CHAPTER 3



Chapter 5

the time value of money

QUESTIONS AND PROBLEMS

Questions for Discussion

1. (a) The statement that $100 today is equivalent to $110 one year

hence assumes that all investors can both borrow and lend

under identical terms, with no transaction costs or taxes, at an

effective risk-less rate of 10 percent per year.

Note that if a person invests $100 for one year and

receives $10 interest plus the initial $100, we say that the

effective rate of interest was 10 percent. Under annual

compounding, the nominal rate would be 10 percent as well,

but if we compounded quarterly, for example, an effective rate

of 10 percent implies a nominal rate of 9.645 percent.

b) The assumptions in part (a) above may be a reasonable

approximation for large and well-established firms, but for

smaller businesses they may not be. While a large firm may

have the market power and investor confidence to be able to

borrow at low rates, a smaller firm often can only acquire funds

at rates that are substantially above the rate obtained on, for

example, bank savings. Thus, smaller firms face larger

interest-rate spreads. Note, also, that the statement is only

correct if we assume that the cash flows entail no risk.

Comparing risky cash flows which occur across different time

periods is much more problematic.

2. (a) The entries in Table 4 are equal to the sum of the inverse of the

entries in Table 1 over the period of the annuity. Thus:

[pic]

b) The easiest method of finding the present value of $1 to be

received in 55 years is based on the fact that:

(1 + i)n = (1 + i)a(1 + i)b where n = a + b

Hence, if the discount rate is 10%, we have, for example:

[pic]

Using a calculator, we find (1.1)55 = 189.06, with the difference

due to rounding.

To find the present value of $1, then, we divide by the compound interest factor of 189.13 and obtain

[pic]

Of course, we could have also divided by [(1.1)25(1.1)30] or by

(1.1)5 eleven times and obtained identical results.

3. (a) The nominal rate is the rate quoted on the debt contract,

relating, for example, future interest payments to the amount of

principal borrowed. The effective interest rate is given as the

internal rate of return, which equates the present value of future

net cash flows with the net amount originally received. The two

may differ because of transaction costs (e.g., service charges)

or interest being compounded more than once a year. Clearly,

the effective interest rate is much more relevant for financial

decision making than the nominal rate.

b) Quoting only a nominal rate is strictly a marketing technique.

Where the nominal rate is lower than the effective rate, quoting

the nominal rate makes the loan appear more attractive, at

least to unsophisticated borrowers. One could classify this

practice as “false and misleading advertising”, and more recent

consumer legislation forces financial institutions to quote

effective interest rates on consumer loans.

c) Service charges and commissions increase the effective

interest cost of a bank loan. As the amount received

decreases, it takes a higher internal rate of return to equate the

present value of future repayments with the initial cash inflow.

4. If there were no limitations on borrowing and lending money at 10 percent and we ignore risk considerations, it would not be reasonable for a wealth-maximizing firm or individual to have a time value of money that differs from this market rate. An individual or firm whose time preference for money was 15 percent, for example, would forego an investment that offered 11 percent even though this investment is attractive (the investor can borrow money at 10 percent, invest it at 11 percent and earn a 1 percent return). Similarly, an individual or firm with a time preference for money of only 6 percent, might accept an investment that offered only 8 percent, although that would not be sensible given the opportunities in the market where the rate is 10 percent.

In subsequent chapters, we will see that when we relax the assumptions initially put forth, these results may not hold.

additional PROBLEMS

Problem 5-1(a)

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Problem 5-1(b)

[pic]

[pic][pic]

Problem 5-2

2. An annuity makes 10 annual payments of $1000 starting three years from now. What is the present value if the interest is 10%?

[pic]

The value of the annuity is $6144.57 as of two years from now (since the first payment is three years from now and the formula assumes that the payments are at the end of the year). Need to now discount this amount back an extra two years to get the present value in today's terms:

[pic]

Problem 5-3

Total needed in 10 years:

[pic]

Equivalent amount needed today (present value):

[pic]

Deposits needed to equal this present value:

[pic]

Problem 5-4(a)

[pic]

Problem 5-4(b)

[pic]

Problem 5-4(b)

[pic]

Problem 5-5

|Frequency | |Effective rate |

|of compounding | | |

|2/yr |[pic] |r = 14.49% |

|3 |[pic] |R = 14.66 |

|5 | |R = 14.80 |

|8 | |r = 14.89 |

|10 | |r = 14.92 |

|20 | |r = 14.97 |

|50 | |r = 15.00 |

|( |[pic] |r = 15.03 |

Graphically, we have:

Problem 5-6(a)

FV = 4,000(1.08)(1.11)6 + 4,000(1.11)6 + 4,000(1.11)5

+ 4,000(1.11)4 + 1,500(1.11)3

+ 1,500(1.11)2 + 1,500(1.11)

+ 1,500

= $35,438.93

Alternatively,

[pic]

Problem 5-6(b)

[pic]

Problem 5-7(a)

What is the present value of $7000 to be received at the end of each year for six years if the interest rate to be used for discounting is 10% compounding annually?

