Time Value of Money
& Its Applications

01Interest: the price of using money

When a lender parts with ₹1,00,000 for a year, the borrower receives the ability to spend it immediately. The lender gives up that ability for a period. Interest compensates the lender for the use of the money. The principal is the original amount. The interest rate expresses the compensation as a proportion of the principal for a stated period.

If the amount repaid after one year is ₹1,09,000, the interest is ₹9,000 and the annual rate is ₹9,000 / ₹1,00,000 = 9%. A rate therefore needs a clock. “9% per year” and “9% per month” describe very different arrangements. It also needs an interest convention: simple, periodically compounded or continuously compounded.

Why rates differ

A rate reflects the alternatives available to the investor and the terms of the claim. The textbook describes interest as compensation for time and risk. In practice, several forces enter that compensation.

Reading connection: Book, chapter 1, printed pp. 3–6; Foundations deck, slides 7–11; Workbook, printed pp. 2–3.

02Simple and compound interest

Begin with the workbook’s example: ₹1,00,000 invested for three years at 9% per year. Keep the principal, rate and period unchanged so the effect of the interest convention is visible.

Simple interest

Simple interest is calculated on the original principal throughout the term. Each year in this example earns ₹9,000. Interest already earned does not itself earn interest.

Substitution: I = 1,00,000 × 0.09 × 3 = ₹27,000. The amount repayable is ₹1,27,000. For nine months at an annual simple rate, n would be 9/12. If a contract uses a day-count basis, use the specified number of days divided by that basis.

Compound interest

Under compound interest, each period’s interest remains in the account. The next period applies the rate to the larger balance. One year multiplies the balance by 1.09. Three years multiply it by 1.09 three times.

Substitution: 1,00,000 × 1.09³ = ₹1,29,502.90. Interest is ₹29,502.90. The additional ₹2,502.90 arises because earlier interest also earns interest.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

Simple-interest conventions occur in some short-term and contractual calculations. Compound accumulation underlies the periodic models developed here. Maturity alone does not decide the convention. A short contract can compound, and a longer contract can specify simple interest. A quoted discount on face value also differs from interest calculated on money advanced. Read the actual terms before selecting the formula.

Reading connection: Workbook, printed pp. 3–5; Foundations deck, slides 15–24; Book, chapter 1, printed pp. 4–13.

03Rate quotations and the payment clock

A nominal annual rate j specifies how the annual quote is divided into compounding periods. An effective annual rate, EAR, measures the actual one-year accumulation under that convention. Use j for a nominal annual quote and r for a rate per period throughout this article.

For ₹1,00,000 at a nominal 9% over three years, quarterly compounding means r = 9%/4 = 2.25% and n = 3 × 4 = 12. More frequent compounding at the same positive nominal annual quote increases the ending amount.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

When an effective annual rate is given

At 12% effective per year, the equivalent monthly rate is 0.948879%, not 1%. Division by 12 is appropriate for a nominal annual rate compounded monthly. Use =(1+12%)^(1/12)-1 for the effective-rate conversion. If a nominal j compounds k times per year but payments occur m times, the equivalent payment-period rate is (1+j/k)^(k/m)-1, assuming this equivalent accumulation convention applies.

Continuous and real rates

Excel uses =100000*EXP(9%*3) for continuous accumulation. If an 8% annual effective return faces 5% inflation, the real return is =(1+8%)/(1+5%)-1 = 2.8571%. Here “nominal” means measured in money before inflation adjustment, a different use of the word from a nominal compounding quote. State which meaning you intend.

Reading connection: Workbook, printed pp. 5–7; Foundations deck, slides 21–24; Book, chapter 1, pp. 14–20, and chapter 6, p. 129.

04What time value of money means

Time value of money is the relationship between an amount available on one date and its equivalent value on another date. Money available today can be invested during the waiting period. Inflation affects purchasing power, uncertainty affects expected receipt, and a preference for present consumption affects the compensation required for waiting.

Use the foundations lecture’s first comparison: ₹10,000 available today can become ₹10,800 after one year at an 8% return. At that rate, ₹10,000 today and ₹10,800 in one year are equivalent values. This does not make every promise to pay ₹10,800 equally attractive. Its risk and other terms must match the rate used.

A comparison becomes meaningful only after the amounts share a date. This idea explains why we value a pension, an EMI, a dividend and a bond coupon using the same mathematics. Before learning that mathematics, identify what cash flows actually occur.

Reading connection: Foundations deck, slides 7–14; Workbook, printed p. 2; Book, chapter 1, pp. 3–6.

05Cash-flow types before valuation

A cash flow is money received or paid on a particular date. Choose a viewpoint before assigning signs. A loan received is an inflow for the borrower and an outflow for the bank. Its repayments have the opposite signs. A timeline records this information before a formula can obscure it.

