Math of Money:Compound Interest Review With Applications

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Compound Interest:

The future value (FV) of a good investment of current value (PV) bucks making interest at a yearly price of r compounded m times each year for a time period of t years is:

FV = PV(1 r/m that is + mt or

where i = r/m is the interest per compounding period and n = mt is the true wide range of compounding durations.

You can re re solve for the value that is present to acquire:

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

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

Observe that the attention won is $28,065.30 – $20,000 = $8,065.30 — significantly more compared to matching easy interest.

Effective Interest price: If cash is spent at a yearly price r, compounded m times each year, the effective rate of interest is:

r eff = (1 r/m that is + m – 1.

Here is the rate of interest that will provide the same yield if compounded only one time each year. In this context r can also be called the rate that is nominal and it is usually denoted as r nom .

Numerical instance: A CD having to pay 9.8% compounded month-to-month includes a nominal price of r nom = 0.098, as well as a rate that is effective of

r eff =(1 + r nom /m) m = (1 + 0.098/12) 12 – 1 = 0.1025.

Hence, we obtain a successful rate of interest of 10.25per cent, because the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the 12 months.

Home loan repayments elements: allow where P = principal, r = interest per period, n = amount of periods, k = quantity of re re payments, R = payment that is monthly and D = financial obligation stability after K re re re payments, then

R = P Р§ r / [1 - (1 + r) -n ]

D = P Р§ (1 + r) k – R Р§ [(1 + r) k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one chooses to spend a lot more than the payment per month, the real question is what amount of months does it just just take through to the home loan is repaid? The clear answer is, the rounded-up, where:

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

where Log could be the logarithm in virtually any base, state 10, or ag ag e.

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

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

Future Value for the Increasing Annuity: it really is an investment this is certainly making interest, and into which regular re payments of a hard and fast amount are produced. Suppose one makes a repayment of R at the conclusion of each period that is compounding a good investment with a present-day value of PV, paying rates of interest at a yearly price of r compounded m times each year, then your future value after t years are going to be

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

where i = r/m could be the interest compensated each period and letter = m Р§ t may be the number that is total of.

Numerical Example: You deposit $100 per thirty days into an account that now contains $5,000 and earns 5% interest each year compounded month-to-month. After a decade, how much money within the account is:

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

Worth of A relationship: allow N = range 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, where year

V = (D/i) + (F – D/i)/(1 i that is + letter

V could be the amount of the worth regarding the dividends plus the last repayment.

You would like to perform some sensitiveness analysis when it comes to « what-if » situations by entering different numerical value(s), in order to make your « good » strategic choice.

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