[pic]

Problem 5-7(b)

What is the present value of $3500 to be paid at the beginning of every six month period for six years if the nominal rate is 10% compounded semi-annually?

[pic]

Note: With semi-annual compounding, the discount rate becomes 10%/2 = 5% per period for 6 ( 2 = 12 periods, with the additional factor (1 + i/m) shifting the annuity one 6-month period closer because of payments at the beginning of each period.

Problem 5-8

The present value of costs associated with painting wooden outside walls every 5 years is as follows:

PV = 2,000 + [pic]

= 2,000 + 3,500+ 2,773.64 + 2,198.03 + 1,741.88 + 1,380.39

= $13,593.94

which is higher than the costs of installing vinyl siding. Therefore, installing vinyl siding should be the preferred alternative.

Problem 5-9(a)

9.

(a) If a bank pays 6% interest compounded annual on a $100 deposit, what will be the value of this (one time) deposit at the end of 10 years?

FV = 1,000(1.06)10 = $1,790.85

Problem 5-9(b)

(b) If another bank pays 6% interest on the same $1000 deposit, but compounds interest rate quarterly, what will be the value of this deposit at the end of 10 years?

FV = 1,000(1 +[pic])10 ( 4 = $1,814.02

The effective annual interest rate is different from 6% because of the interest to accrue to interest payments. It can be found from:

[pic] = 0.064

Problem 5-9(c)

[pic]

Problem 5-10

With monthly compounding, we obtain at the end of 5 years:

[pic]

which provides an effective annual yield of:

[pic]

With continuous compounding we obtain at the end of 5 years:

[pic]

which provides an effective annual yield of:

[pic]

The $9.55 difference in future values and .028 percentage points difference in effective yields are minimal. Continuous compounding is not a significant advantage over monthly compounding.

Problem 5-11

******This question is the highest point of Chapter 5

My way of solving this problem:

I assume that the mortgage payments are made bi-weekly and so are the computation of interest rates by the bank: I assume that when the debtor makes a repayment of annuities (26 times a year), the creditor calculates the interest rates(26 times a year).

[pic] biweekly

Please, recall that the above A is an annual Annuity or mortgage repayment amount: A/26 is a bi-weekly mortgage repayment.

Bi-weekly mortgage repayment = A/26 = $ 1068 (in approximation)

The Textbook way of solving this problem:

Mortgages in Canada are always compounded semi-annually. Therefore, m=2 in the effective rate formula. Also, with bi-weekly payments, f=26 in the same formula (26 payments per year).

Therefore, by using the formulas for the effective annual interest rate and the effective period interest rate respectively:

Effective annual interest rate is:

[pic]

Effective bi-weekly interest rate is:

[pic]

Let’s use the formula: This time the formula should be in terms of bi-weekly perspectives, not annual perspectives (for the annuity part and the interest rate part). Suppose that the bi-weekly annuity or the bi-weekly mortgage repayment is $ A’, and the bi-weekly effective interest rate is r’. As there are 26 times payments a year, there will be 25X26 times of bi-weekly payments throughout 25 years of period.

Therefore, the formula, as seen from the bi-weekly time perspectives, is

[pic]

Please, note that in the above formula, A’ is a bi-weekly mortgage repayment or Annuity (as opposed A = annual mortgage repayment or annual annuity in the formula of my way), and r’ is the bi-weekly interest rate. n’ = nx 26 is the number of times of bi-weekly payments made throughout the period of 25 years.

(b)

After 26 payments, there will be

(25 x 26) – 26 = 624 payments remaining to be made

[pic]

(c) After 25 payments, the balance remaining would be:

[pic]

Interest over the following two weeks would be:

($247,830.30)(0.0039) = $977.39

So, with the 26th payment, $977.39 goes to interest and (1068.69 – 977.39) = $91.30 goes to reduce the balance outstanding on the mortgage.

Problem 5-12(a)

A five year loan of $35000 is to be repaid in five annual payments of $10000. What is the effective interest rate on this loan?

[pic]

From tables, n = 5, (a513%) = 3.517

(a514%) = 3.433

Interpolating, we have:

[pic] yielding r = 13.20%

Using a computer, the exact solution is found to be r = 13.2016%.

Problem 5-12(b)

[pic]

From tables, (a203%) = 14.88

(a204%) = 13.59

Interpolating, we have:

[pic] yielding r = 3.68% per quarter year

Thus, (1 + .0368)4 = 1.1555 yielding an effective annual interest rate of 15.55%.

Using a computer, the exact solution is r = 3.6670%, yielding an effective annual interest rate of 15.4948%.

