# Level 1 CFA® Exam:

TVM – Single Cash Flow Problems

In this lesson, you’ll learn how to compute the future value and the present value of a single cash flow (lump sum). We will discuss:

- some basic formulas,
- compounding and its impact on future and present values,
- other variables affecting future and present values, and
- the difference between stated annual interest rate and effective annual rate (EAR).

Suppose we have USD 1,000 that we want to invest. Because we have it already, we can say the amount is the present value (PV) of our investment. We're investing our money for one year at an expected rate of return of 10%. How much money we will receive in a year?

We will receive USD 1,100 [\(=1000\times{(1+10\%)}\)] in a year, an amount we call the future value (FV) of the investment.

Important!

Bear in mind that unless a question reads otherwise, a provided interest rate is always an annual rate.

Back to our example, the USD 1,000 is the principal on which interest is calculated. The formula we used applies to simple interest only, which means that interest is calculated only on the principal, proportionally to the lifetime of the investment. In practice, however, compounding is more common, so we usually deal with compound interest rather than simple interest. Compound interest calculation consists of adding interest accrued after the end of each compounding period to the principal. Each consecutive period, we invest a greater amount and so we receive greater interest.

Here is the general formula for future value (FV) using compounding:

\(FV_N=PV\times(1+r)^N\)

- \(FV_N\) - future value
- PV - present value
- r - periodic interest rate
- N - number of periods

Remember:

the more frequent the compounding, the higher the future value,

the higher the present value, the higher the future value,

the higher the number of periods, the higher the future value,

the higher the interest rate, the higher the future value.

We have USD 1,000 and invest it for 2 years. The annual interest rate is 10% and interest is compounded annually. Let's calculate the amount of our savings after the first and the second year respectively and calculate how much interest we will receive after each year.

After a year we get:

\(FV_1=1,000\times{(1+10\%)}=1,000\times1.1=1,100\)

\(\text{interest}_1=1,100-1,000=100\)

Because of compounding, after the first year, we invest the whole amount including both the principal in the amount of USD 1,000 and the interest of USD 100. After two years we get:

\(FV_2=1,100\times{(1+10\%)}=1,100\times1.1=1,210\)

\(\text{interest}_2=1,210-1,100=110\)

As you can see, thanks to investing a larger amount after the first year, the interest we earned in the second year was greater (USD 110 vs USD 100).

We can calculate the amount we will receive at the end of the second year using directly the formula for the future value:

\(FV_2=1,000\times{(1+10\%)^2}=1,000\times1.1^2=1,210\)

Important!

At a given interest rate, the future value increases as the present value increases and the life of the investment gets longer. With a given life of the investment, the future value increases when the present value increases and the interest rate increases.

The initial investment is USD 100,000. You decided to deposit it for three years with an interest rate of 4% a year compounded annually. What's the value of the investment in three years' time?

The present value equals USD 100,000, interest rate is 4%, N is equal to 3 years.

Once again, we need to use the formula for the future value:

\(FV_3=100,000\times{(1+4\%)^3}=100,000\times1.04^3=112,486.40\)

### Future Value for Multiple Compounding Periods and Continuous Compounding

star content check off when doneIn practice, interest is not only compounded at yearly intervals. Interest can be compounded with any frequency. For a compounding period different that one year, the following formula should be used:

\(FV_N=PV\times\Bigl(1+\frac{r_S}{m}\Bigr)^{m\times N}\)

- \(FV_N\) - future value
- PV - present value
- \(r_S\) - stated annual interest rate
- N - number of years
- m - number of compounding periods per year
- \(\frac{r_S}{m}\) - periodic interest rate

Compound interest is about adding interest accrued after the end of each compounding period to the principal. Each consecutive period we invest a greater amount and so we receive greater interest.

As we know, an interest rate is usually specified for a period of one year. It's called a stated annual interest rate or a quoted interest rate and is denoted by \(r_S\). The symbol \(m\) means the number of compounding periods per year, for example, if interest is compounded on a monthly basis, there are 12 compounding periods, so \(m=12\). This means that interest will be added to the principal at the end of each month. Compounding periods longer than a year are very uncommon, but supposing an interest is compounded e.g. every 4 years, then \(m=0.25\).

If we increased the number of compounding periods ad infinitum (i.e. by going from monthly, to daily, to hourly, to one-second compounding periods, etc.), we would arrive at continuous compounding.

Here’s the formula for continuous compounding:

\(FV_N=PV\times e^{r_S\times N}\)

- \(FV_N\) - future value
- PV - present value
- \(r_S\) - stated annual interest rate
- N - number of years
- e - Euler's number

(...)

Another crucial concept is the effective annual rate.

Effective annual rate (EAR) is the interest rate that an investor actually earns annually taking into account the frequency of compounding.

