01 Time Value Of Money 07

Author: Dilip ThosarFor BVUAmplify DITMAugust2008

Time Value Of Money Slide no. 1

April 8, 2023

Time Value of Money



April 8, 2023

Time Value Of Money• Present and Future Value of Money

– Uncertainty of Realization– Opportunity cost of investment

• Risk Free Interest Rate

• Discounting

• Interest Rate (annual)

• Cost Of Money (Capital)

• Compounding

• USE OF SPREADSHEETS : MS-Excel



April 8, 2023

Future Value, Discounting Factor• “FV0”= PV

• FVFV11 = = PV + PV * R + PV * R

= PV * (1+R)= PV * (1+R)

• Discounting Factor D = PV/FV

DD11 = 1/ (1+R) = 1/ (1+R)



April 8, 2023

Compounding FV2 = FV1 *(1+R)

FVFV22 = PV*(1+R) = PV*(1+R)22

= PV*(1 + 2*R + R2) = PV*(1+2R) + (PV*R)*(1+R) interest on principal + interest on interest

= PV*(1+2R) + (PV*R)*(1+R)• Compound Interest & Simple InterestCompound Interest & Simple Interest• IIc2c2=(1+R)=(1+R)22-1 = 2R+R-1 = 2R+R2 2 IIs2s2= 2R= 2R

PV = FVPV = FV2 2 / (1+R)/ (1+R)22

DD22 = 1/ (1+R) = 1/ (1+R)22



April 8, 2023

Compounding & Discounting Factors

Ci,R = 1 / (1+R)i

Di,R = 1 / (1+R)i

• Numerical Exercise : Generate your own discounting and compunding factor tables for various i and R values.



April 8, 2023

FV at end of N years

FVFVNN = PV * (1+R) = PV * (1+R)NN

PV = FVPV = FVN N / (1+R)/ (1+R)NN

DDNN = 1/ (1+R) = 1/ (1+R)NN



April 8, 2023

Periodic (Intra-year) Compounding• R = Annual Interest Rate• n=No. of periods in year• r = Period Interest Rate = R/n

• Reff Effective Annual Interest Rate = (1+r)(1+r)nn - 1= (1+R/n)n - 1

• FVFVr=1r=1 = PV*(1+r) = PV*(1+r)nn = PV*(1+R/n) = PV*(1+R/n)nn

R= 12%

Period n r=R/n Reff=(1+r)^n -1

Annual 1 12.00% 12.00%

Semi-Annual 2 6.00% 12.36%

Quarterly 4 3.00% 12.55%

Monthly 12 1.00% 12.68%

Weekly 52 0.23% 12.73%

Daily 365 0.03% 12.75%



April 8, 2023

Periodic (Multi-year) compounding• Future value at end of N years (N*n periods)

FVFVNN = PV * (1+ = PV * (1+RReffeff))NN

= PV * (1+ = PV * (1+ (1+r)(1+r)n n -1-1 ))NN

= PV * ((1+r)= PV * ((1+r)n n ))NN

= PV * (1+r)= PV * (1+r)n*Nn*N

FVFVNN = PV*(1+R/n) = PV*(1+R/n)n*Nn*N



April 8, 2023

PV of a stream of returns

• Return Ci at the end of ith year has FVi = Ci

• PV of this return PVi = FVi * Di = Ci / (1+R)i

• PV of the stream = Σ PVi

• PV = ΣN

i=1 Ci / (1+R)i

C1 C2 C3 CN

PV = -------- + -------- + -------- + ... + -------- (1+R)1 (1+R)2 (1+R)3 (1+R)N



April 8, 2023

Annuity• When all Ci are equal, Ci = C

• Annuity Present Value

PVn = ΣN

i=1 Ci / (1+R)i

• Annuity Factor PV/C

= AN,R = ΣN

i=1 (1 / (1+R)i)

AN,R = (1-1/(1+R)i)/R

• Annuity Factor Lookup Table



April 8, 2023

Perpetuity• Stream of constant return every year for ever

• Annuity without termination

• PV ∞ = C * Σ∞

i=1 (1 / (1+R)i)

Σ∞

i=1 (1 / (1+R)i) = 1/R !!!

• PV ∞,R,C = C / R

• Perpetuity Factor = PV∞ / C = P = 1/R

• Perpetuity Factor Table



April 8, 2023

Growing Perpetuity• Perpetual Stream of INCREASING annual

returns

• If the diminishing of value due to R has a bigger than the increase in the annuity, the PV works out to be finite

• For a growth rate of G for the annual return,

PV ∞,R,C,G = C / (R-G)



April 8, 2023

Net Present Value NPV• NPV = PV(inflows) – PV(outflows)

• For an investment decision, NPV = Investment (at t=0) – PV

(returns)

• For an Annuity Purchase, NPV = Price of Annuity – PV of

returns



April 8, 2023

The lighter side• A young attorney who had taken over his

father’s practice rushed home elated one night.

• “Dad, listen,” he shouted, “I’ve finally settled that old Birla lawsuit.”

• “Settled it!” cried his astonished father. “Why, I gave that to you as an annuity for life.”



April 8, 2023

Numericals : A1. Draw up a Table showing Discounting

factors (PV of a future Re. 1) for various interest rates R and periods N (no. of years)

2. Draw up a Table showing Annuity factors (PV of an annuity of Re. 1 p.a.for N years) for various interest rates R and periods N (no. of years)

3. Draw up a Table showing Perpetuity factors (PV of a perpetuity of Re. 1 p.a.) for various interest rates R and periods N (no. of years)



April 8, 2023

Numericals: B1. On a contract, you have a choice of receiving Rs 25000 six years from

now or Rs 50000 twelve years from now. At what implied interest rate would you be indifferent between the two contracts?

