still want the money now, but if nothing you can do will make more than $100, or if there is a high element of risk to what you can do that will make $100 that might in fact cause you to lose money, then you might take the increased, guaranteed money later.

There are two fundamental formulas used to calculate the time value of money; the "future value" and the "present value" formulas. They're basically the same formula, rearranged to solve for different values. The future value formula answers the question, "how much money will I have if I invest a certain amount now, at a given rate of return, for a specified time"?. The formula is FV = PV * (1+R)^N, where FV is the future value (how much you'll have later), PV is the present value (how much you'll have now), R is the periodic rate of return (the percentage that your money will grow in each unit period of time, say a month or a year), and N is the number of unit periods of time in the overall time span.

Now, you asked what "compounding" is. The theory is very simple; if you put an amount of money (the "principal") into an investment that pays you a rate of return (interest), and don't touch the account (in effect reinvesting the interest you earn in the account back into the same account), then after the first period during which interest is calculated and paid, you'll earn interest on not just the original principal, but the amount of interest already earned. This allows your future value to grow faster than if you were paid "simple interest", where interest is only ever paid on the principal (for instance, if you withdrew the amount of interest you earned each time it was paid). That's accounted for in the future value formula using the exponent term; if you're earning 8% a year on your investment, then after 1 year you'll have 108% of your original investment, then after two years you'll have 1.08² = 116.64% (instead of just 116% which you'd get with simple interest). That .64% advantage to compounding doesn't sound like much of an advantage, but stay tuned; after ten years you'll have 215.89% (instead of 180%) of your original investment, after 20 you'll have 466.10% (instead of 260%) and after 30 your money will have grown by over 1000% as opposed to a measly 340% you'd get with simple interest.

The present value formula is based on the same fundamental formula, but it's "solved" for the PV term and assumes you'll know the FV amount. The present value formula answers questions like "how much money would I have to invest now in order to have X dollars at a specific future time?". That formula is PV = FV / (1+R)^N where all the terms mean the same thing, except that R in this form is typically called the "discount rate", because its purpose here is to lower (discount) a future amount of money to show what it's worth to you now.

Now, the discount rate (or yield rate) used in these calculations isn't always the actual yield rate that the investment promises or has been shown to have over time. Investors will calculate the discount rate for a stock or other investment based on the risks they see in the company's financial numbers or in the market as a whole. The models used by professional investors to quantify risk are rather complex (the people who come up with them for the big investment banks are called "quants", and the typical quant graduates with an advanced math degree and is hired out of college with a six-figure salary), but it's typically enough for the average investor to understand that there is an inherent risk in any investment, and the longer the time period, the higher the chance that something bad will happen that reduces the return on your investment. This is why the 30-year Treasury note carries a higher interest rate than the 10-year T-note, which carries higher interest than the 6-month, 1-year and 5-year T-bills.

In most cases, you as an individual investor (or even an institutional investor like a hedge fund manager for an investment bank) cannot control the rate of return on an investment. The actual yield is determined by the market as a whole, in the form of people buying and selling the investments at a price that, coupled with the investment's payouts, determines the yield. The risk/return numbers are instead used to make a "buy/don't buy" decision on a particular investment. If the amount of risk you foresee in an investment would require you to be earning 10% to justify it, but in fact the investment only pays 6%, then don't buy it. If however, you'd be willing to accept 4% on the same investment given your perceived level of risk, then you should buy.

Answer 4

Compound interest means that the interest in each time period is calculated taking into account previously earned interest and not only the initial sum. Thus, if you had $1000 and invested it so that you'd earn 5% each year, than if you would withdraw the earnings each year you in 30 years you would earn 0.05*30*1000 = $1500, so summarily you'd have $2500, or 150% profit.

However, if you left all the money to earn interest - including the interest money - then at the end of 30 years you'd have $4321 - or 330% profit.

This is why compound interest is so important - the interest on the earned interest makes money grow significantly faster. On the other hand, the same happens if you owe money - the interest on the money owed is added to the initial sum and so the whole sum owed grows quicker.

Compound interest is also important when calculating interest by time periods. For example, if you are told the loan accumulates 1% interest monthly, you may think it's 12% yearly. However, it is not so, since monthly interest is compounded - i.e., in February the addition not only February's 1% but also 1% on 1% from January, etc. - the real interest is 12.68% yearly. Thus, it is always useful to know how interest is compounded - both for loans and investments - daily, monthly, yearly, etc.