Ask: How many payments occur? Are the amounts equal? Are the intervals equal? When is the first payment? Is there a final date? The six common patterns below answer those questions. At this stage, classify the pattern without calculating its value.

A growing finite annuity combines a fixed number of payments with growth in their amounts. A deferred annuity starts after a waiting period. A bond combines coupons and principal. Retirement combines saving and withdrawal phases. More complex problems can therefore be decomposed into these simpler patterns.

The diagrams show positive amounts to make timing easy to read. In a spreadsheet that describes an actual investment, deposits are outflows and receipts are inflows. Retain the same viewpoint throughout.

Reading connection: Foundations deck, slides 25–34; Session 2, slides 2, 12, 19, 23, 25; Workbook, printed p. 7; Book, chapter 3, pp. 54–60.

06Present value, future value and the two factors

Present value (PV) is the equivalent amount at the chosen starting date. Future value (FV) is the equivalent amount at a specified later date. “Present” can mean the retirement date when we value withdrawals there. It need not mean today’s calendar date.

The compounding factor

At 8% for three years, the factor is 1.08³ = 1.259712. Each rupee at time 0 becomes ₹1.259712 at time 3. A factor has no currency unit. Multiplying a currency amount by it gives another currency amount.

The discounting factor

At the same rate and horizon, the discounting factor is 1/1.259712 = 0.793832241. A rupee due in three years is worth about ₹0.793832 today. This is the reverse of accumulation, because the two factors multiply to exactly 1.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

Longer time and a higher positive rate increase a compounding factor and reduce a discounting factor. For a fixed positive rate, the discounting factor approaches zero as the date moves farther away. The compounding factor grows without bound. These limits will explain why an infinite payment stream can have a finite present value.

Reading connection: Session 2, slides 3–6; Workbook, printed pp. 5–6; Book, chapter 1, p. 6, and chapter 2, pp. 23–30.

07Excel conventions that keep the model consistent

Calculate the relationship by hand first. Use Excel to reproduce it and to build a schedule that exposes mistakes. The sign convention is especially important: the money paid into an investment and the money received from it have opposite signs.

type=0 means payments at the end of each period. type=1 means payments at the beginning. Enter a zero for an absent payment or balance. A lump sum has pmt=0. Use 8% or 0.08 for an 8% rate. The value 8 means 800%.

Each Excel layout in this guide is a separate blank worksheet unless stated otherwise. Put labels in column A and their inputs or formulas in column B. In cash-flow tables, use the explicit column headings given. Formula separators vary by regional settings, so some Excel installations need semicolons instead of commas. Currency formatting changes display, not the value used in the calculation.

The equation is useful for checking whether the signs describe a coherent exchange. For r=0, it becomes PV + n×PMT + FV = 0. For NPV, keep the time-zero cash flow outside the function. A missing period within a regular series requires a zero entry so later amounts keep their correct dates.

Reading connection: Book, appendix B, printed pp. 247–253; Foundations deck, slides 33–34; Session 3, slides 17 and 39.

08Single lump sum: one payment, one date

Use the same ₹10,000, 8% and three-year inputs as Session 2. There are two questions, depending on which amount is known. The timeline determines which direction to use.

Future value of ₹10,000 invested today

The interest earned is ₹2,597.12. All of the starting amount remains invested for all three years.

Present value of ₹10,000 due in three years

An investment of ₹7,938.32 today, using the unrounded amount internally, would grow to ₹10,000 at the assumed 8% rate. This calculation discounts a future ₹10,000. It is a different cash flow from investing ₹10,000 today.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

B5 and B6 both return ₹12,597.12. B7 and B8 both return ₹7,938.32. B9 returns ₹10,000. To solve instead for time, use =LN(FV/PV)/LN(1+r). To solve a periodic rate, use =(FV/PV)^(1/n)-1. For example, =RATE(3,0,-10000,12597.12) returns 8%. Those rearrangements use the same compound relationship.

Reading connection: Session 2, slides 3–6; Foundations deck, slides 37–38 and 47–48; Book, chapters 1–2, pp. 7–13 and 23–38.

09Uneven cash flows: value each date separately

Session 2 uses receipts of ₹2,000, ₹3,000, ₹4,000, ₹5,000 and ₹6,000 at the ends of years 1–5. The rate is 10% per year. Equal spacing permits a common periodic rate, but the unequal amounts prevent us from using one constant annuity payment.

Total PV = ₹14,443.38. Total FV at year 5 = ₹23,261.20. The final ₹6,000 has zero time to earn interest when measured at its own receipt date. The first ₹2,000 compounds for four years, not five.

