# Time Value of Money

Updated 2018-06-07

# Lesson 5: Time Value of Money

Previously we’ve dealt economic problems assuming current costs and benefits (money doesn’t change value over time). A more realistic / complex problem should account for how the value of money changes over time.

### Compound

Interest is calculated periodically on the unpaid amount, thus accumulates based on the compounding periods (interest on top of interest).

Period Value at start of period Interest at end of period Future value at end of period
1 $$P$$ $$P\cdot i$$ $$P+P\cdot i$$
2 $$P+P\cdot i$$ $$(P+P\cdot i)\cdot i$$ $$P+P\cdot i+(P+P\cdot i)\cdot i$$
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$n$$ $$\boxed{P(1+i)^{n-1}}$$ $$\boxed{i\cdot P(1+i)^{n-1}}$$ $$\boxed{P(1+i)^n}$$

Generalizes to single payment compound amount: $$F=P(1+i)^n$$(single payment implies no payment is made during the periods and only paid at the very end). And interest at a particular period: $$I_n=i\cdot P(1+i)^{n-1}$$.

#### Discount

Compounding is using compound interest rate to determine future value $$F​$$ given present value $$P​$$. Discounting is using the discount rate to determine present value $$P​$$ given future value $$F​$$.

• Used in discounted cash flow (DCF) analysis
• Discount rate is the interest rate used in DCF analysis
• Greater uncertainty implies higher discount rate

Example: compound interest

Given 1000 today, put it in a saving’s account with 2% interest. How much would it be in 10 years? Applying the compounding interest formula: \begin{aligned} F&=P(1+i)^n\\ &=(1000)(1.02)^{10}\\ &=1219 \end{aligned} We earn extra219 in 10 years.

Example: discount

We want to have 100,000 in our 5% savings account in 40 years. How much money should we put in now? Applying the inverse compounding interest formula to get discount formula: \begin{align} P&=F(1+i)^{-n}\\ &=(100000)(1.05)^{-40}\\ &=14205 \end{align} We need to put in \14,205 right now to get \$100,000 in 40 years. ### Nominal Nominal interest rate, denoted by $$r$$ is the annual interest rate without considering any effect of any compounding. Example: 2% interest semiannually Then the nominal interest rate $$r$$ is $$2\times 2\%=4\%$$. ### Effective Effective interest rate per year is denoted by $$i_a$$. This is the annual interest rate when accounting for compounding during the year. Example: Suppose we deposit$100, and we still have 2% interest semiannually like the previous example.

Then the value at the end of the first year is $$P(1+i)^2=100(1+0.02)^2=104.04$$. Effectively, the interest value is \$4.04. The effective interest rate, $$i_a$$ is thus $$4.04\div 100=4.04\%$$.

Notice how the effective interest rate is slightly different from nominal interest rate.

The conversion between effective and nominal interest rate is:

\boxed{ \begin{aligned} i_a&=\left(1+\frac rm\right)^m-1\\ &=\left(1+i\right)^m-1 \end{aligned} }

Where $$r$$ is the nominal interest rate, $$m$$ is the number of compounding sub-periods per time period, and $$i$$ is the effective interest per compounding period.

## Cashflow Equivalence

Money is values differently over time. Comparing the cash flow values in different time can determine when they are equivalent.

Equivalence implies different amount of money have the same value in different times with respect to an interest rate.