Time Value of Money
“Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.” – Albert Einstein.
All of us are aware of the proverb “ A bird in hand is worth two in the bush”. The same concept gets extended to the money as well, and we all would prefer to receive RS.100 today, instead of getting Rs.100 next year. This is because the amount of Rs.100 would fetch some interest at least Rs.40 if remains in Savings Account. Thus there is a compensation for waiting. Alternatively, the interest can be called the opportunity cost for the sacrifice of foregoing the use of money today. In short the Time adds value to the money & this is the aspect we will be discussing in this chapter. Time Value of Money ( TVM in short ) is one of the major concepts in the Financial Mathematics and Financial Management which is used to analyse, compare and evaluate different investment or loan or annuity options available. The rate at which the compensation is received in the ‘Rate of Interest’ or ‘ Rate of Return’ in the case of investments.
The following terms are extensively used under TVM concept:
1. Present Value ( PV ) – P – This is the Present Value of the Investment
2. Period ( NPER ) or N – Number of Period for which the Investment is made
3. Rate ( I ) – Rate of Interest at which the investment is made
4. Future Value ( FV ) -A – Resultant amount at the end of the period( PV + Interest)
5. Payment ( PMT ) -Series of equal installments instead of Single payment of PV
The TVM problems can be classified into just 4 categories under two types
a. Lump Sum or Series of Equal installments to
b. Lump Sum or Series of Equal Repayments ( annuities ) as Future Value
Simple Interest: Simple interest provides just the interest on the amount irrespective of whether the interest is withdrawn or not. For example, if Rs.10000 is invested under Simple Interest of 8%p. a. for 3 years , the total amount would be Rs.12400 as follows
Interest @8% p. a for 3 years = 10000*8%= 800*3 Rs. 2400
Total amount Rs.12400
Total amount A or FV = P + P * N * I = 10000+10000*3*0.08
Compound Interest: Under the Compound Interest, the interest also earns interest as though the same has been reinvested at the same interest rate. For example, if Rs.10000 is invested for 3 years under compound interest@ 8% p a compounded yearly, then the Total amount would be Rs.12597.12
Principal amount Rs.10000
Value after 1 year with interest @ 8% (RS.800 ) Rs.10800
Value after 2 years with interest @ 8%(Rs.864 ) Rs.11664
Value after 3 Years with interest @ 8%( Rs.933.12) Rs.12597.12
Here Rs.64 in the second year is the result of interest on the interest of Rs.800 and Rs.133.12 in the third year is the interest on 1664 ( 800+800+64) all calculated @ 8% p. a
FV or A = P(1+I)^N = 10000( 1.08)^(3 ) =10000*1.269712 =12597.12
In the above case, the interest is added back to the principal yearly – i.e. the interest is compounded yearly. It is possible the compounding could Half- yearly, quarterly, monthly or even daily. The increase in the frequency of compounding increases the total amount, which can be seen from the following example. Let us calculate the FV of an amount of Rs.10000 invested at 12% p a various compounding options for 1 Year. While PV remains the same at Rs.10000 in all cases, the Rate of Interest ( I ) & the Period ( N ) changes as below:
Annual Compounding: Here I =12, N = 1
FV = 10000*1.12^( 1 ) = 11200
Semi-annual compounding: Here the I = 12/2 = 6, & N= 2 (since there will be 2 such periods)
FV =10000*1.06^(2 ) = 10000*1.1236= 11236.00
Quarterly compounding: Here I = 12/4 = 3(being Qtly), N = 4 (compounding is Qtly)
FV = 10000*1.03^(4 ) = 10000*1.1255 =11255
Monthly Compounding: Here I = 12/12 = 1, N = 12 (as compounding is mthly)
FV = 10000*1.01(12) = 10000*1.126825 =11268.25
Nominal Rate of Return & Effective Rate of Interest: Nominal Rate of Interest is the simplest form of interest where the interest is compounded annually. Effective Rate of Interest takes into account the power and the effect of compounding when the frequency of compounding increases such as Half-yearly, Quarterly, Monthly, etc. For example the Nominal Rate of Interest is 12% p.a in all the above cases, the Effective Rate of Interest is 12.36%,12.55%, & 12.68% when the interest is compounded half-yearly, Quarterly & Monthly as the case may be.
