Time Value of Money

Time Value of Money numerical with solutions

1. Future value (FV) of a single cash flow = PV × (1+r) ⁿ = PV × FVF _{r, n}

2. Future value of Annuity = Annuity amount × [(1+r) ⁿ – 1] ÷ r

                                            = Annuity amount × FVAF _{r, n}

3. Future value of Annuity due = [Annuity amount × [(1+r) ⁿ – 1] ÷ r] × (1+r)

                                                   = Annuity amount × FVAF _{r, n} × (1+r)

4. Present value (PV) of a single cash flow = FV ÷ (1+r) ⁿ = FV × PVF _{r, n}

5. Present value (PV) of Annuity = Annuity amount × [1 – {1 ÷ (1 + r) ⁿ}] ÷ r

   = Annuity amount × PVAF _{r, n}

6. Present value (PV) of Annuity due = [Annuity amount × [1 – {1 ÷ (1 + r) ⁿ}] ÷ r] × (1+r) = Annuity amount × PVAF _{r, n} × (1+r)

7. Present value of perpetuity = Cash flow ÷ r

8. Present value of growing perpetuity = Cash flow ÷ (r – g)

9. Present value of growing annuity = [Cash flow ₁ ÷ (r-g) ] × [ 1- { (1+g) ÷(1+r)}ⁿ ]

10. Present value of growing annuity due = [Cash flow ₁ ÷ (r-g) ] × [ 1- { (1+g) ÷ (1+r)}ⁿ ] × (1 + r)

11. Future value of growing annuity = Present value of growing annuity × (1+r) ⁿ  = [Cash flow ₁ ÷ (r-g) ] × [ (1+r)ⁿ – (1+g)ⁿ]

12. Future value of growing annuity due = Present value of growing annuity × (1+r) ⁿ × (1 +r) = [Cash flow ₁ ÷ (r-g) ] × [ (1+r)ⁿ – (1+g)ⁿ] × (1 +r)

In case, r = g

13. Present value of growing annuity = CF₁ × [n ÷ (1 + r)]

14. Present value of growing annuity due = CF₁ × [n ÷ (1 + r)] ×(1 + r)

15. Future value of growing annuity = CF₁ × n × (1+r) ⁿ⁻¹

16. Future value of growing annuity due = CF₁ × n × (1+r) ⁿ⁻¹ × (1 + r)

Finding growth rates

FV = PV (1 +g) ⁿ

g (growth rate) = _(FV/PV)^-n _ ₁

17. Effective annual rate = (1 + r/m) ^m – 1

Where m = no. of periods

In case of quarterly, no. of period = 4

In case of half yearly, no. of period = 2

In case of monthly, no. of period = 12

In case of daily, no. of period = 365

PVF_{r, n} = PVIF_{r, n} = Present value interest factor at rate of interest r after n periods.

PVFA_{r, n} = PVIFA_{r, n} = Present value interest factor for an annuity at rate of interest r after n periods.

FVF_{r, n} = FVIF_r,n =  CVIP_r,n = CVP_r,n  = Future or compound value interest factor at rate of interest r after n periods.

FVFA_{r, n} = FVIFA_{r, n} = CVIPAr, n = CVPA_{r, n} = Future or compound value interest factor for an annuity at rate of interest r after n periods.

1. Mr. X invested Rs. 7, 10,000 at the beginning of the year one at the rate of 10% compounded annually. Calculate how much he will receive after the end of 1st, 2nd and 3rd year of his investment?

Future value at the end of year 1 FV₁ = PV (1+r) ¹ = 710000 * (1+0.1)¹ = 710000 * 1.1= 781000

Or FV₁ = PV × FVF_{r, n} = 710000 × PVF_{.1, 1} = 710000 × 1.1 = 781000

A	B	A × B
Present value at year 0	FVFr,n  ( r= 0.1, n=1,2,3) taken from PVF table	Future Value at the end of
Rs. 7,10,000	FVF0.1, 1 = 1.1	Year 1 – FV1 = PV × FVF0.1, 1 = Rs. 7,10,000 × 1.1 = Rs. 7,81,000
Rs. 7,10,000	FVF0.1, 2 = 1.21	Year 2 – FV2 = PV × FVF0.1, 2 = Rs.  7,10,000 × 1.21 = Rs. 8,59,100
Rs. 7,10,000	FVF0.1, 3 = 1.33	Year 3 – FV3 = PV × FVF0.1, 3 = Rs.  7,10,000 × 1.33 = Rs. 9,45,010

2. Mr. X invested Rs. 1,000, Rs. 2,000 and Rs. 5,000 at the starting of 1st, 2nd and 3rd year. What will be compounded value of his investment at the end of 3rd year when interest is provided at the rate of 12%.

