Time Value of Money

Learning Objectives

  • Calculate equivalent sums of money using discount rates

  • Able to use the internal rate of return (IRR) to quantify an energy investment

  • Able to use the capital recovery function (CRF) to estimate a loan

    payment

Concepts

  • Time Value of Money

  • Equivalence and Comparison Principle

  • Net Present Value (NPV)

  • Future Value

  • Discounting

  • Discount rate

  • Interest rate

Equivalence principle

  • Given a choice between money now and money later, most demand a larger value at a later date

  • When someone is indifferent between sum 1 now and sum 2 at a fixed later date, the sums are considered equivalent

  • This equivalence can be expressed using a discount rate

Equivalence principle as unit conversion

  • You can think of using the discount rate as a unit conversion where you have units of dollars now and units of dollars in the future

  • The equals sign in this case signifies the indifference of a consumer or an investor.

  • However, our unit conversion is based on a more complicated model than a usual unit conversion.

Discount rate vs. Interest rate

  • Discount rate usually refers to personal preferences

  • Interest rate is usually a real rate charged by a bank

Mathematical Model

In both the case of a discount rate and an interest rate, we use an exponential model. Our inflation calculations that assume a constant inflation rate also use an exponential model.

There is some controversy over the use of this discount rate model in climate change discussions.

David Roberts has an entertaining article on this topic.

Monthly vs. Yearly interest rates

  • Many types of loans advertise a yearly interest rate, but charge

    interest monthly.

  • The yearly interest rate is the APR or annual percentage rate

  • To find the monthly rate divide this by twelve

  • $i$ is the annual percentage rate

  • $n$ is the number of periods in months

FV=PV(1+i/12)nFV = PV (1 + i/12)^{n}

Cash flow diagrams

Internal Rate of Return

  • Tells us at what interest rate a cash flow has a net present value of

    zero

  • We will look at this on a spreadsheet

  • This doesn't have a closed-form solution

  • Usually solved by a computer

Finding the IRR is the equivalent of asking, here is a loan and payments, what was the interest rate you got?

Inflation

Inflation

  • The cost of goods usually rises over time

  • This rate is monitored by the Consumer Price Index

  • As prices rise, the value of money decreases

Inflation

1+r0=1+r1+f1 + r_0 = \frac{1+r}{1+f}

  • $r_0$ is the effective rate of interest

  • $r$ is the nominal rate of interest

  • $f$ is the inflation rate

For small inflation rates,

r0≈r−fr_0 \approx r - f

Discount Rate and Net Present Value

Present Value (USD)=Future Amount (USD)(1+Discount Rate)number of years in future\textrm{Present Value (USD)} = \frac {\textrm{Future Amount (USD)}} {(1 + \textrm{Discount Rate})^{\textrm{number of years in future}} }

P=F(1+i)nP = \frac {F} {(1 + i)^{n} }

Present Value Notation

Single payment PV=C(1+i)nPV = \frac{C}{(1+i)^n}

Stream of payments

PV=C0+C11+i+C2(1+i)2+⋯+CN(1+i)NPV = C_0 + \frac{C_1}{1+i} + \frac{C_2}{(1+i)^2} + \cdots + \frac{C_N}{(1+i)^N}

Compact notation PV=∑n=0NCn(1+i)nPV = \sum_{n=0}^{N} \frac{C_n}{(1+i)^n}

NPV Spreadsheet Example

  • Excel considers first value in the NPV function to be year 1

Capital Recovery Function

Capital Recovery Factor

Suppose we make a loan. We want to know what the yearly payment is so that the present value of all payments is equal to the loan amount.

This formula allows us to calculate this payment.

CRF=i(1+i)n(1+i)n−1CRF = \frac {i(1+i)^n}{(1+i)^n-1}

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