Excel cash-flow worksheet

Put the rate 10% in H2 and horizon 5 in H3. Use row 1 for headings: A Year, B Cash flow, C Discount factor, D Present value, E Compounding factor, F Future value. Enter years 1–5 in A2:A6 and the five amounts in B2:B6.

C2 =(1+$H$2)^-A2
D2 =B2*C2
E2 =(1+$H$2)^($H$3-A2)
F2 =B2*E2
Copy C2:F2 through row 6.
D7 =SUM(D2:D6)
F7 =SUM(F2:F6)
H5 =NPV(H2,B2:B6)
H6 =H5*(1+H2)^H3
H7 =F7-H6

D7 and H5 agree. F7 and H6 agree. H7 is zero apart from floating-point precision. If an investment costs ₹14,000 today, its NPV at 10% is =NPV(H2,B2:B6)-14000 = ₹443.38. NPV measures the value after considering that initial cost. Keep it outside Excel’s NPV arguments.

Quarterly extension from the lecture

Receipts of ₹10,000, ₹12,000, ₹15,000, ₹14,000, ₹18,000, ₹20,000, ₹22,000 and ₹25,000 occur at quarter-ends. Interpret the 12% annual quote as nominal with quarterly compounding. The rate is 3% per quarter and there are eight periods.

=NPV(12%/4,10000,12000,15000,14000,18000,20000,22000,25000)

The present value is ₹1,17,085.72. For irregular calendar dates, record the dates explicitly and use =XNPV(rate,values,dates). Its year fractions use actual date differences divided by 365. That is a different timing model from treating every row as one equal period.

Reading connection: Session 2, slides 7–11; Foundations deck, slides 39–40 and 49–50; Book, chapter 2, pp. 31–34, and chapter 3, pp. 42–50.

10Ordinary annuity: adding the factors

An ordinary annuity consists of equal amounts at equal intervals, beginning one period after the valuation date. The shortcut comes from adding the factors for those separate lump sums. Start with the Session 2 example: five year-end payments of ₹10,000 and an annual rate of 8%.

Present value interest factor for an annuity: PVIFA

PVIFA(8%,5) = [1 − 1.08⁻⁵]/0.08 = 3.992710037. Therefore PV = ₹10,000 × 3.992710037 = ₹39,927.10. The factor is the value today of five future payments of ₹1 each.

Future value interest factor for an annuity: FVIFA

FVIFA(8%,5) = [1.08⁵ − 1]/0.08 = 5.86660096. Therefore FV at year 5 = ₹10,000 × 5.86660096 = ₹58,666.01. The first payment earns four years of interest, while the last earns none. Total contributions are ₹50,000, and the excess is accumulated interest.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

For a zero rate, both annuity factors equal n. Avoid dividing by zero in the formula. Use =IF(B3=0,B4,(1-(1+B3)^-B4)/B3) for PVIFA. The corresponding FV formula uses n at zero as well. Excel PV and FV already handle the zero-rate case.

Reading connection: Session 2, slides 12–18; Session 3, slides 3–9; Book, chapter 3, pp. 54–60.

11Monthly savings, quarterly deposits and target payments

The annuity mathematics stays the same when the clock changes. Convert both the rate and the count. Session 3 provides useful examples that connect a present loan amount, an accumulated balance and the payment required for a goal.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

The monthly target case produces ₹4,347.09, with contributions of ₹5,21,651.38 and growth of ₹4,78,348.62. A SIP describes periodic investing. It does not guarantee the assumed investment return. The equation gives the saving required under the stated constant-return scenario.

Loan PV: =-PV(12%/12,36,10000,0,0)
Quarterly FV: =FV(8%/4,20,-25000,0,0)
Annual target payment: =-PMT(8%,10,0,1000000,0)

Reading connection: Session 3, slides 10–15; Foundations deck, slides 42 and 51–54; Book, chapter 6, pp. 126–132.

12Annuity due: payments at the beginning

An annuity due moves each of the same n payments one period earlier. Five annual payments begin at time 0 and end at time 4. If FV is measured at time 5, every payment earns one more period of interest than its ordinary-annuity counterpart. If PV is measured at time 0, every payment is discounted for one fewer period.

For ₹10,000, 8% and five payments, PV due = ₹39,927.100371 × 1.08 = ₹43,121.27. FV due = ₹58,666.0096 × 1.08 = ₹63,359.29. The advantage comes from timing, with no change in the payment amount or quoted rate.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

Rental in advance

Session 3 asks for quarterly rent paid in advance on an asset worth ₹10,00,000, over 20 quarters, at 3% per quarter. Rental = 10,00,000 / [PVIFA(3%,20) × 1.03] = ₹65,257.97. Excel: =-PMT(3%,20,1000000,0,1). If rent were paid in arrears, the payment would be ₹67,215.71. Earlier payment supports a lower amount for the same asset value.

