**Quantitative Methods: Basic Concepts**

**Reading 6. The Time Value of Money**

**Learning Outcome Statements**

a. interpret interest rates as required rates of return, discount rates, or opportunity costs;

b. explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk;

*CFA Curriculum, 2020, Volume 1*

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### Subject 1. The Time Value of Money and Interest Rates

**time value of money**(TVM) refers to the fact that $1 today is worth more than $1 in the future. This is because the $1 today can be invested to earn interest immediately. The TVM reflects the relationship between present value, future value, time, and interest rate. The time value of money underlies rates of return, interest rates, required rates of return, discount rates, opportunity costs, inflation, and risk. It reflects the relationship between time, cash flow, and interest rate.

There are three ways to interpret interest rates:

**Required rate of return**is the return required by investors or lenders to postpone their current consumption.**Discount rate**is the rate used to discount future cash flows to allow for the time value of money (that is, to bring a future value equivalent to present value).**Opportunity cost**is the most valuable alternative investors give up when they choose what to do with money.

In a

*certain world*, the interest rate is called the risk-free rate. For investors preferring current to future consumption, the risk-free interest rate is the rate of compensation required to postpone current consumption. For example, the interest rate paid by T-bills is a risk-free rate of interest.

In an

*uncertain world*, there are two factors that complicate interest rates:

**Inflation**: When prices are expected to increase, lenders charge not only an opportunity cost for postponing consumption but also an inflation premium that takes into account the expected increase in prices. The**nominal cost of money**consists of the**real rate**(a pure rate of interest) and an**inflation premium**.**Risk**: Companies exhibit varying degrees of uncertainty concerning their ability to repay lenders. Lenders therefore charge interest rates that incorporate**default risk**. The return that borrowers pay thus comprises the nominal risk-free rate (real rate + an inflation premium) and a default risk premium.

**Compounding**is the process of accumulating interest over a period of time. A compounding period is the number of times per year that interest is paid.

**Continuous compounding**occurs when the number of compounding periods becomes infinite; interest is added continuously.

**Discounting**is the calculation of the present value of some known future value. Discount rate is the rate used to calculate the present value of some future cash flow. Discounted cash flow is the present value of some future cash flow.

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**User Contributed Comments**
32

You need to log in first to add your comment. ###### kasvi

interest rate is risk fee rate.

###### gauravcsc

No Kasvi---i do not agree with ur statement that interest rate is Risk free rate.It is risk free only for very secured bonds.else interest in normal market = Real / Risk Free Rate + Default Risk Premium + Inflation Risk Premium

###### arpit11

the charge regarding the postponed consumption can also be taken as a charge for using a dollar than later.

i think as far as the premium for inflation is considered, its also possible that if lenders expect inflationary pressures in the future they may be eager to lend now than later and thus actually reduce the inflation premium or take it as negative.

###### kautilya

Int. Rate = Real Rate + Inflation Premium + Risk Premium.

###### karoll

If we should be more specific

Interest Rate = Default Risk Rate + Inflation Premium + Default Risk Premium + Liquidity Premium + Maturity Premium

###### falvi

Interest rate = Risk + Risk free rate = Risk (1 + nominal rate)

means for secured debts risk is zero so risk free rate is than equals to interest rate.

but for unsecured debts risk can comprise of many factors which includes: Infination risk, default risk, currency risk, risk risk, market risk etc.

so if we see above FV=PV(1+i) for $1

###### spattewar

interest rate = nominal rate + risk premium

nominal rate contains the inflation factor since inflation is macro economical term and is indepedant on the market type. whereas the risk premium depends upon the market type.

