QF5213 Chap.2 Present Value, Future Value and Compounding
Present Value, Future Value and Compounding
Why a dollar has a date attached
Finance treats an amount of money as incomplete until you know when it arrives. The Week 2 class gives three reasons a dollar today is worth more than a dollar later: today's dollar can be put to work earning interest, rising prices shrink what a later dollar can buy, and a promise to pay may not be kept.
Because of this, cash flows that land on different dates cannot simply be added. They first have to be translated to a single date, and the two translation tools are present value and future value.
The translation always starts with a timeline. Today is time 0, later points mark the end of each period, outflows are negative and inflows positive.
Drawing the line before touching a formula prevents the most frequent slip in this topic, which is counting one period too many or too few.
Simple growth, compound growth and the single-sum formula
Simple interest is paid only on the original deposit, so the balance rises by the same dollar amount each year.
Compound interest is paid on the running balance, so each year's interest also earns interest and the yearly increase keeps getting bigger. Over two or three years the difference looks trivial; over several decades it becomes the dominant part of the balance, and a higher rate magnifies it further.
The compound relationship links a present amount to a future one through the factor one plus the rate, raised to the number of periods.
Multiplying by that factor is compounding; dividing by it is discounting, and the result answers a practical question: what is the most you should pay today for a sum promised later, given the return you require?
When the cash flows are uneven, such as an initial outlay followed by inflows of different sizes, no shortcut applies.
Each flow is moved to the same date on its own and the results are added, keeping the minus sign on the outlay.
Valuing the stream at time 0 and then compounding the total forward gives exactly the same answer as compounding each flow to the later date separately, a useful check under exam conditions.
When interest is credited more than once a year
A quoted annual rate can be credited annually, semiannually, monthly or daily.
Dividing the rate by the number of credits per year gives the rate per period, and multiplying the years by the same number gives the count of periods. More frequent crediting means interest starts earning interest sooner, so the future value climbs as frequency rises, though each further increase adds less and the values approach a ceiling set by continuous compounding.
For a saver that is good news; for a borrower carrying a card balance charged monthly it is the reverse.
The equal-payment shortcuts known as annuities, and the conversion of quoted rates into effective annual rates, follow in the next chapter, and both rest on the single-sum mechanics set out here.
Both tests in the course are multiple choice, open book and two hours long, so speed with the factor and the timeline matters more than memorised wording.
What this chapter covers
- 01
Three reasons money today outranks the same money later
- 02
Drawing a timeline with end-of-period dates and signed cash flows
- 03
Simple interest against compound interest, year by year
- 04
Future value and present value of a single sum
- 05
Valuing an uneven stream of cash flows on one common date
- 06
Semiannual, monthly, daily and continuous compounding
Worked example · free
Pricing an early settlement under semiannual compounding
- 1Draw the timeline and convert to the rate per period and the number of periods.
- 1Compute the compounding factor for the full horizon.
- 2Discount the $45,000 to time 0.
- 2Repeat with annual compounding and explain the direction of the difference.
Key terms
- Time Value of Money
- The principle that a dollar received today is worth more than a dollar received later, because of interest, inflation and payment risk.
- Future Value
- The amount a sum invested now grows to at a stated later date, found by compounding at the period interest rate.
- Present Value
- The value today of an amount due in the future, found by discounting it at the required rate of return.
- Compound Interest
- Interest earned on both the original principal and the interest already accumulated, so yearly interest grows over time.
- Simple Interest
- Interest earned only on the original principal, which adds the same dollar amount in every period.
- Discounting
- The process of moving a future cash flow back to an earlier date by dividing by the compounding factor.
- Compounding Frequency
- The number of times per year interest is credited, which sets the rate per period and the number of periods.
- Continuous Compounding
- The limiting case of ever more frequent crediting, where a sum grows by the factor e raised to rate times years.
Present Value, Future Value and Compounding FAQ
What is the difference between present value and future value?
Present value states what a later amount is worth at time 0, while future value states what an amount held now will grow to at a chosen later date. They are the same relationship read in opposite directions: compounding multiplies by one plus the rate raised to the number of periods, and discounting divides by that factor.
Why does compound interest grow faster than simple interest?
Under simple interest only the original deposit earns interest, so the balance climbs by a fixed amount each year. Under compound interest the interest credited in earlier years joins the base and earns interest too, so each yearly increment is larger than the one before, and the gap widens with time and with the rate.
How do I find the present value of uneven cash flows?
Discount every flow separately to time 0 using its own number of periods, then add the results, entering any outlay as a negative number. If a later valuation date is needed, compound the single total forward rather than redoing each flow, since both routes give the identical figure.
Does more frequent compounding always mean a bigger future value?
Yes, for a given quoted annual rate and a positive rate, crediting interest more often lets interest start earning interest sooner, so the ending balance rises. The gains shrink with each increase in frequency and level off at the continuous compounding value, which is why monthly and daily results sit very close together.
Is frequent compounding good or bad for a borrower?
It works against a borrower, because the debt balance grows by the same mechanism that helps a saver. A credit card that charges a quoted rate monthly ends up costing more over a year than the same quoted rate charged once, so borrowers should compare offers on a like-for-like compounding basis.
Exam move
Take three single-sum questions and solve each twice: once by multiplying or dividing by the compounding factor on a calculator, once with the table factor, and confirm the answers match to the cent. Then draw one uneven stream of four flows, value it at time 0 and at its last date, and check that compounding the time 0 total forward reproduces the second figure.
Finish by recomputing one answer at monthly compounding to feel the frequency effect.
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