Beginning-of-month and month-end labels refer to the timeline’s boundaries. A SIP placed on the first day of a calendar month is an annuity due only when that date is the beginning of the model’s payment period. Define the dates rather than relying on a label.

Reading connection: Session 2, slides 19–22; Session 3, slides 16–20; Book, chapter 3, pp. 58–60.

13Perpetuity: infinitely many payments, a finite PV

A level perpetuity pays the same amount C every period forever, with the first payment one period from the valuation date. It has no maturity date. At a positive discount rate, each distant payment contributes progressively less to present value.

The annuity factor approaches 1/r as the number of payments grows. It does not approach zero. Each individual discount factor approaches zero, while their sum approaches a finite positive limit.

Scholarship endowment from Session 3

A scholarship of ₹1,00,000 is paid at every year-end. Assume a constant 8% annual return. Required endowment = 1,00,000 / 0.08 = ₹12,50,000. Under the model, its annual earnings exactly fund the payment, leaving the nominal capital intact. Inflation can still erode what the scholarship and capital buy.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

B4 is ₹12,50,000. B5 plus B6 reproduces that value. An immediate first scholarship adds one payment today, giving ₹13,50,000. There is no FV “at the end of forever.” At a specified finite date N, however, we can value the remaining perpetuity and separately accumulate any receipts retained up to that date.

Fixed dividend example

The Session 2 preferred-share example assumes a perpetual ₹8 dividend and a 10% required return. Its model value is ₹8/0.10 = ₹80 per share. Excel uses =8/10%. A real security’s redemption, dividend discretion and credit terms must fit the assumed pattern.

Reading connection: Session 2, slides 23–24; Session 3, slides 21–22; Workbook, printed p. 6; Book, chapter 3, pp. 51–54.

14Growing cash flows: perpetual and finite horizons

Growth changes the amount of successive payments. Discounting changes the date at which those payments are valued. Keep the growth rate g and the discount rate r separate.

Growing perpetuity

C₁ is the first payment one period from today. Session 2 gives C₁ = ₹5,000, g = 5% and r = 12%. The next payments are ₹5,250 and ₹5,512.50. PV = 5,000/(0.12−0.05) = ₹71,428.57. Excel uses =5000/(12%-5%). Session 3 uses ₹60,000 growing at 5%, discounted at 11%, which gives ₹10,00,000.

The ratio (1+g)/(1+r) controls convergence. With positive cash flows and conventional rates above −100%, it must be below 1. If g≥r, the infinite sum has no finite value under this model. A negative answer obtained by mechanically using r−g is not an economic valuation.

Growing finite annuity

Retirement spending usually has a finite planning horizon. Payments may grow with inflation over that horizon. Because there are only n payments, the series can be valued even when g exceeds r.

For payments beginning immediately, call the first amount C₀. The next is C₀(1+g), and there are n payments at times 0 through n−1.

If r=g, use the limit. Ordinary PV = n×C₁/(1+r). Due PV = n×C₀. The finite-horizon formula’s apparent division by zero is a removable mathematical issue. It does not mean the finite series has infinite value.

Excel schedule for any growing series

Set H2=60000 (first payment), H3=11% (return), H4=5% (growth), H5=5 (payments), H6=0 for ordinary or 1 for due. In rows 2–6 enter payment numbers 1–5 in column A.

B2 =A2-$H$6                 [payment time]
C2 =$H$2*(1+$H$4)^(A2-1)   [cash flow]
D2 =C2/(1+$H$3)^B2         [PV contribution]
E2 =C2*(1+$H$3)^($H$5-B2)  [FV at time n]
Copy through row 6.
D7 =SUM(D2:D6)
E7 =SUM(E2:E6)

For the ordinary version, PV is ₹2,42,589.01. The same schedule, with H6=1, values the first ₹60,000 immediately. Excel has no dedicated growing-annuity function, so explicit cash flows provide a transparent check. The bracketed descriptions above are annotations, not part of the Excel formula.

Reading connection: Session 2, slides 25–27; Session 3, slides 23–24; Session 4, slides 9 and 13–16; Book, chapter 8, pp. 177–184.

15Deferred annuities: locating the intermediate value

A deferred annuity starts after a waiting period. Its ordinary-annuity PV sits one period before its first payment. This placement is the bridge to retirement modelling.

Consider an original illustration: three annual receipts of ₹10,000 arrive at times 4, 5 and 6. The annual rate is 8%. First value those three receipts at time 3, using a three-payment ordinary annuity. Then discount that intermediate lump sum from time 3 to time 0.

