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Reading 06: Time Value of Money (Funding a Future Obligation (Step 1:…
Reading 06: Time Value of Money
01. Interest Rate
Formula
Required rate of return = Real Risk-free Rate + Expected Inflation + Default Risk Premium + Liquidity Premium + Maturity Risk Premium.
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.
Interpretation
a. Required rate of return
b. Present value discount rate.
c. Opportunity cost of consuming rather than saving.
02. Time Value formulas
Present Value
Single Cashflow
\(PV_{n}=FV\times \left ( 1+r \right )^{-n}\)
Annuity
Ordinary Annuity
\( PV_{A}=\frac{A}{r}\times \left [ 1-\frac{1}{\left ( 1+r \right )^{n}} \right ]\)
Annuity Due
\( PV_{A}=A+\frac{A}{r}\times \left [ 1-\frac{1}{\left ( 1+r \right )^{n-1}} \right ]\)
More than 1 compounding period per year
\( PV_{n}=FV\times \left ( 1+\frac{r}{m} \right )^{-n\times m} \)
Perpetuity
\( PV=\frac{A}{r}\)
Future Value
Single Cashflow
\( FV_{n}=PV\times \left ( 1+r \right )^{n}\)
Continuous compounding
\( FV_{n}=PV\times e^{r\times n}\)
Annuity
Annuity Due
\( FV_{A}=\left ( 1+r \right )\times A\times \frac{\left ( 1+r \right )^{n}-1}{1}\)
Ordinary Annuity
\( FV_{A}=A\times \frac{\left ( 1+r \right )^{n}-1}{r}\)
More than 1 compounding period per year
\(FV_{n}=PV\times \left ( 1+\frac{r}{m} \right )^{n\times m} \)
Effective Annual Rate
Compound more than 1 time per year
\(EAR=\left ( 1+\frac{r}{m} \right )^{m}-1\)
Continuous compounding
\( EAR=e^{n \times r}-1\)
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.