2. ABC Ltd’s last dividend was Rs 4 per share and these are expected to grow forever at a rate of 5% per annum. Its share price is Rs 60. What is the required rate of return? If the investor’s required rate of return is 12%, what price would he be willing to pay? If the growth rate falls to 3%, what will the new price be?

3. A zero coupon 1000 par value bond is currently selling for Rs 322. It matures in exactly ten years. What is the discount rate on this bond?

4. You have just won the Maharashtra state lottery and have three reward options to choose from. You can select a lump sum payment of 61 million now, 10 annual end of the year payments of 9.5 million or 30 annual end of the year payments 0f 5.5 million.. Which option would you choose if the interest rate is (a) 7% (b) 8% (c) 9%

5. You have just been left an inheritance of $ I million, which is earning an interest of 5% per annum. You plan to withdraw $ 100,000 per year. How long will the inheritance last?

Microsoft Office Excel Worksheet



April 8, 2023

Numericals :B (contd.)6. You have been offered share of a business. The business is expected

to generate cash flows of $ 1 million, growing at 5% forever. What is the value of the business if the discount rate is 15%. How much should the business grow annually to justify a valuation of 1,25,00,000.

7. Your uncle is quite impressed with your MBA degree. He wants you to help him in his post retirement planning. Currently, he has Rs 10 lacs with him. He will retire after 10 years. During these ten years, he will save Rs 150,000 per annum. He expects to live for 15 years post retirement. How much can he consume every year post retirement? Assume interest rate of 10%. How much should he save over the next ten years if he wants Rs 800,000 per annum post retirement?

8. You are the manager of a professional soccer team and are negotiating a contract with your rival team’s star player. You can afford to pay $1.5 million over annually over the next three years. The player’s agent believes that he will not accept anything below nominal value of 5 million. Can you meet the agent’s demand without relaxing your financial constraint. Interest rate is 10%.



April 8, 2023

Numericals :B (contd.)9. You have been hired to run a pension fund. The fund currently has 10 million and expects

to get 3 million per annum for the next 5 years and 2million annually for five years after that. At 8%,a. How much will the Fund have after ten years? b. If the Fund is required to pay a perpetuity starting at the end of the 11th year, what would be the

annual cash flow of the perpetuity?10. You are an investment advisor who has been approached by a client for help on his

financial strategy. He has savings of $250,000. He is 55 years old and expects to work for ten years more. During these ten years he expects to earn 100,000 per annum. Assume an interest rate of 5%. a. Once he retires ten years from now, he would like to withdraw $80,000 per annum for the next 25

years. ( His actuary tells him he will live until 90 years.) How much would he need ten years from now to be able to do this.

b. How much of his income would he need to save each year for the next ten years to be able to afford these planned withdrawls ( $80,000 per year).

c. Assume interest rates decline to 4% ten years from now. How much, if any, would your client have to lower his withdrawl every year assuming that he still plans to withdraw cash each year for the next 25 years.

11. Early Bird has decided to start saving for his retirement. He plans to invest $2000 every year beginning from his 21st birthday. He will continue this for 10 years and then stop. But, his savings continue to earn interest until his retirement at 60 years. Late Bird also plans to save $ 2000 every year. However, she will start her savings from her 31st birthday and continue to save till he retires at 60 years. The interest rate for both is 7%. Who is better off at retirement and by how much?



April 8, 2023

Numericals :B (contd.)12.You borrow 10000 at 14% for four years for your car. The loan is repayable in

four equal annual instalments payable at the end of each year. a. What is the annual payment that will completely amortise the loan over four years?b. Of each equal payment, what is the amount of interest and the amount of principal

repayment?

13.You wish to buy a photocopier and the supplier has quoted a price of 11,000 cash or 3000 per year for five years. If your cost of capital is 12%, which alternative would you prefer? What if the cost of capital is 8%.

14.New York State has started a lottery scheme wherein it will give away 40 million (2million per year for the next 20 years) to the winner. It plans to donate 50% of the collections of 36 million to charities. If the discount rate is 10%, will it save anything from the scheme or will it have to fund the charity donations from other sources?

15.Savings in pension fund up to 2000 per annum are tax free. You are starting your career at 25 years. You plan to retire at 65 years. Your investments are likely to yield 8% per annum. How much money will you have at retirement? What amount will you have if your annual returns are taxed at 25%.

Microsoft Office Excel Worksheet



April 8, 2023

Numericals C1. An RBI bond 2002, of face value Rs. 10,000 yields Rs.

14,802.50 at the end of 5 years. All interest is taxable @20% on realization. What are the pre-tax and effective post-tax yield rates of the bond?

2. A 5-year fixed deposit scheme at the ICICI bank provides an interest @8% p.a. compounded quarterly. Is it better to invest in this scheme or an RBI bond 2002?

3. A secure scheme offers unit certificates which pay back Rs.2,000/- every year starting from the end of the tenth year from purchase of unit, till the death of the buyer. Assuming a life of 80 years, a present age of 20 years, and an risk-free interest rate of 6% for all those years, would it be preferable, rather than investing for the risk-free interest, to buy this certificate at a price of

a. Rs. 5,000/- ? c. Rs. 20,000/- ?b. Rs. 10,000/- ? d. Rs, 50,000/- ?



April 8, 2023

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01 Time Value Of Money 07

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