Time Value of Money

Time Value of Money

Syllabus tag: KASNEB CPA | Intermediate Level | CA22 Financial Management | Topic 1 Time Value of Money

Lesson objectives

By the end of this topic, you will be able to:

• Compound a present amount forward and discount a future amount back
• Value an annuity, both its present value and its accumulated future value
• Value a level perpetuity and a growing perpetuity
• Adjust for compounding more often than once a year
• Convert a nominal rate to an effective annual rate and compare offers

Why this matters

A shilling today is worth more than a shilling next year. Not mainly because of inflation — because today's shilling can be invested and start earning. Every later topic in this paper rests on that one idea.

Words to know

• Present value (PV) — what a future amount is worth today.
• Future value (FV) — what an amount today will grow to by a given date.
• Discounting — moving a future amount backwards to today.
• Compounding — moving a present amount forwards to a future date.
• Annuity — a series of equal cash flows at equal intervals.
• Perpetuity — an annuity with no end date.
• Nominal rate — the quoted annual rate, before compounding is allowed for.
• Effective rate — the rate actually earned once compounding is counted.

Compounding: today to the future

FV = PV × (1 + r)^n

KES 250,000 invested for 3 years at 10% a year:

FV = 250,000 × 1.10^3 = KES 332,750

Simple interest would give only 325,000. The extra 7,750 is interest earning interest, which is the whole point of compounding.

Discounting: the future back to today

PV = FV ÷ (1 + r)^n

KES 800,000 receivable in 5 years, discounted at 8%:

PV = 800,000 ÷ 1.08^5 = KES 544,467

So a promise of 800,000 in five years is worth about 544,467 today. Offered 600,000 in cash instead, you should take the cash.

Annuities

Equal amounts at equal intervals. Two formulas, depending on direction.

a) Present value of an annuity — what a stream of future receipts is worth now

PV = A × [1 − (1 + r)^-n] ÷ r

KES 120,000 a year for 4 years at 12%:

PV = 120,000 × [1 − 1.12^-4] ÷ 0.12 = KES 364,482

b) Future value of an annuity — what regular savings accumulate to

FV = A × [(1 + r)^n − 1] ÷ r

KES 45,000 saved at the end of each year for 5 years at 9%:

FV = 45,000 × [1.09^5 − 1] ÷ 0.09 = KES 269,312

:::checkpoint You pay in 45,000 a year for five years — 225,000 in total — and end with 269,312. Explain where the extra 44,312 comes from, and why it is smaller than five full years of interest on the whole 225,000. :::

Perpetuities

A cash flow with no end date:

PV = A ÷ r

KES 84,000 a year for ever at 14% is worth 84,000 ÷ 0.14 = KES 600,000.

The total received is unlimited, yet the value is finite, because each later receipt is discounted more heavily than the one before.

If the cash flow grows at a constant rate g:

PV = A1 ÷ (r − g)

A dividend of KES 6.36 next year, growing at 6% for ever, required return 15%:

PV = 6.36 ÷ (0.15 − 0.06) = KES 70.67

This is the Gordon growth model. It returns in Cost of Capital and again in Business Valuation. Note the condition: g must be below r, or the formula gives a negative or infinite answer with no economic meaning.

Compounding more than once a year

Divide the rate, multiply the periods. KES 400,000 at 10% nominal for 4 years, compounded quarterly:

FV = 400,000 × (1 + 0.10 / 4)^16 = KES 593,802

Using 1.10^4 instead gives 585,640 and understates the effect.

The effective annual rate makes competing offers comparable:

EAR = (1 + r / m)^m − 1

At 16% nominal compounded quarterly: (1 + 0.04)^4 − 1 = 16.99%

So "16% compounded quarterly" really costs 16.99% a year. A rival offer of 16.5% compounded annually is the cheaper loan, despite the higher headline rate.

Where the examiner takes this

Time value questions rarely stand alone at Intermediate level. They arrive inside investment appraisal (discounting project cash flows), cost of capital (valuing a dividend stream), and lease-versus-buy decisions. Make the mechanics automatic here and the rest of the paper gets easier.

:::checkpoint Two lenders quote the same 12% nominal rate. One compounds annually, the other monthly. Without calculating, say which loan costs more and why, then name the single figure you would ask both lenders for to settle it. :::