Business mathematics

Question

A finance director estimates that his company will have to spend 0.25 million on new machinery in two years from now. Two alternative method of providing the money are being considered both assuming an annual rate of interest of 10%.

A Single sum of money A to be set aside and invested now, with interest compounded every six months. How much should this  single sum be, and what is the effective annual rate of interest?

B to be put into a reserve fund  every six months, starting now. If interest is compounded  every six months, what should B be, in order that the 0.25 million will be available in two years from now?

Answer 1

The finance director needs 0.25 million in two years to purchase machinery. The interest rate is 10% annually, compounded every six months. Two options are provided:

1. Option A: Set aside a single lump sum today (compounded semi-annually).
2. Option B: Deposit equal amounts, B, into a reserve fund every six months, starting now.

We need to find the single sum for Option A, the effective annual rate of interest, and the equal semi-annual deposits B for Option B.

Step 1: Understanding Interest Compounding

The annual rate of 10% with semi-annual compounding means that every six months, the rate is:

Semi-annual interest rate=10%2=5% or 0.05\text{Semi-annual interest rate} = \frac{10\%}{2} = 5\% \text{ or } 0.05Semi-annual interest rate=210%​=5% or 0.05

The number of compounding periods in two years (since interest compounds every six months) is:

n=2×2=4 periods.n = 2 \times 2 = 4 \text{ periods}.n=2×2=4 periods.

Option A: Single Sum Set Aside Today

For a lump sum (present value), the formula is:

FV=PV×(1+r)nFV = PV \times (1 + r)^nFV=PV×(1+r)n

Where:

1. FVFVFV is the future value (0.25 million).
2. rrr is the semi-annual interest rate (0.05).
3. nnn is the number of periods (4).
4. PVPVPV is the present value we need to calculate.

Rearranging to solve for PVPVPV:

PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}PV=(1+r)nFV​

Substitute the values:

PV=250,000(1+0.05)4PV = \frac{250,000}{(1 + 0.05)^4}PV=(1+0.05)4250,000​ PV=250,0001.21550625≈205,378PV = \frac{250,000}{1.21550625} \approx 205,378PV=1.21550625250,000​≈205,378

So, the single sum to set aside today is approximately $205,378.

Effective Annual Rate of Interest (EAR):

The formula for EAR with semi-annual compounding is:

EAR=(1+rnominalm)m−1\text{EAR} = \left(1 + \frac{r_{\text{nominal}}}{m}\right)^m - 1EAR=(1+mrnominal​​)m−1

Where:

1. rnominal=10%r_{\text{nominal}} = 10\%rnominal​=10% (annual nominal rate).
2. m=2m = 2m=2 (compounding periods per year).

Substitute:

EAR=(1+0.102)2−1\text{EAR} = \left(1 + \frac{0.10}{2}\right)^2 - 1EAR=(1+20.10​)2−1 EAR=(1.05)2−1=1.1025−1=0.1025 or 10.25%\text{EAR} = (1.05)^2 - 1 = 1.1025 - 1 = 0.1025 \text{ or } 10.25\%EAR=(1.05)2−1=1.1025−1=0.1025 or 10.25%

The effective annual rate of interest is 10.25%.

Option B: Equal Semi-Annual Deposits

For equal deposits made every six months into a reserve fund, we use the future value of an ordinary annuity formula:

FV=B×(1+r)n−1rFV = B \times \frac{(1 + r)^n - 1}{r}FV=B×r(1+r)n−1​

Where:

1. FV=250,000FV = 250,000FV=250,000
2. r=0.05r = 0.05r=0.05 (semi-annual rate)
3. n=4n = 4n=4 (number of periods)
4. BBB is the semi-annual deposit.

Rearranging to solve for BBB:

B=FV⋅r(1+r)n−1B = \frac{FV \cdot r}{(1 + r)^n - 1}B=(1+r)n−1FV⋅r​

Substitute the values:

B=250,000⋅0.05(1.05)4−1B = \frac{250,000 \cdot 0.05}{(1.05)^4 - 1}B=(1.05)4−1250,000⋅0.05​

First, calculate the denominator:

(1.05)4=1.21550625,so (1.05)4−1=0.21550625(1.05)^4 = 1.21550625, \quad \text{so } (1.05)^4 - 1 = 0.21550625(1.05)4=1.21550625,so (1.05)4−1=0.21550625

Now solve for BBB:

B=250,000⋅0.050.21550625≈12,5000.21550625≈58,000B = \frac{250,000 \cdot 0.05}{0.21550625} \approx \frac{12,500}{0.21550625} \approx 58,000B=0.21550625250,000⋅0.05​≈0.2155062512,500​≈58,000

The equal semi-annual deposit, BBB, is approximately $58,000.

Final Answers:

1. Option A: The single sum to set aside today is approximately $205,378.
2. The effective annual rate of interest is 10.25%.
3. Option B: The equal semi-annual deposit into the reserve fund is approximately $58,000.

Let me know if you'd like me to break down any part further!