PV₃ = ₹25,770.97. PV₀ = ₹20,457.83. A separate check discounts the receipts by four, five and six years individually.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

The same payments can be treated as an annuity due valued at time 4, then discounted four periods. That route gives the same PV₀. Write the intermediate date beside every subtotal to avoid discounting from the wrong year.

Reading connection: Book, chapter 6, printed pp. 119–125; Session 3, slide 25; original numerical illustration.

16Loan amortisation: the payment and every balance

A fully amortising loan is paid off through a sequence of instalments. Each instalment covers interest on the opening balance and repays part of the principal. The loan amount equals the PV of its repayments at the contractual periodic rate.

Use the foundations lecture’s loan structure, expressed here as an INR classroom example: ₹5,00,000, a nominal annual rate of 7.5% compounded monthly, 60 month-end payments, no fees and no balloon balance. The currency relabelling is not an exchange-rate conversion.

Substitution: EMI = 5,00,000 × 0.00625/[1−1.00625⁻⁶⁰] = ₹10,018.97. The borrower receives ₹5,00,000 at time 0 and pays the EMI at months 1–60.

The first interest charge is ₹3,125.00. The first principal repayment is ₹6,893.97. In the final month almost all the payment reduces principal. Total payments are ₹6,01,138.46, and total interest is ₹1,01,138.46. These totals describe cash paid. Compare the effective borrowing cost as well when evaluating alternative loan terms.

Excel loan worksheet

Inputs: B2=500000, B3=7.5%, B4=12, B5=5. Calculate B6=B3/B4, B7=B4*B5, and B8=-PMT(B6,B7,B2,0,0). Row 12 contains headings A Month, B Opening, C EMI, D Interest, E Principal, F Closing. Row 13 is month 1.

A13 =1
B13 =$B$2
C13 =$B$8
D13 =B13*$B$6
E13 =C13-D13
F13 =B13-E13
A14 =A13+1
B14 =F13
Copy C13:F13 to C14:F14.
Copy A14:F14 down through row 72 (month 60).
B10 =SUM(E13:E72)-B2
B11 =F72
G13 =-IPMT($B$6,A13,$B$7,$B$2,0,0)
H13 =-PPMT($B$6,A13,$B$7,$B$2,0,0)
Copy G13:H13 through row 72.

B10 and B11 should be near zero. Column G must agree with D, and H with E. Keep the exact calculated EMI in the schedule and round only display. A lender who rounds each actual instalment to paise may adjust the final payment. A payment in advance uses a different first-period schedule: the immediate payment contains no elapsed-period interest.

Reading connection: Foundations deck, slide 34, adapted to INR; Session 2, slide 16; Book, chapter 5, printed pp. 95–104.

17Retirement case 1: Priya’s level withdrawals

Retirement joins two timelines. Savings accumulate before retirement. Withdrawals consume the corpus after retirement. Work backwards from the withdrawal need to the corpus, then to the saving required. The corpus is simultaneously the PV of retirement withdrawals and the FV target for pre-retirement saving.

Session 4 gives Priya’s age as 28 and retirement age as 58. Current expenses are ₹40,000 a month. Inflation before retirement is 7%. Retirement lasts 22 years. The post-retirement annual return is 7.5%. Saving earns a nominal 12% compounded monthly. Withdrawals occur at the start of each retirement year. Assume no existing savings.

Step 1. Project the first annual withdrawal

The first withdrawal is ₹36,53,882.42. This uses 30 years of pre-retirement inflation. For this introductory case, that annual rupee amount then remains level throughout retirement. It does not continue to rise at 7%.

Step 2. Value the withdrawals on the retirement date

Substitution: ₹36,53,882.42 × 10.61719101 × 1.075 = ₹4,17,03,515.15, about ₹4.17 crore. The factor 1.075 reflects the first withdrawal on the retirement date, before any post-retirement interest is earned.

Step 3. Calculate the monthly saving

Monthly saving = ₹11,932.46. The first saving payment is one month from now. “Start saving today” describes the decision to begin the plan, while the mathematical schedule assumes month-end contributions.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

Excel withdrawal schedule

In row 17 place headings A Age, B Opening before withdrawal, C Withdrawal, D After withdrawal, E Interest, F Next opening. Row 18 starts at age 58. Each row represents the following year.

A18 =58
B18 =$B$9
C18 =$B$8
D18 =B18-C18
E18 =D18*$B$6
F18 =D18+E18
A19 =A18+1
B19 =F18
Copy C18:F18 to C19:F19.
Copy row 19 through row 39 (age 79).

The last withdrawal leaves a balance near zero. This is a deterministic teaching model. Level rupee withdrawals lose purchasing power during retirement if inflation continues. Rahul’s next case addresses that limitation explicitly.

