### Time Value of Money

I am amazed how long it's taken me to finally understand the time value of money!

Today it finally clicked.

FV of 1 invested with r interest rate = (1 + r)^t where t is the number of periods.

PV of 1 paid out in the future = 1/(1+r)^t where t is the number of periods.

These both assume compound interest.

PV = FV/(1+r)^t

Balancing this equation to solve for r gives

r = (FV/PV)^(1/t) -1

Annuity = a sequence of evenly spaced, level cash flows.

Perpetuity = payment stream lasts forever.

Cash payment from perpetuity = interest rate X present value

C = r X PV

PV = C/r

Suppose some worthy person wishes to endow a chair in finance at your university. If r = 10% and aim is to provide 100K Cash payment annually, how much must be set aside today?

PV = 100K/.10 = 1 million

Present value of t-year annuity = C[1/r - (1/(r(1+r)^t))] <- "Annunity Factor"

"Amortizing Loan " - part of the monthly payment is used to pay interest on the loan and part is used to reducet he amount of the loan.

FV = PV X (1+r)^t

FV = [(1+r)^t -1]/r (For a $1 annuity)

1 + effective annual rate = (1 + monthly rate)^12

for example if monthly is 1%, annual is 12.68%\

If you're quoted an APR and there are m compounding periods in a year, then $1 will grow to $1 X (1+APR/m)^m -- m is number of periods in year.

Real future value of investment. Interest rate in real dollars.

1 + real interest rate = (1 + nominal interest rate)/(1 + inflation rate)

Real interest rate ~ nominal interest rate - inflation rate.

Today it finally clicked.

FV of 1 invested with r interest rate = (1 + r)^t where t is the number of periods.

PV of 1 paid out in the future = 1/(1+r)^t where t is the number of periods.

These both assume compound interest.

PV = FV/(1+r)^t

Balancing this equation to solve for r gives

r = (FV/PV)^(1/t) -1

**FV of multiple cash flows**- just take sum of each payment.Annuity = a sequence of evenly spaced, level cash flows.

Perpetuity = payment stream lasts forever.

**PV of a Perpetuity**Cash payment from perpetuity = interest rate X present value

C = r X PV

PV = C/r

Suppose some worthy person wishes to endow a chair in finance at your university. If r = 10% and aim is to provide 100K Cash payment annually, how much must be set aside today?

PV = 100K/.10 = 1 million

**PV of an Annuity**Present value of t-year annuity = C[1/r - (1/(r(1+r)^t))] <- "Annunity Factor"

"Amortizing Loan " - part of the monthly payment is used to pay interest on the loan and part is used to reducet he amount of the loan.

**FV of a Annuity**FV = PV X (1+r)^t

FV = [(1+r)^t -1]/r (For a $1 annuity)

**Effective Annual Interest Rate = not the same as APR!**

1 + effective annual rate = (1 + monthly rate)^12

for example if monthly is 1%, annual is 12.68%\

If you're quoted an APR and there are m compounding periods in a year, then $1 will grow to $1 X (1+APR/m)^m -- m is number of periods in year.

**Inflation**Real future value of investment. Interest rate in real dollars.

1 + real interest rate = (1 + nominal interest rate)/(1 + inflation rate)

*Useful approximation*Real interest rate ~ nominal interest rate - inflation rate.

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