Solution:

Method 1	Method 2
A	B	C = A × B	FV = PV × (1+ r)n
Money invested at the beginning of year	FVF r,n
1- Rs. 1000	FVF 0.12, 3 = 1.405	Rs. 1000×1.405= Rs. 1,405	= Rs. 1000× (1+ .12)3= Rs. 1,405
2 – Rs. 2000	FVF 0.12, 2 = 1.254	Rs. 2000×1.254= Rs. 2,508	= Rs. 2000× (1+ .12)2=2,508
3 – Rs. 5000	FVF 0.12, 1 = 1.120	Rs. 5000×1.120= Rs. 5,600	= Rs. 5000× (1+ .12)1 = 5,600
compound value of his investment at the end of 3rd year when interest is provided at the rate of 12%	1405 + 2508 + 5600 = Rs. 9513	1405 + 2508 + 5600 = Rs. 9,513

3. Mr. X has invested an amount of Rs. 15,000 each at the end of 1st, 2nd and 3rd year. Calculate the compound value of his investment at the end of 3rd year if interest is provided at a rate of 9 % compounded annually.

Solution:

FV at the end of year 3 = Annuity × FVAF _{0.09, 3} = Rs. 15,000 × 3.278 = Rs. 49,170

4. A has invested Rs. 7,000 for 3 years at an interest rate of 12 % per annum compounded semiannually. What amount he will get after 3 years?

Solution:

FV = PV × FVF _{0.06, 6} = Rs. 7,000 × 1.419 = Rs. 9,933

(In case of semiannual compounding divide r by 2 and multiply n by 2)

5. Vitthal has invested Rs. 25, 000 now for 3 years at the rate of 8 % per annum compounded quarterly. What amount he will get after 3 years?

Solution: FV = PV × FVF _{0.02, 12} = Rs. 25,000 × 1.268 = Rs. 31,700

(In case of quarterly compounding divide r by 4 and multiply n by 4)

6. Ravi wants to deposit Rs. 10, 00,000 in a bank for a year. He has received following offers of rate of interest from different banks

SBI-10.75% p.a. compounded weekly

PNB-11% p.a. compounded monthly

HSBC-11.25% p.a. compounded quarterly

ICICI – 11.2% p.a. compounded half yearly

HDFC- 11.5% p.a. compounded yearly.

In which bank should he deposit his money?

Solution:

7. Mr. Ajai will receive Rs. 30,000 after 3 years. How much he has invested now if rate of interest is 10%.

Solution:

Money invested today = Present value = FV × PVF _{.1, 3} = Rs. 30,000 × 0.751 = 22,539. 

= FV / (1+r)ⁿ = Rs. 30000/ (1.1)³ = 22539

8. Shyam will receive Rs. 6,000, Rs. 4,000 and Rs. 2,000 at the end of 1st, 2nd and 3rd year. Find present value of these cash flows considering discounting rate be 15 %.

Solution:

	A	B	C= A×B
Year	FV	PVFr,n	PV = FV × PVFr,n
1	Rs. 6,000	PVF.15, 1 = 0.870	Rs. 5,220
2	Rs. 4,000	PVF.15, 2 = 0.756	Rs. 3,024
3	Rs. 2,000	PVF.15, 3 = 0.658	Rs. 1,316
Present value of future cash flows	=5220+3024+1316 = Rs. 9560

9. Find the present value of the annuity consisting of a cash inflow of 17,000 per year for 3 years discounting rate being 18 %

Solution:

Present value of annuity = Annuity × PVAF _{.18, 3} = Rs. 17,000 × 2.174 = Rs. 36, 958

10. How much Rina has to invest to yield Rs. 10,000 p.a. in perpetuity if opportunity cost of capital (r) is 11 %.

Solution:

Amount to be invested by Rina = Present value of perpetuity = Cash flow ÷ r = Rs. 10,000 ÷.11

= Rs. 90,909

11. Rajni makes recurring deposit of Rs. 13,000 in the beginning of each of 5 years starting now at 12% p.a. how much she will get after 5 years?

