Time Value of Money

Interest is charged on the principal for one year:

$I = P \times r \times t = 2000 \times 0.18 \times 1 = \text{Rs } 360$

You pay Rs 360 as interest on the bank loan (so Rs 2,360 is repaid in total).

Compounding is the process of finding the future value of money when interest is earned not only on the original principal but also on the interest already accumulated, so the amount grows faster each period. The formula is $FV = PV(1+r)^n$.

Example: Rs 1,000 deposited for 2 years at 10% p.a. grows to $1000 \times (1.10)^2 = \text{Rs } 1,210$, whereas simple interest would give only Rs 1,200.

(a) Present value with $PV = \dfrac{CF_t}{(1.10)^t}$:

Present value $\approx$ Rs 2,188.60.

(b) Future value at the end of year 5 with $FV = CF_t(1.10)^{5-t}$:

Future value $\approx$ Rs 3,524.77. (Check: $2{,}188.60 \times 1.10^5 = 3{,}524.77$.)

Meaning. An annuity is a series of equal cash payments or receipts made at regular intervals (every month or year) for a fixed number of periods. Examples are loan instalments, monthly rent and insurance premiums.

Types of annuity

1. Ordinary annuity (annuity in arrears). Each payment is made at the end of every period, for example a loan instalment paid at the end of each year. $FV = A\,\frac{(1+r)^n - 1}{r}, \qquad PV = A\,\frac{1-(1+r)^{-n}}{r}$

2. Annuity due (annuity in advance). Each payment is made at the beginning of every period, for example rent paid in advance. Its value is the ordinary annuity value multiplied by $(1+r)$, because every payment earns one extra period of interest.

3. Perpetuity. An annuity that continues forever, with no end date, for example a fixed payment from an endowment. Its present value is $PV = \frac{A}{r}$

A fully amortized loan is repaid by equal year-end payments, each containing interest on the outstanding balance and a principal repayment.

(a) Annual payment

$PMT = \frac{P \cdot r}{1-(1+r)^{-n}} = \frac{1{,}000{,}000 \times 0.12}{1-(1.12)^{-5}} = \frac{1{,}000{,}000}{3.6048} = \text{Rs } 277{,}409.73$

Amortization schedule (Rs)

(b) Year 2 interest fraction:

$\frac{101{,}110.83}{277{,}409.73} = 0.3645 \approx 36.45\%$

(c) Year 3 principal fraction:

$\frac{197{,}454.77}{277{,}409.73} = 0.7118 \approx 71.18\%$

(d) In the final (fifth) year, interest is about Rs 29,722.47 and principal repaid is about Rs 247,687.26, together making the payment of Rs 277,409.73 and clearing the loan.

(b) Decrease

Present value is found by discounting a future cash flow, $PV = \dfrac{FV}{(1+i)^n}$. Since the discount rate $i$ appears in the denominator, a rise in $i$ makes the denominator larger and hence the present value smaller.

Given:

• Present value (PV) = Rs. 50,000
• Rate of interest (i) = 10% per annum = 0.10
• Time (n) = 3 years

Formula: $FV = PV(1+i)^n$

Calculation of accumulated amount: $FV = 50{,}000(1+0.10)^3$ $FV = 50{,}000(1.10)^3$ $FV = 50{,}000 \times 1.331$ $FV = \text{Rs. } 66{,}550$

Calculation of compound interest: $CI = FV - PV$ $CI = 66{,}550 - 50{,}000$ $CI = \text{Rs. } 16{,}550$

Answer:

• Amount accumulated after 3 years = Rs. 66,550
• Compound interest earned = Rs. 16,550

Given:

• Annual cash inflow (A) = Rs 20,000
• Required rate of return (i) = 12% = 0.12
• Number of years (n) = 5

To find: Present value of annuity (PVA)

Formula: $PVA = A \times \frac{1-(1+i)^{-n}}{i}$

Substitution: $PVA = 20{,}000 \times \frac{1-(1+0.12)^{-5}}{0.12}$

$PVA = 20{,}000 \times \frac{1-(1.12)^{-5}}{0.12}$

$(1.12)^{5} = 1.7623 \quad \Rightarrow \quad (1.12)^{-5} = \frac{1}{1.7623} = 0.5674$

$PVA = 20{,}000 \times \frac{1-0.5674}{0.12}$

$PVA = 20{,}000 \times \frac{0.4326}{0.12}$

$PVA = 20{,}000 \times 3.6048$

$PVA = \text{Rs } 72{,}095.52 \text{ (approx.)}$

Conclusion: The present value of the annuity is Rs 72,095.52. This means the series of five year-end receipts of Rs 20,000 each is worth Rs 72,095.52 today at a 12% discount rate.

Given: Loan $P$ = Rs 5,00,000; nominal rate = 12% p.a. compounded monthly; repayment in 5 equal annual installments.