Answer 5

Here are some really excellent video tutorials on these topics:

Introduction to Compound Interest

Introduction to Present Value

Answer 6

A real simple definition or analogy of present Value would be the "Principal" or "Loan Amount" being lent and the future value as being returning the Principal along with cost of borrowing

The (1+i)^n is the interest you earn on present value

The (i+i)^-n is the interest you pay on future value

The first one is the FVIF or future value of a $1

The second one is the PVIF or present value of a $1

Both these interest factors assume interest is paid annually, if the interest payment is made more often within the payment year then interest factors look this way

(1+i/m)^mn

(i+i/m)^-mn

m is the frequency of interest payment, the higher this frequency the more of interest you pay or earn and you pay or earn the most interest when compounding occurs on each small fraction of time

This entails

m->∞
i/m->0
(1+i/∞)^∞ -> e^i

here e is the Euler's e

Thus the interest factors turn to this

FVIF = e^it
PVIF = e^-it

The preceeding examples only considered a single repayment at future date.

Now if you were obliged to make periodic loan repayments say in amount of $1 for n number of periods. Then the present value of all such periodic payment is the "Principal" or amount you borrowed. This is the sum of discounted periodic payments as

(1+i)^-1 + (1+i)^-2 + (1+i)^-3 + ... + (1+i)^n-1 + (1+i)^-n

if we replace 1/1+i with x then this turns out to be geometric series of the form

x + x^2 + x^3 + ... + x^n-1 + x^n

This simplifies to

(1-x^n) / (1 - x)

replacing (1/1+i) for x we get

[1-(1+i)^-n] / (1-1+i)
PVIFA = [1-(1+i)^-n] / i

which is the present value of periodic payment in amount of $1

The future value of periodic payments in amount of $1 can be arrived at multiplying the PVIFA by (1+i)^n giving

FVIFA = (1+i)^n[1-(1+i)^-n] / i
FVIFA = [(1+i)^n-(1+i)^n(1+i)^-n] / i
FVIFA = [(1+i)^n-(1+i)^n-n] / i
FVIFA = [(1+i)^n-(1+i)^0] / i
FVIFA = [(1+i)^n-1] / i

Once again the interest is compounded per annum and for intra-year compounding you would have to at first find the annual effective yield AEY to use as the effect rate is the PVIFA and FVIFA calculation

AEY = (1+i/m)^m -1

for continuous compounding

AEY = e^i - 1

All the calculation discussed thus far did not take inflation into consideration, if we were to adjust the amounts for a growth of g% then the present value of a $1 would be as follow

PVIFG = (1+g)^(n-1) . (1+i)^(-n)

FVIFG = (1+g)^(n-1) . (1+i)^(n)

PVIFGA = [ 1 - (1+g)^n . (1+i)^-n ] / [i - g]

FVIFGA = (1+i)^n[ 1 - (1+g)^n . (1+i)^-n ] / [i - g]
FVIFGA = [ (1+i)^n - (1+i)^n. (1+g)^n . (1+i)^-n ] / [i - g]
FVIFGA = [ (1+i)^n - (1+g)^n ] / [i - g]

Once again you would have to use AEY if compounding frequency of interest is intra-year

Now assume that each loan repayment increases or decreases by an extra amount Q per period. To find the present value of series of payments P that increase or decrease per period by an amount Q we would do the following calculations

PV = P PVIFA(i%, n) + Q/i [ PVIFA(i%, n) - n PVIF(i%,n) ]

Here

PVIFA(i%, n) = [ 1 - (1+i)^-n ] / i

and

PVIF(i%, n) = (1+i)^-n

All of these calculations have been available in tadXL add-in for finance and incrementally being offered as JavaScript financial functions library tadJS. Please note that the tad series of the financial functions library for various environments such as Excel, JavaScript, PHP, Ruby, Microsoft.net and others are property of the author writing this post. All of these libraries except one for Excel are available for FREE for public use.

And the future value of such payments with increments may be found by multiplying the PV by (1+i)^n as follows

FV = P PVIFA(i%, n)(1+i)^n + Q/i [ PVIFA(i%, n)(1+i)^n - n PVIF(i%,n)(1+i)^n ]
FV = P FVIFA(i%, n) + Q/i [ FVIFA(i%, n) - n ]

Here

FVIFA(i%, n) = [ (1+i)^n - 1 ] / i

Answer 7

s&p 500 CAGR: 7%

5 years: 1000 * 1.07 ^ 5 = 1 402

10 years: 1000 * 1.07 ^ 10 = 1 967

80 years: 1000 * 1.07 ^ 80 = 224 234

Just math. CAGR: compound annual growth rate