Solution:

Future value of Annuity due = Annuity × FVAF_{0.12, 5} × (1+.12) = Rs. 13,000 × 6.353 × 1.12

= Rs. 92,500

12. What amount should be invested now to get an amount of Rs. 25,000 in the beginning of next 5 years at 9 % p.a. rate of interest?

Solution:

Present value of Annuity due = Annuity × PVAF_{0.09, 5} × (1+.09) = 25000 × 3.890 × 1.09

 = Rs. 1,06, 002.5

13. TCS wants to offer scholarship of Rs. 35,000 per year to 100 disabled sports persons  starting from one year now and it will increase at a constant rate of 5% every year Find the present value of this scholarship if rate of interest is 7%.

Solution:

Present value of perpetuity with constant growth rate = CF ÷ (r-g) = Rs. 35,000 ÷ (0.07-0.05)

= Rs. 17, 50, 000

Present value of scholarship given to 100 disabled sports persons = Rs. 17,50, 000 × 100

= Rs. 17, 50, 00, 000

14. TCS wants to offer scholarship of 35000 per year to 100 disabled sports persons  starting from one year now for next 10 years and it will increase at a constant rate of 5% every year Find the present value of this scholarship if rate of interest is 7%.

15.  A machinery is to be replaced after 7 years which will cost Rs. 10, 00, 000 at the end of 7 years. However, Rs. 75,000 can be realized by selling the existing machinery at that time. Calculate the amount to be deposited every year to get the amount required to purchase new machinery after 7 years considering deposit yields a rate of 6 %.

Solution:

Replacement cost of machinery at the end of 7 year after deducting realized value of existing machinery = Rs. 10, 00, 000 – Rs. 75, 000 = Rs. 9, 25, 000

Future value of Annuity = Annuity × FVAF_{0.06, 7}

→ Annuity = FV/ FVAF_{0.06, 7} = Rs. 9, 25, 000 / 8.394 = Rs. 1, 10, 198

Amount to be deposited every year to get Rs. 9, 75, 000 at the end of year 7 Rs. 1, 10, 198.

16. Mahesh took a loan of Rs. 15, 00,000 to be paid in ten equal annual installments at rate of 15 %. Calculate annual installment payable by him.

Solution:

PV of annuity = Annuity × PVAF _{0.15, 10} => Annuity = PV of annuity / × PVAF _{0.15, 10}  

=> Annuity or Annual installment = Rs. 15, 00,000 /5.019 = Rs. 2, 98,864

17. A financial institution offered that it will pay a lump sum amount of Rs. 12,000 at the end of 10 years to investors who pay 1,000 annually for 10 years. Find out implicit rate of interest in the offer.

Solution: FV = Annuity × FVAF _{r, 10}

=> FVAF _{r, 10} = FV/Annuity = Rs. 12,000 /Rs. 1,000 = 12

Looking into FVAF table at 10 years, implicit rate of interest is 4 %.

18. SBI has issued deep discount bonds of Rs. 15,000 each which will mature after 7 years for 35,000. What is implicit rate of interest of these DDRBs.

19. Profit of Tesla has grown from Rs. 10 crores to Rs. 100 crores in 5 years’ time. Calculate compound rate of growth company has maintained in profit during this period. Assume that profit has grown evenly throughout this period.

Solution:

Growth rate = (FV/PV) ^1/5 – 1 = (100/10) ^1/5 – 1 = 1.5849 – 1 = 0.5849 = 58.49 %

20. Find the number of years required to accumulate a sum of Rs. 5,000 with an initial investment of 1,200 at 10% rate of interest.

Solution:

FV = PV × (1+r) ⁿ PV × FVF_{0.1, n}

FVF_{0.1, n} or(1+r) ⁿ = Rs. 5000 / Rs. 1200 = 4.17

(1.1) ⁿ = 4.17

Looking at FVF table at 10 % no. of years for value of 4.17 is close to 15 years

So, n = 15 years