###### Dieguis

no, the interest rate contains monetary policy factors (expansive or not), in some countries of the European Union the real interest rates are negatives becouse de CPI inflation is bigger than the nominal interest rate

###### joher

interest rate is the rate that is determined by demand and supply for diffent set financial instruments

###### YEROS

Nominal cost of money=real cost+inflation premium

###### mela

interest rate is the price of parting with your money

###### kuzzie

risk free interest rate is the rate of borrowing that is free of risk. the interest is sure to be received. an example is government bonds and securities that is sold in the market. the nominal or real interest rate plus inflation premium gives the money interest rate. interest rate is also called cost of capital

###### StanleyMo

hope this help: inflation premium:

An Inflation Premium to offset the possibility that inflation may erode the value of the money during the term of the loan. A unit of money (dollar, peso, etc) will purchase progressively fewer goods and services during a period of inflation, so the lender must increase the interest rate to compensate for that loss

###### Bluejay

An interest rate is composed of the nominal interest rate + default premium + liquidity premium + maturity premium.

The nominal interest rate is the rate of a risk free assest such as a T-Bill + inflation.

###### mundia

interest rate is the price you pay me for allowing you to use my money which i would otherwise have used for a gain or a loss on me but i allow you to use it for a gain on me.(it allows me to exclude any chances of losing if i invest)

###### Ratego

My take is simple;

Interest rate=real risk free rate+inf prem+risk premiums(which might include all or more risk premiums stated above)

###### cashpatel

Always know that as interest rate is ordinary income for lenders and Investors. If you invest $1.00 today and interest rate is 1% on it then you will gain 1% worth of money in retun by investing $1.00 today.

###### natkhat

its just a difference of money value (i.e.what value will your money fetch after a particular time gap)

###### Masterguru

now its clear ir=real risk free rate+inf. premium+risk prem.

###### indrasenareddy

can some one throw light on maturity premium and on liquidity premium

###### latenttruth

"Maturity Premium" reflects the fact that longer the time to maturity of a security the greater the sensitivity of its price to a given change in interest rates.

For example, for a given change in interest rates, a three year bond which pays $100 at the end of each year will undergo a larger percentage change in price than a two year bond which pays $100 at the end of each year.

"Liquidity Premium" captures the potential loss that may arise if an asset cannot be sold at its fair value in the market.

###### studentx

so what is the duration of the T-bill used to compute RRR?

###### nidha

I think the 10 year govt bond rate is considered as risk free rate of return. Is it right?

###### chipster

risk-free rate = theoretical rate of return with zero investment risk.

Nidha, think about it...10 yr duration is too long. Interest rate on 3 mo US Treasury Bill is usually used as the risk-free rate.

###### johntan1979

Too many sooks spoil the coup

###### To-be-CFA

Nominal risk-free rate = Real risk-free rate + Inflation premium

Required rate of return = Nominal risk-free rate + Risk premium

###### Stacerz02

How are government bonds risk free if the US is sitting on trillions of debt? Makes zero sense.

###### jmorris

speaking under correction...i think the idea is that the goverment is most likely never to default (given the level of political instability). i wish to believe it cant be 100% zero investment risk. maybe we can leave an error term between 0 and 5%. on the other hand probability of goverment overthrow and radical reforms is less likely to happen in a period that is less than three years. there is political risk which can actaul lead to total loss. i actually also wish to know how political instability of a given country is accounted for.

###### Beckhama

@Jmorris

Economists created many indexes to help quantify stability risks in emerging markets, but I think we can all agree that they're imperfect.

There are open sources of information available that can help investors assess risks--CIA's World Factbook and the Department of Foreign Affairs and International Trade just to name a few.

If you're looking for a much more detailed analysis, you'll have to consult "experts".

###### AcksonM

I wonder if the certain world underlining the real rate of return actually exists. How is it (real rate of return) determined in the first place, is it hypothetical?

###### Allen88

If anyone could help me out, I am not an expert or anything, so hear me out. What exactly is it meant by the real rate?

I get the risk premium, the inflation premium, and the RRR, but I can't really seem to understand what do they mean by real rate?

Sorry if the question sounds stupid, again, I'm a beginner!

###### MathLoser

@Stacerz02: Government bonds (T-Bills, T-notes, T-bonds) are risk-free because:

1/ Backed by the government. Where did they get the money to pay you? From taxes.

2/ Ability to "print money" of the government.