Present value, the discount factor, and why d is not i

CM1 · Actuarial Mathematics · 7 min read

1Discounting is one idea

If a rupee today grows to (1 + i) in a year, then a rupee in a year's time is worth 1 ÷ (1 + i) today. That is all discounting is: dividing by the accumulation factor instead of multiplying by it.

The quantity gets a name because it appears constantly:

and over t years the present value of 1 is vt. Three things to carry away about v: it is always less than 1 for a positive interest rate, it is the price today of a rupee later, and multiplying by vt and multiplying by (1 + i)t undo each other exactly — which is the check the worked example runs.

2The second rate, and the mechanism behind the confusion

The rate of discount, written d, measures the same year from the other end. The rate of interest asks: on the amount I have now, how much do I gain? The rate of discount asks: of the amount I will end up with, how much is the deduction?

Same money, same year, different denominator — and a larger denominator makes a smaller rate, so d is always less than i. That is the whole mechanism, and everything else is algebra on it.

Because the deduction is taken off the end amount, d = 1 − v, and from there:

Both directions are worth knowing, because questions are set in both. A sanity check that costs nothing: having converted, confirm that d came out smaller than i. If it did not, the conversion went the wrong way.

3Where it stops being a notation question: commercial discounting

A bank buying a bill before it matures does not usually say "I will charge you i per cent interest". It deducts a percentage of the face value and hands over the rest. That deduction is a rate of discount, not a rate of interest, and the difference is money.

The lender's return is the deduction divided by what the lender actually paid out — which is the face value less the deduction. So the effective rate of interest is higher than the quoted discount rate, always, and by more the larger the rate. The worked example turns a quoted 20% into a true 25%.

This is not a historical curiosity. Bill discounting, invoice discounting and trade finance are quoted this way, so a borrower comparing a discounted bill with a term loan quoted as interest is comparing two different measures, and the comparison is wrong unless one is converted.

4Simple discount, and the trap in it

Over periods shorter than a year, commercial practice often deducts d proportionally to the time: a deduction of d × t of the face value. That is simple discount, and it is the mirror of simple interest — convenient, conventional, and inconsistent in exactly the same way, so it must not be carried into a multi-year calculation.

Two half-year deductions at a simple discount rate are not the same as one annual deduction, and a question that mixes the conventions is testing whether you noticed which one it gave you.

5Why an actuary is made to learn this

Every reserve, every premium and every price in the rest of CM1 is a present value, so v is the most-used symbol in the paper. The reason d earns its own place is that real instruments are quoted in it: treasury bills, commercial paper and discounted invoices are all priced as a deduction from the maturity amount. An actuary who reads a quoted discount rate as an interest rate will value the instrument wrongly, and in the same direction every time — too favourably.

A worked example

Two invented examples, built for this page. The rate is deliberately a round 25% so that every factor is exact and nothing has to be rounded.

1. 1Part 1: the three quantities at i = 25%

   Quantity	Working	Value
   Accumulation factor for one year	1 + 0.25	1.25
   Discount factor, v	1 ÷ 1.25	0.8
   Rate of discount, d	1 − 0.8	0.2, or 20%

   Check d the other way, from the formula: d = i ÷ (1 + i) = 0.25 ÷ 1.25 = 0.2. And back again: i = d ÷ (1 − d) = 0.2 ÷ 0.8 = 0.25. The two conversions are inverses, and d came out below i, as it must.

2. 2Part 1 continued: a present value, and accumulating it back

   What is ₹1,00,000 due in three years worth today at 25%?

   PV = 1,00,000 × v³ = 1,00,000 × 0.8³ = 1,00,000 × 0.512 = ₹51,200

   Now check it by running the money forwards from the printed ₹51,200, one year at a time at 25%:

   Time	Working	Balance
   0	—	₹51,200
   1	51,200 × 1.25	₹64,000
   2	64,000 × 1.25	₹80,000
   3	80,000 × 1.25	₹1,00,000

   It lands exactly on the ₹1,00,000 we started from. That is the check worth building the habit on: discounting and accumulating are the same operation in opposite directions, so a present value that will not accumulate back is wrong.

   The same thing in one line, using the printed figures: 51,200 × 1.253 = 51,200 × 1.953125 = ₹1,00,000.

3. 3Part 2: a discounted bill, where the two rates stop being notation

   An invented trade bill. A supplier in Surat holds a bill for ₹1,00,000 payable in one year and sells it to a financier who quotes "a discount rate of 20%".

   The financier deducts 20% of the face value and pays over the rest:

   proceeds = 1,00,000 × (1 − 0.20) = ₹80,000

   So the financier pays ₹80,000 now and receives ₹1,00,000 in a year. The return on the money actually laid out is

   20,000 ÷ 80,000 = 0.25, or 25%

   The quoted rate was 20%. The rate of interest earned is 25%. Both numbers are correct descriptions of the same transaction, and they differ because one is measured on the ₹1,00,000 at the end and the other on the ₹80,000 at the start.

   Check it against Part 1, where the figures are already printed: at i = 25% the discount factor is 0.8, so the present value of ₹1,00,000 due in one year is 1,00,000 × 0.8 = ₹80,000 — the financier's proceeds exactly. The bill priced at a 20% discount rate and the bill priced at a 25% interest rate are the same bill at the same price.

4. 4Part 2 continued: what the supplier should compare it with

   Suppose the supplier's bank offers an ordinary one-year loan at 22% interest. Which is cheaper?

   The bill looks cheaper at 20% and is not. Converted to the same measure, the bill costs 25% and the loan costs 22%, so the loan is cheaper by three percentage points. Read the other way round, using d = i ÷ (1 + i), the loan's 22% interest is a discount rate of

   0.22 ÷ 1.22 = 0.1803, or 18.03%

   against the bill's 20%. Either conversion gives the same ranking, which is the point: the comparison is only meaningful once both are in one measure, and it reverses the answer a borrower would reach by comparing 20% with 22% as printed.

   Check the conversion by going back: i = 0.1803 ÷ (1 − 0.1803) = 0.1803 ÷ 0.8197 = 0.2200. It returns the 22% it came from.

Where this fits in CM1

This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM1 topics explained

• Simple and compound interest, and the principle that rules out one of them
• Nominal rates payable p times a year, and the loan that costs more than it says
• Force of interest explained, and how it relates to i, v and d
• When the interest rate changes: steps, and a force that varies
• Continuous payment streams, and why they are worth more than they look
• Annuity functions in CM1: immediate, due, deferred and temporary
• Perpetuities and increasing annuities: payments that never stop, or never stay level
• Money rates, real rates and inflation: the exact link, not the subtraction
• Real yields from an inflation index, and index-linked payments
• Loan schedules: the prospective and retrospective methods
• Life table functions explained: l(x), d(x), q(x) and p(x)
• Net premiums and the equivalence principle, worked from first principles
• Reserves in CM1: prospective, retrospective, and why they agree
• Thiele's differential equation, explained term by term

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.