Problem 5-12(c)

[pic]

From tables, (a513%) = 3.517

(a514%) = 3.433

Interpolating, we have:

[pic] yielding r = 13.41%

Using a computer, the exact solution is r = 13.4080%.

Problem 5-13(a)

[pic]

Problem 5-13(b)

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Problem 5-14(a)

Present value of annuity of 15 years at 10% with payments at the beginning of the year:

[pic]

[pic]

Problem 5-14(b)

Present value of additional annuity, discounted to the time of retirement:

[pic]

Amount to be deposited:

[pic]

Mr. Smith would have to deposit an additional $104.57 per year.

Problem 5-15

PV10 = (1 + .05)-10 = $0.61

PV30 = (1 + .05)-30 = $0.23

Problem 5-16(a)

16. Meena Lele

Alternative A

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With this alternative, Ms. Lele's annual income would increase by $2,155.63.

Alternative B

Selling expense of house = 265,000(.05) = $13,250

Net proceeds from sale of house = 265,000 - 13,250 = $251,750

The annual amount that could be drawn if this amount were invested at 8% is:

[pic]

Incremental net income after deduction for rental accommodation:

= 23,594.19 - 19,000

= $4,594.19

Ms. Lele should choose Alternative B as it gives her the greater increase in income.

Problem 5-16(b)

Ms. Lele would have to consider that under Alternative A she would retain an equity interest in her home. In an inflationary environment, with continually increasing real estate prices, this equity interest could prove to be quite valuable.

Under Alternative A, Ms. Lele would have to either start paying back her loan if she lives past the twenty-five years and keeps the house or sell the house at that time and pay off the loan with the proceeds. If house prices have risen an average of 3% per year (implying that in 25 years her house is worth more than twice its current value) then she could sell at that time and invest the remaining proceeds to provide continued income. The drawback is that for the next 25 years she would have a much lower annual income than she could receive under Alternative B.

Under Alternative B, however, after 25 years Ms. Lele will have no form of income on which to live (or to supplement other income), and she may have difficulty supporting herself without that income. While she would enjoy a higher income for 25 years, she may regret it if she lives longer.

Problem 5-17

Effective interest rate per month:

[pic]

Mortgage payment:

[pic]

The next problem is figuring out when the mortgage will be paid off. Normally, to figure out the balance remaining you would take the present value of the mortgage payments that are remaining to be paid. In this case, because of the extra $2000 per year you are paying, this might become difficult. The reason that the extra payment makes the calculations more complicated (and very tricky to do without a spreadsheet) is that you do not know how many of the extra payments will be made. For instance, if the mortgage were fully paid off half way through the 17th year, then you would not use the 18th bonus payment on the mortgage. One approach (there are others) is to calculate the PV (as of the day you buy the house) of the payments that had been made up to each month, including the PV of any extra $2000 payments. Then take $155,000 (the PV of the mortgage) and subtract the PV of the payments that had been made so far. This difference is a PV (as of the day the house is purchased) so then compound it forward the appropriate number of months and that gives the balance remaining. You can then simply look through the months until you find in which month the balance remaining goes to zero. Some of the numbers are:

|Months since bought house |Extra payment |Cumulative PV of regular |PV of cumulative extra |Balance remaining at end of|

| | |payments |payments |month |

|1 | |$1,426.69 |$ - |$154,888.59 |

|2 | |$2,841.27 |$ - |$154,776.22 |

|3 | |$4,243.83 |$ - |$154,662.90 |

|4 | |$5,634.49 |$ - |$154,548.60 |

|5 | |$7,013.33 |$ - |$154,433.32 |

|6 | |$8,380.47 |$ - |$154,317.06 |

|7 | |$9,736.00 |$ - |$154,199.80 |

|8 | |$11,080.01 |$ - |$154,081.53 |

|9 | |$12,412.62 |$ - |$153,962.25 |

|10 | |$13,733.90 |$ - |$153,841.95 |

|11 | |$15,043.97 |$ - |$153,720.63 |

|12 |$2,000.00 |$16,342.91 |$1,805.45 |$151,598.26 |

|13 | |$17,630.82 |$1,805.45 |$151,457.71 |

|14 | |$18,907.80 |$1,805.45 |$151,315.96 |

|. |. |. |. |. |

|. |. |. |. |. |

|. |. |. |. |. |

|203 | |$138,258.71 |$14,950.63 |$10,112.52 |

|204 |$2,000.00 |$138,511.34 |$15,301.77 |$6,760.22 |

|205 | |$138,761.83 |$15,301.77 |$5,379.20 |

|206 | |$139,010.18 |$15,301.77 |$3,986.36 |

|207 | |$139,256.43 |$15,301.77 |$2,581.59 |

|208 | |$139,500.59 |$15,301.77 |$1,164.79 |

|209 | |$139,742.67 |$15,301.77 |($264.14) |

After 209 months (17 years and 5 months) the balance would be negative, indicating that the mortgage is fully paid off.

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