Interest
by Max Yi Ren
Disclaimer
This lecture note is created based on existing materials from the ODE lab at University of Michigan.
Interest can be thought as the “rental charge” for the use of money.
Simple interest:
- Principal ($P$) = dollar value at the time given
- (Discount) Rate ($i$) = interest earned per unit principal per unit time
- Future value ($FV$) = value at a future point
Formulas:
- One period: $FV = P + Pi = P(1+i)$
- Multiple ($n$) periods: $FV = P(1+i)^n$
Present value
Future value can be converted into present value.
Present value allows the evaluation of multiple value amounts arriving at different times to be equivalently valued at a given point in time.
Present value $PV = FV/(1+i)^n$
Series compound future value
Series of equal payments ($R$) given at regular time periods \(FV = [\frac{(1+i)^n-1}{i}]R\)
Practice: Calculate the future value of $R = 5,000$ at the end of each year for 5 years at $6\%$.
Series present value
Series of equal payments ($R$) given at regular time periods
\[PV = [\frac{(1+i)^n-1}{i(1+i)^n}]R\]Practice: Calculate the present value of $R = 5,000$ at the end of each year for 5 years at $6\%$.
Gradient series
$R$ now changes $g\%$ at the end of each year
\(PV = [\frac{1-(1+g)^n(1+i)^{-n}}{i-g}]R\), when $g=i$, we have \(PV = \frac{nR}{1+i}\).
Net present value
With a series of $R$s.
\(NPV = \sum_{t=0}^N \frac{R_t}{(1+i)^t}\), where $R_t$ is the investment at time $t$.
Using Excel for calculation
- Future value = FV(rate,nper,pmt,pv,type)
- Present value = PV(rate,nper,pmt,fv,type)
- Net present value = NPV(rate,value1,value2, …)
- Non-uniform net present value = XNPV(rate,values,dates)
Simple investment economics
How do we evaluate alternative financing and investment decisions for launching a new product?
Investment economics or capital budgeting seeks to answer this question.
- Net present value (NPV)
- Annual cost
- Internal rate of return (IRR)
- Breakeven analysis
How to calculate NPV
see above.
How to calculate annual cost
Calculate the annual investment cost using \(R = [\frac{i(1+i)^n}{(1+i)^n-1}]PV\), or using the Excel function pmt(rate,nper,pv,fv,type); the annual maintenance cost, income, etc.
How to calculate IRR
Discount rate: the rate at which the present value of a series of investments is equal to the present value of the returns on those investments.
For an investment that requires and produces a number of cash flows over time, Internal Rate of Return is the discount rate that makes the net present value of those cash flows equal to zero.
For a set of transactions over time, the IRR on your transactions will give you a picture of your overall rate of return.
Given the NPV formula:
\(NPV = \sum_{t=0}^N \frac{R_t}{(1+i)^t}\),
IRR is the discount rate which sets the NPV of the given cash flows made at the given times to zero. This is a nonlinear equation that must be solved numerically. Note that some of the cash flow $R_t$s are expenses.
How to perform break-even analysis
Find how many periods are needed to equal the present value (PV) of the investment cost
Activity 1: Interest
Use this problem to practice the above analysis tools.