Reading connection: Session 4, slides 4–12; Session 3, slides 25–28; Book, chapter 6, printed pp. 126–138.

18Retirement case 2: Rahul’s inflation-linked income

Rahul is 40 and retires at 60. Current annual expenses are ₹10 lakh. Inflation is 6% before and after retirement. He plans 25 withdrawals beginning at age 60, with each later withdrawal 6% larger. The post-retirement return is 9% effective annually. Existing savings of ₹30 lakh and new year-end annual savings earn 10% effective annually before retirement.

Step 1. Calculate the first withdrawal at age 60

The first withdrawal is ₹32,07,135.47. Here the subscript 0 means the retirement date. The withdrawal at age 61 is C₀×1.06. The final withdrawal is at age 84, with exponent 24.

Step 2. Value a growing annuity due

Using full precision gives ₹5,85,28,761.62, about ₹5.85 crore. Multiplication by 1.09 moves the ordinary growing-annuity pattern one period earlier while preserving the same first amount.

Step 3. Check the corpus using a real rate

The real-rate method expresses every retirement payment in purchasing power at age 60. It produces exactly the same corpus as the nominal growing-withdrawal method. Use the real rate’s due factor 1+rreal in this route. Mixing a nominal due factor with a real annuity factor would give a different, incorrect result.

Step 4. Deduct accumulated existing savings and fund the gap

Existing savings grow to ₹2,01,82,499.85. The gap is ₹3,83,46,261.77. The required year-end annual saving is ₹6,69,511.34. This calculation assumes the full ₹30 lakh remains invested and available for retirement.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

Excel retirement cash-flow worksheet

In row 20 use A Age, B Years after retirement, C Withdrawal, D Discount factor, E PV at 60. Rows 21–45 contain the 25 withdrawals.

A21 =60
B21 =A21-60
C21 =$B$9*(1+$B$4)^B21
D21 =(1+$B$6)^-B21
E21 =C21*D21
A22 =A21+1
Copy B21:E21 to B22:E22.
Copy row 22 through row 45 (age 84).
E46 =SUM(E21:E45)
E47 =E46-$B$10

E47 should be near zero. For a balance schedule, start with the corpus, withdraw C₀ immediately, then earn 9% on the remaining funds before the next withdrawal. Repeat with payments growing at 6%. The fund is exhausted after the 25th withdrawal under the assumptions. If the existing savings exceed the model corpus, the required additional saving is zero and the surplus should be shown separately.

Reading connection: Session 4, slides 13–16; Book, chapter 6, printed pp. 130–138; growing-annuity framework from Session 4, slide 9.

19Retirement variations: existing savings, longevity and legacy

The main cases can be modified without inventing a new valuation rule. Each additional requirement becomes another dated cash flow or another unknown in the same balance equation.

Meera: a target with existing savings

Session 3 gives Meera, age 40, a ₹60 lakh target at 60, ₹5 lakh already saved and a 10% annual return. Existing savings first grow to ₹33,63,749.97. The target left for new savings is ₹26,36,250.03. Equal year-end saving = ₹46,027.94.

B2 =500000*(1+10%)^20
B3 =6000000-B2
B4 =-PMT(10%,20,0,B3,0)

Mr. Kumar: what can a corpus pay?

Session 4 gives a corpus of ₹1.5 crore and a nominal annual 9% rate compounded monthly. At r=0.75% and n=300 months, a 25-year month-end pension is ₹1,25,879.45. The capital-preserving level perpetuity pays ₹1.5 crore × 0.75% = ₹1,12,500 per month. Both are nominal amounts under a constant-rate assumption.

25-year pension: =-PMT(9%/12,25*12,15000000,0,0)
Perpetual monthly income: =15000000*9%/12
Months with ₹1.5 lakh withdrawal: =NPER(9%/12,-150000,15000000,0,0)

With ₹1.5 lakh withdrawn monthly, the continuous solution for the period count is 185.53 months, about 15.46 years. A real monthly schedule makes 185 full withdrawals and then a smaller final withdrawal in month 186. If the payment equals the interest earned, the idealised nominal principal remains intact. If payment is lower, the balance grows.

Income plus a legacy

A retiree wants ₹6 lakh at each year-end for 20 years and ₹50 lakh left at the end of year 20. At 8%, the required corpus is the sum of the income annuity and the discounted legacy.

PV of income = ₹58,90,888.44. PV of legacy = ₹10,72,741.04. Total = ₹69,63,629.48.

=-PV(8%,20,600000,5000000,0)

Each scenario is a planning model with a fixed horizon and return. Longevity beyond the selected horizon, variable returns, inflation and unexpected expenses can change the outcome. These uncertainties belong in scenario analysis after the base schedule has been understood and reconciled.

