At what rate of compound interest will rupees 20000 become 24200 after 2 years?

Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is:

FV = PV(1 + r/m)mtorFV = PV(1 + i)n

where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

One may solve for the present value PV to obtain:

PV = FV/(1 + r/m)mt

Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt   = 20,000(1 + 0.085/12)(12)(4)   = $28,065.30

Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest.

Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is:

reff = (1 + r/m)m - 1.

This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom.

Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of:

r eff =(1 + rnom /m)m   =   (1 + 0.098/12)12 - 1   =  0.1025.

Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then

R = P � r / [1 - (1 + r)-n]

and

D = P � (1 + r)k - R � [(1 + r)k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where:

n = log[x / (x � P � r)] / log (1 + r)

where Log is the logarithm in any base, say 10, or e.

Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then

FV = [ R(1 + r)n - 1 ] / r

Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be

FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / iwhere i = r/m is the interest paid each period and n = m � t is the total number of periods.

Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is:

FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
5,000(1+0.05/12)120 + [100(1+0.05/12)120 - 1 ] / (0.05/12) = $23,763.28

Value of a Bond:

Let N = number of year to maturity, I = the interest rate, D = the dividend, and F = the face-value at the end of N years, then the value of the bond is V, whereV = (D/i) + (F - D/i)/(1 + i)N

V is the sum of the value of the dividends and the final payment.

You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision.

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Here P1 = Rs.20000 and r = 10%
So, Amount after 1 year
= `"P"(1 + "r"/100)`

= `20000(1 + 10/100)`

= `20000 xx (110)/(100)` = 22000
Thus, P2 = Rs.22000 and r = 10%
Amount after 2 year
= `"P"(1 + "r"/100)`

= `22000(1 + 10/100)`

= `22000 xx (110)/(100)` = 24200
Thus, P3 = Rs.24200 and r = 10%
Amount after 3 year
= `"P"(1 + "r"/100)`

= `24200(1 + 10/100)`

= `24200 xx (110)/(100)` = 26620
Hence, Amount = Rs.26620
Also, C.I.
= A - P
= Rs.26620 - Rs.20000
= Rs.6620.

Answer (Detailed Solution Below)

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P = 20000

A = 28800

T = 2 years

Formula for compound interest,

A = P{1 + R/100}T

⇒ Putting all values in above formula, we get-

⇒ 28800 = 20000{ 1 + R/100}2

⇒ 28800/20000 = {1 + R/100}2

⇒144/100 = {1 + R/100}2

\(\Rightarrow 1 + \frac{{\rm{R}}}{{100}} = \frac{{12}}{{10}}\)

⇒ R = 20%

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