Since interest is compounded monthly but installments are paid yearly, first convert the nominal rate into the effective annual rate.

Step 1: Effective annual rate

Monthly rate = $\dfrac{12\%}{12} = 1\% = 0.01$

$i = (1+0.01)^{12} - 1 = 1.126825 - 1 = 0.126825 \approx 12.68\%$

(a) Annual installment

$EMI = \frac{P \times i}{1-(1+i)^{-n}} = \frac{5{,}00{,}000 \times 0.126825}{1-(1.126825)^{-5}}$

$(1.126825)^{5} = 1.8167 \Rightarrow (1.126825)^{-5} = 0.5504$

$EMI = \frac{63{,}412.52}{1-0.5504} = \frac{63{,}412.52}{0.4496} \approx \text{Rs } 1{,}41{,}058$

Each annual installment is about Rs 1,41,058.

(b) Loan amortization schedule (first two years)

Interest each year = opening balance $\times$ 12.6825%; principal repaid = installment $-$ interest.

Interest falls and the principal component rises each year because interest is charged on a declining balance.

(If the question is read as 12% compounded annually, then $i = 0.12$ and $EMI = \dfrac{5{,}00{,}000 \times 0.12}{1-(1.12)^{-5}} \approx$ Rs 1,38,705.)

Periodic rate $= \dfrac{\text{annual rate}}{m}$ and effective annual rate $EAR = \left(1 + \dfrac{i}{m}\right)^m - 1$, where $m$ is the number of payments in a year.

(a) Half-yearly interest ($m = 2$)

$\text{Periodic rate} = \frac{12\%}{2} = 6\% \text{ per half-year}$

$EAR = (1.06)^2 - 1 = 1.1236 - 1 = 12.36\%$

Interest paid each half-year $= 5{,}00{,}000 \times 0.06 = \text{Rs } 30{,}000$.

(b) Quarterly interest ($m = 4$)

$\text{Periodic rate} = \frac{12\%}{4} = 3\% \text{ per quarter}$

$EAR = (1.03)^4 - 1 = 1.12551 - 1 = 12.55\%$

Interest paid each quarter $= 5{,}00{,}000 \times 0.03 = \text{Rs } 15{,}000$.

The more frequent the payment, the higher the effective rate: 12.36% for half-yearly and 12.55% for quarterly, both above the stated 12%.

Use the compound interest formula:

$FV = PV(1+r)^n = 4{,}00{,}000 \times (1.06)^4$

$(1.06)^4 = 1.262477$

$FV = 4{,}00{,}000 \times 1.262477 = \text{Rs } 5{,}04{,}990.78$

At the end of 4 years he will have about Rs 5,04,991, of which Rs 1,04,991 is interest.

To compare the two offers we must bring them to the same point in time. Here both are compared at the end of year 3 (future value) and again today (present value).

Method 1: Future value of Rs 5,00,000 received today

$FV = PV(1+r)^n = 5{,}00{,}000 \times (1.08)^3$

$(1.08)^3 = 1.259712$

$FV = 5{,}00{,}000 \times 1.259712 = \text{Rs } 6{,}29{,}856$

Method 2: Present value of Rs 5,60,000 received after 3 years

$PV = \frac{FV}{(1+r)^n} = \frac{5{,}60{,}000}{1.259712} = \text{Rs } 4{,}44{,}546$

Decision

• Rs 5,00,000 today grows to Rs 6,29,856 in 3 years, which is more than Rs 5,60,000.
• Equivalently, Rs 5,60,000 after 3 years is worth only Rs 4,44,546 today, which is less than Rs 5,00,000.

Both methods give the same conclusion, so I would choose Rs 5,00,000 today. Money received earlier can be invested at 8% and the gain (Rs 69,856 more at the end of year 3) is larger than the extra Rs 60,000 offered for waiting. This is the time value of money.

Each cash flow is assumed to arrive at the end of its year and is compounded to the end of year 5 at 10% using $FV = CF \times (1.10)^{5-t}$.

Compounding factors: year 1 = $(1.1)^4 = 1.4641$, year 2 = $(1.1)^3 = 1.3310$, year 3 = $(1.1)^2 = 1.2100$, year 4 = $1.1000$, year 5 = $1.0000$.

Project A (equal cash flows, an ordinary annuity)

$FV_A = 10{,}000 \times \frac{(1.1)^5 - 1}{0.1} = 10{,}000 \times \frac{1.61051 - 1}{0.1} = 10{,}000 \times 6.1051$

$FV_A = \text{Rs } 61{,}051$

Project B (uneven cash flows)

$FV_B = \text{Rs } 61{,}424.80$

Conclusion. Both projects have a total cash inflow of Rs 50,000, but Project B has the higher future value (Rs 61,425 against Rs 61,051) because more of its cash comes in the earlier years in a larger amount and so earns more interest.