Reading connection: Session 3, slides 29–33; Session 4, slides 17–22; Book, chapter 6, printed pp. 133–138.

20Share valuation: dividends and the required return

The dividend discount model values the dividends a shareholder expects to receive. A future sale price represents the value, at the sale date, of dividends expected after that date. A consistent model therefore avoids adding an unexplained sale price on top of the full perpetual dividend stream.

Dₜ is the dividend per share at time t. kₑ is the required return on equity. P₀ is the model value per share immediately after the dividend at time 0, if any, has been paid. Use dividends, rather than accounting earnings, as the cash flows in this model.

Zero growth

A perpetual ₹8 annual dividend at a 10% required return gives ₹80 per share. This is the level perpetuity already studied. A fixed contractual dividend rate does not remove the possibility of nonpayment.

Constant growth: the Gordon model

Session 2 gives a just-paid dividend D₀=₹4, growth g=5% and kₑ=12%. The next dividend is ₹4×1.05=₹4.20. Value is ₹4.20/(0.12−0.05)=₹60. Dividing the just-paid ₹4 by the spread would omit the first year’s growth.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

B6 is ₹60 and B7 is 12%. In the stable model, required return equals dividend yield plus price growth: kₑ=D₁/P₀+g. This is a model relationship, not a guarantee of a realised holding-period return.

Where kₑ and g may come from

Session 4 uses rf=6%, beta=1.5 and a 6% market risk premium to give kₑ=15%. With a 60% payout, retention is 40%. At a stable ROE of 15%, the corresponding growth estimate is 6%. This growth relationship assumes a sustainable reinvestment and return pattern. Retaining more earnings adds value only when the economics of reinvestment support it. Growth alone is not sufficient.

DDM is most useful when expected distributions can be modelled sensibly. A firm paying no current dividend may still have value because of later distributions. Setting all future distributions to zero while claiming a positive dividend-based value is inconsistent.

Reading connection: Session 2, slides 24–27; Session 4, slides 24–26 and 36; Book, chapter 8, printed pp. 174–187.

21Two-stage and three-stage dividend models

High growth may be reasonable for a limited period. A mature company’s long-run growth assumption is usually different. A staged model values each forecast dividend explicitly, then uses a stable-growth model for dividends beyond the forecast period.

Two-stage lecture case

Session 4 gives D₀=₹10, dividend growth of 20% for four years, then 6% forever. The required return is 12%. First calculate D₁ through D₄. Then calculate D₅ using the stable 6% rate. The terminal value P₄ belongs at year 4 because it values dividends starting in year 5.

PV of explicit dividends = ₹47.67. PV of terminal value = ₹232.81. Total share value = ₹280.48. About 83.0% of this estimate comes from the terminal stage. That share is a result of these assumptions, not a universal percentage for all stocks.

Excel dividend worksheet

Inputs: H2=10 (D₀), H3=20% (initial growth), H4=4 (initial years), H5=6% (stable growth), H6=12% (required return). Row 1 headings: A Year, B Growth, C Dividend, D Discount factor, E Present value. Rows 2–5 represent years 1–4.

A2 =1
B2 =$H$3
C2 =$H$2*(1+B2)
D2 =(1+$H$6)^-A2
E2 =C2*D2
A3 =A2+1
B3 =$H$3
C3 =C2*(1+B3)
Copy D2:E2 to D3:E3.
Copy row 3 through row 5.
H8 =C5*(1+H5)              [D5]
H9 =H8/(H6-H5)            [P4]
H10 =H9/(1+H6)^H4         [PV of P4]
H11 =SUM(E2:E5)+H10       [P0]
H12 =NPV(H6,C2:C5)+H10    [independent Excel layout]

H11 and H12 give ₹280.48. Do not discount P₄ for five years merely because its numerator is D₅. Its location is the year before that dividend. The bracketed labels above explain the cells and are not entered as part of formulas.

Three-stage extension from Session 4

D₀=₹4. Growth is 25% for years 1–3, 15% for years 4–6 and 5% from year 7 onward. The required return remains 12%.

D₇ = ₹12.475928. P₆ = D₇/(12%−5%) = ₹178.23. Discount P₆ six years and add the six dividend PVs. Total value = ₹122.90.

Put years 1–6 in A2:A7. Put 25% in B2:B4 and 15% in B5:B7.
C2 =4*(1+B2)
C3 =C2*(1+B3)               [copy through C7]
D2 =C2/(1+12%)^A2           [copy through D7]
H2 =C7*(1+5%)
H3 =H2/(12%-5%)
H4 =SUM(D2:D7)+H3/(1+12%)^6

The same method supports additional stages or year-specific growth. The stable stage requires kₑ>g. A finite first stage can have g>kₑ because only finitely many payments are being discounted. Test the terminal assumptions: small changes in the required-return-minus-growth spread can materially change estimated value.