Here's the fomrula to be used in your level 1 CFA exam:

\(EAR=(1+r)^m-1\)

- \(EAR\) - Effective Annual Rate
- \(m\) - number of compounding periods per year
- \(r\) - periodic interest rate

Important!

If interest is compounded more frequently than once a year, the EAR will be always greater than the stated annual interest rate.

The difference between the EAR and the stated annual interest rate:

Suppose we decide to put USD 200 in two deposit accounts, USD 100 in one account and USD 100 in the other. The first account pays 5% annually with interest compounded once a year and the other pays 5% annually but compounded monthly.

In the first case, the future value of the investment after a year will amount to USD 105.

In the other case, our interest will be compounded every month, so the future value of the investment after a year will amount to:

\(100\times(1+\frac{5\%}{12})^{12}=105.12\)

Since interest in the second case is compounded monthly, you can earn 12 cents more.

In both cases, the stated annual interest rate is 5%. EAR in the first case equals:

\(EAR=(1+5\%)^1-1=5\%\)

which is the same value as the stated annual interest rate.

However, in the second case with monthly compounding, the effective annual rate equals:

\(EAR=(1+\frac{5\%}{12})^{12}-1=5.12\%\)

As you can see, in the second case EAR exceeds the stated annual interest rate by 0.12 percentage points.

The effective annual interest rate for continuous compounding can be calculated using the following formula:

\(EAR=e^{r_S}-1\)

- EAR - Effective Annual Rate
- e - Euler's number
- \(r_S\) - stated annual interest rate

You need to consider two investments with the same initial outlay. The first will last 4 years, interest is compounded daily (assume a year has 365 days), and the annual interest rate is 10%. The other will also last 4 years, however, interest will be compounded semi-annually and the annual rate is 11%. Which investment offers a higher rate of return? What is the difference between the effective annual rates of the two investments?

(...)

Now that we know how to calculate the future value, we can deal with the concept of present value. Once you find out how to calculate present value, you will be able to answer the following question: How much is the amount which we are to pay or receive in the future worth now?

The present value of an investment:

\(PV=\frac{FV_N}{(1+r)^N}\)

- PV - present value
- \(FV_N\) - future value
- r - periodic interest rate
- N - number of periods

Remember:

the more frequent the compounding, the lower the present value,

the higher the future value, the higher the present value,

the higher the number of periods, the lower the present value,

the higher the interest rate, the lower the present value.

As you may have noticed, the formula for the present value is derived from the formula for the future value.

Suppose you know that in 10 years' time you are going to need USD 300,000 to buy a house (300,000 is the future value). How much is this amount worth today? In other words, how much money we need today, to receive USD 300,000 in 10 years at a given interest rate. Let's assume that the stated annual interest rate is 5% compounded annually.

The present value is:

\(PV=\frac{FV_{10}}{(1+r)^{10}}=\frac{300,000}{1.05^{10}}=184,173.98\)

So, to be able to buy a house for USD 300,000 in 10 years, we need to invest around USD 184,000 today for ten years at an annual interest rate of 5%.

Important!

At a given interest rate, the present value increases as the future value increases or the life of the investment gets shorter. With a given life of the investment, the present value increases when the future value increases and the interest rate decreases.

### Present Value for Multiple Compounding Periods and Continuous Compounding

star content check off when donePV for different compounding periods:

\(PV=\frac{FV_N}{\Bigl(1+\frac{r_S}{m}\Bigr)^{m\times N}}\)

- PV - present value
- \(FV_N\) - future value
- \(r_S\) - quoted annual interest rate
- N - number of years
- m - number of compounding periods per year
- \(\frac{r_S}{m}\) - periodic rate

PV in the case of a continuous compounding:

\(PV=\frac{FV_N}{e^{\bigl(r_SN\bigr)}}\)

- PV - present value
- \(FV_N\) - future value
- \(r_S\) - stated annual interest rate
- N - number of years
- e - Euler's number

(...)

- Compound interest calculation consists of adding interest accrued after the end of each compounding period to the principal.
- If we increased the number of compounding periods ad infinitum (i.e. by going from monthly, to daily, to hourly, to one-second compounding periods, etc.), we would arrive at continuous compounding.
- An interest rate is usually specified for a period of one year. It's called a stated annual interest rate or a quoted interest rate and is denoted by \(r_S\).
- Effective annual rate (EAR) is the interest rate that an investor actually earns annually taking into account the frequency of compounding.
- The more frequent the compounding, the higher the future value.
- The more frequent the compounding, the lower the present value.
- The higher the present value, the higher the future value.
- The higher the future value, the higher the present value
- The higher the number of periods, the higher the future value.
- The higher the number of periods, the lower the present value.
- The higher the interest rate, the higher the future value.
- The higher the interest rate, the lower the present value.
- The more frequent the compounding, the higher the future value.
- The more frequent the compounding, the lower the present value.