Reading 06: Time Value of Money

Reading 06: Time Value of Money

01. Interest Rate

Formula

02. Time Value formulas

Interpretation

a. Required rate of return

b. Present value discount rate.

c. Opportunity cost of consuming rather than saving.

Required rate of return = Real Risk-free Rate + Expected Inflation + Default Risk Premium + Liquidity Premium + Maturity Risk Premium.

Present Value

Future Value

Single Cashflow

Annuity

More than 1 compounding period per year

Perpetuity

Single Cashflow

Continuous compounding

Annuity

Effective Annual Rate

Compound more than 1 time per year

Continuous compounding

Annuity Due

$PV=\frac{A}{r}$

$ FV_{n}=PV\times \left ( 1+r \right )^{n}$

More than 1 compounding period per year

$FV_{n}=PV\times \left ( 1+\frac{r}{m} \right )^{n\times m} $

$ FV_{n}=PV\times e^{r\times n}$

$EAR=\left ( 1+\frac{r}{m} \right )^{m}-1$

$ EAR=e^{n \times r}-1$

Ordinary Annuity

$ FV_{A}=A\times \frac{\left ( 1+r \right )^{n}-1}{r}$

$ FV_{A}=\left ( 1+r \right )\times A\times \frac{\left ( 1+r \right )^{n}-1}{1}$

$PV_{n}=FV\times \left ( 1+r \right )^{-n}$

$ PV_{n}=FV\times \left ( 1+\frac{r}{m} \right )^{-n\times m} $

Ordinary Annuity

Annuity Due

$ PV_{A}=\frac{A}{r}\times \left [ 1-\frac{1}{\left ( 1+r \right )^{n}} \right ]$

$ PV_{A}=A+\frac{A}{r}\times \left [ 1-\frac{1}{\left ( 1+r \right )^{n-1}} \right ]$

Real risk-free rate: is a theoretical rate on a single-period loan that has no expectation of inflation.

Default risk: risk that a borrower will not make the promised payment in timely manner

Liquidity risk: risk of receiving less than fair value for an investment if it must be sold for cash quickly.

Maturity risk: Longer-maturity investment have more volatile price than shorter one.

03. Uneven Cashflows Problem

Step 1: Draw Time Line

Step 2: At the same point in time, Solve for PV or FV of each cash flow

Step 3: Sum all the PVs or FVs

When Compounding Periods are Other than Annual.

SET the compounding period (P/Y) in Calculator accordingly (BUT MUST RESET AFTER THE CALCULATION), and solve as normal, or

Set P/Y = 1 (aka. Compound annually) but

I/Y = Annual interest rate ÷ m (m is the number of compounding period per year)

N = Number of year × m

Funding a Future Obligation

Step 1: Reading the problem carefully, identifying key words regarding using BGN instead of END (“annuity due”, “today”, “beginning”).

Step 2: Draw time line, including the corresponding payments.

Step 3: Identifying the moment in time (t) that all the PVs, FVs will be calculate.

Step 4: Solve the problem.