Reading connection: Session 4, slides 27–41; Book, chapter 8, printed pp. 181–186.

22Coupon bonds and zero-coupon bonds

A bond brings the patterns together. A plain coupon bond pays a regular coupon and a face amount at maturity. A zero-coupon bond pays only its maturity amount. Identify the contractual cash flows first, then discount them at the yield convention specified for the problem.

Coupon rate determines payment; yield determines value

Adapt the book’s five-year, 10% coupon, 8% yield example to a ₹1,00,000 face amount. Coupons are semiannual, the quoted yield is nominal compounded semiannually, and valuation is immediately after a coupon date. There are ten payments, each coupon is ₹5,000 and the periodic yield is 4%.

Coupon PV = ₹40,554.48. Principal PV = ₹67,556.42. Price = ₹1,08,110.90. The price exceeds face value because the 10% coupon is attractive relative to the 8% required yield under the same periodic convention.

Excel working

Enter the following in a blank worksheet. Values shown are rounded for reading. Keep formula cells at full precision.

B12 and B13 agree at ₹1,08,110.90. B14 is approximately 108.1109 per ₹100 face. B15 recovers 8%. If the coupon rate equals the yield on the same basis, the bond prices at par on a coupon date. A yield above the coupon rate produces a discount.

Zero-coupon bond

For ₹1,00,000 due in five years and a 10% nominal annual yield compounded semiannually, price = 1,00,000/1.05¹⁰ = ₹61,391.33. Excel: =-PV(10%/2,5*2,0,100000,0). The gain to redemption is ₹38,608.67, conditional on receiving payment as promised.

Even though the zero pays no coupons, its quoted yield still has a compounding convention. If the quote instead meant 10% effective annually, price would be 1,00,000/1.10⁵ = ₹62,092.13. These are different rate quotes. The contract or market convention determines the appropriate interpretation.

Excel cash-flow audit for the coupon bond

On the same worksheet, row 19 headings are A Period, B Coupon, C Principal, D Total cash flow, E Discount factor, F PV. Rows 20–29 represent periods 1–10.

A20 =1
B20 =$B$7
C20 =IF(A20=$B$9,$B$2,0)
D20 =B20+C20
E20 =(1+$B$8)^-A20
F20 =D20*E20
A21 =A20+1
Copy B20:F20 to B21:F21. Copy row 21 through row 29.
F30 =SUM(F20:F29)
F31 =F30-$B$12

F31 should be near zero. These examples assume valuation on a coupon date, no default and no embedded options. Settlement between coupon dates requires accrued interest and a day-count convention. A single yield is also a summary: a spot-rate approach discounts each maturity with its corresponding rate. Holding to maturity does not guarantee a compound realised return equal to quoted YTM unless coupons can be reinvested appropriately and promised payments arrive.

Reading connection: Book, chapter 7, printed pp. 142–158, especially examples 7.1 and zero-coupon discussion pp. 152–154; INR adaptation and original cash-flow worksheet.

23Practice and a reusable modelling routine

For each question, record the valuation date, draw the cash flows and name the pattern. Work through the substitution before entering Excel. Then build a second route: direct cash-flow sums against the shortcut, or a balance schedule against the financial function.

A model you can audit

A numerical result becomes useful when you can explain why the cash-flow pattern fits, why the rate matches it and what assumption would change the answer. Rebuild a loan schedule from a blank sheet. Change Priya’s payments from level to indexed. Recalculate a dividend model with a lower stable growth rate. Each variation strengthens the same underlying skill.

Nothing can beat self-learning and practice.

Reading connection: Workbook, printed p. 8; Session 3, slides 36–39; Session 4, slides 43–45; original concept questions.

24Sources and reading guide

This article is an original synthesis for The Mountain Path Academy. Its classroom examples primarily follow Prof. V. Ravichandran’s supplied lectures and workbook. The textbook supports the mathematical development and the applications. Diagrams have been newly drawn for this article. No textbook page or slide image is reproduced.

Book PDF page numbers are 22 greater than printed page numbers in the chapters used here. The workbook’s PDF page numbers are one greater than its printed page numbers. Slide numbers count from the first slide, including covers. The separate editorial note records numerical and wording discrepancies found during source review. This draft recalculates amounts from the stated inputs and retains unrounded intermediate values.

Excel notation follows the financial-function conventions described in the book and checked against Microsoft’s function documentation. Functions and formulas are teaching tools: the model’s dates and assumptions determine whether a result answers the intended question.

Reading connection: All six supplied sources.