**Brief Concept:**

Time Value of Money (TVM) refers to calculation of effect of compounding of interest on an investment and measurement of its present value/ future value. For the purpose of tackling questions in exam, it can be best understood by plotting timelines and their respective cash flows with +/- differentiation (inflows/ outflows). Understanding of discounting factors for calculation of present value is also one of the critical requirements for the exam.

## **LOSS as per CFA Curriculum:**

There are six key learning outcomes from the topic as per CFA study session:

### 1. Interest rates as Required Rate of Return/ Discount Rate/ Opportunity cost:

- Equilibrium interest rates/ market rates are the required rate of return (RRR) for an investor.
- The RRR/ interest rates can be used as discount rates for calculation of present value of an investment. RRR can also be interpreted as the opportunity cost of current consumption.

### 2. Interest rate as sum of real risk-free rate, expected inflation, and risk premiums:

- Real risk free rate of interest is the theoretical rate on a single period investment that has no expectation of inflation in it.
- US T-bill rates are nominal risk free rates as they contain expected inflation premium.
- Other securities in addition to inflation premium, also contain risk premiums associated with specific types of risk i.e. default risk, liquidity risk, maturity risk etc.
- Accordingly the RRR can be defined as RRR = Nominal Risk free rate+ default risk premium” liquidity premium+ maturity risk premium+…(other risk premiums)

### 3. Effective annual rate (EAR):

- The annual rate of interest that an investor actually realizes as a result of compounding over periods is called as EAR. The EAR is adjusted over different compounding periods.
- It is determined as
**EAR = (1+ periodic rate)**^{ m} – 1

### 4. Time value of money problems for different frequencies of compounding:

- The problems can include semiannual/ quarterly/ monthly/ daily compounding. Ceteris paribus as the compounding period increases, the EAR increases.
- The limit of shorter and shorter compounding is called continuous compounding. To convert an annual stated rate to the EAR with continuous compounding, following formula can be used:
**EAR = e**^{r} - 1

### 5. Future value (FV) and present value (PV) calculation:

- FV of a single cash flow is calculated as:
**FV = PV (1+ I/Y)**^{N } (Here I/Y is rate of return per compounding year)
- An annuity is a stream of equal cash flows that occurs at equal intervals over a given period. Annuity can be of two types: a) Ordinary annuity- annuity payments occur at the end of each compounding period and b) Annuity Due- annuity payments occur at the beginning of each compounding period
- Perpetuity is a financial instrument that pays a fixed amount of money at a defined set of intervals over an infinite period of time. The PV of perpetuity is:
**PV= PMT/ (I/Y)**
- One crude way of calculating the PV of annuities is the sum of PVs of all the annuity payments. The same can be easily calculated by using PV calculation function in financial calculator. However, if it is not an annuity i.e. the cash flows are uneven then PV of each tranche of cashflow will give the correct figure.
- A critical concept here is PV of any series of CFs is equal to the sum of the PV of individual CFs.

### 6. Use of a time line in modeling and solving time value of money:

- Timeline comes in handy for solving problems related to uneven CFs and finding out value of investment at an intermittent period.
- The same can also be used for calculation of EMIs for mortgages. This can also be done using the financial calculator.

**Use of financial calculator:**

- Financial calculators (TI BA II Plus/ HP 12 C) are critical in solving the TVM problems. The candidates are required to be through with the calculation of PMT/ PV/ FV/ (I/Y) in the output which will save a lot of time during the exam.
- Further the functions would be helpful in calculating the EMI/PMT value and solving fixed income problems (coupon rate/ yield/ face value/ market value calculation) during the exam.

**Type of questions expected in the topic area:**

The typical questions expected from this topic area include calculation of

- RRR for various compounding periods
- PMT or I/Y of ordinary annuity/ annuity due
- Evaluation of options between 2 or more investment alternatives
- Bond price given the coupon, tenor and face value.