Holding period return(HPR): Generally, the rate of return is expressed in terms of rate per annum. It is possible that a particular investment is held for certain odd period say 7months 10 days or 1 year 25 days, etc and if the investor wants to know the return he has earned on the investment for the period the investment is held, it is termed as Holding Period Return.
Calculation of holding period return is just academic as this will only indicate the gross earning from the investment and it may not be useful in comparing different products.
Real Rate Of Return ( RRR ): We all know that inflation affects everyone in some form or other. This applies to the Return on the Investment, as well. While the return on investment offers a compensation for parting with the money the inflation reduces the effect of such compensation. Suppose the investment gives a return of 10% and the inflation is 5%, the real return is roughly 5% ( 10-5 ). But the Real Rate of Return after Inflation is given by the formula:
In the case referred above, the RRR is 4.7619% [(1.10/1.05)-1] and not 5% as arrived at in the previous paragraph.
To understand the formula, let us assume that we have Rs.1000 and we are able to get 10 meals @ Rs.100 each today. After one year due to accumulation of Interest Rs.1000 would become Rs1100, taking into account the interest of Rs.100. Due to inflation of 5%, the meal would cost 105 next year. Thus Rs.1100 would be sufficient to provide 10.47619 meals ( 1100/105). Thus the Real Rate of Return comes to 0.47619 for Rs.1000 which works out to 0.047619 per 100 or 4.7619%
Internal Rate of Return( IRR ) & XIRR : In all cases discussed earlier, the return percentage on the investment is constant and the return is uniform. But in can happen that the returns or cash inflows are different during the period under study. Let us assume the case where an investment is made on a Project which obviously cannot give uniform return year after year. If the Investment is taken as ‘Cash outflow’ ( -ve ) and returns are taken as ‘ Cash inflows’ ( +ve ), then, Internal Rate of Return ( IRR ) is that rate of discount, which makes the Present Value of all the return to Zero. We all know that the NPV of an investment is given by the formula:
If NPV is the Net Present Value of all the cash flows connected with the investment starting from year ‘0’ to year ‘n’, then ‘r’ is that rate if interest which makes NPV=0. Alternatively, at the rate of IRR, the Net present Value of all Costs would be equal to the Net Present Value of the Benefits.
Generally, IRR is the best option to evaluate and compare a set of project which have different outflows that give different returns spread over different periods of time. The calculation of IRR through the above formula would be tedious and has to be done on trial & error basis & using the approximation thereafter. Hence the calculations are generally arrived through the Calculators or Worksheet.
While it would be simple to calculate IRR where the Outflows and inflows are on specific dates though out the time period under study, the calculations as above cannot give the right answer where the outflows and inflows occur on various dates during the year. In such cases, we use the ‘XIRR’ formula from the Spreadsheet, by incorporating the dates in one column & the Cash flows in the next column. In short, IRR uses one variable viz. cash flows, the XIRR uses ‘two variables’, viz. dates and cash flows.
Compound Annual Growth Rate (GAGR): Compound Annual Growth Rate ( GAGR ) is the rate at which the investment has grown over the period had all the intermediate cash flows were reinvested at this rate. It is the nth root of the gross percentage of return over n years as shown below
The concept can be understood with an example. Suppose you had invested a sum of Rs.10000 in a Fund on 1st April 2010 and it has become 13500 on 1st April 2011 , 15000 on 1st April 2012 & finally to Rs. 18000 on 1st April 2013. The CAGR in this case would be
-1 = 1.8 ^(1/3 )-1 =1.21644 – 1 = 0.21644 = 21.644%
This means that the investment of Rs.10000 has grown at the Compounded Annual Growth rate of 21.644% during 3 years under study, which can be re-checked with the following
FV = 10000*1.21644^(3) =10000*1.7999999 = 18000 ( which is the value realized )