Chapter 3

The time value of money

 

Solutions to questions

1.    A rate of return is the ratio of net cash inflows to net cash outflows produced by a financial contract. It is often expressed as a percentage. The ‘financial contract’ involved may be, for example, an investment in shares, land or bonds. An interest rate is a rate of return produced by debt of one form or another. Thus, an interest rate is one type of rate of return.
2.   Simple interest is a method of calculating interest in which the interest is computed on the basis of the original sum borrowed. Compound interest is a method of calculating interest in which interest is computed on the basis of the original sum borrowed, plus interest owing but unpaid at the date of calculation.
3.    The statement is true. The three different terms—‘present value’, ‘price’ and ‘principal’—all refer to the value of a financial contract at a given date (typically, today). ‘Present value’ tends to be used when the valuation is the result of a discounting procedure; ‘price’ tends to be used when the valuation is the result of a market transaction in a security, and ‘principal’ tends to be used when the valuation refers to an amount lent by way of a standard loan. However, these circumstances are not independent. For example, a price may be offered and agreed to because a discounting procedure indicates to market participants that the price is the correct valuation.
4.    The term ‘nominal interest rate’ can mean an interest rate where interest is charged more frequently than the quoted period. For example, ‘6 per cent per annum payable quarterly’ is a nominal interest rate because interest is charged quarterly—that is, four times each year—but the quoted period is a year. The term ‘nominal interest rate’ can also mean an interest rate before taking out the effects of inflation. For example, if a lender receives interest at the rate of 5 per cent per annum over a period of time in which the inflation rate is 7 per cent, the lender’s real interest received in this case is negative, but the nominal return is 5 per cent.

Solutions to problems

 
1.    Nicholas’s interest is:
 
2.    Nicholas’s principal is now $2 132.50. So the interest is:
 
3.    Using Equation 3.2:
      
4.    Using Equation 3.2:
        
            where d is the loan term in days.
       Solving,
        
       \ d = 47
       The loan term was 47 days.
5.    Days between 2 April and 16 May = 28 (April) + 16 (May) = 44
       Therefore, using simple interest, the repayment is S where:

 
6.        Following the Australian conventions set out in Section 3.3.4, we require the exact number of days in the term of the deposit. There are 23 days remaining in February, then 31 in March, 30 in April and 5 in May; the total is 89 days. Using simple interest, the amount of interest is given by:
 


 
 
8.    (a)   Using Equation 3.4:
                    
              The interest component is $1 158.92.
 
       (b)   Interest each year is 0.08 × $1 000 = $80. Ten years' interest is 10 × $80 = $800. The interest in part (a) includes ‘interest on interest’, as each year's interest is reinvested. In part (b), interest is withdrawn each year, so there is no compounding effect.
 
9.    Using Equation 3.4:
       S   =   P (1 + i)ⁿ
            =   $65 000 (1.147)³
            =   $98 085.23
 
10.  (a)   Using Equation 3.4:
                        S   =   P (1 + i)ⁿ
                            =   $87 000 (1.0735)³
                            =   $107 628.03
 
       (b)   Using Equation 3.4:
                        S   =   P (1 + i)ⁿ
                            =   $87 000 (1.0735)⁶
                            =   $133 147.05
 
11.  (a)   (i)      The after-tax interest rate is:
                                        (1 – 0.45)(12.4 per cent)
                                    =   6.82 per cent
                        Using Equation 3.4:
                               S   =   $10 000 (1.0682)¹⁰
                                    =   $19 343.08
              (ii)      The after-tax interest rate is:
                                    =   (1 – 0.30) (12.4 per cent)
                                    =   8.680 per cent
                        So,  S   =   $10 000 (1.0868)¹⁰
                                    =   $22 987.74
              (iii)     The after-tax interest rate is:
                                    =   (1 – 0.15) (12.4 per cent)
                                    =   10.54 per cent
                        So,  S   =   $10 000 (1.1054)¹⁰
                                    =   $27 239.22
              (iv)            S   =   $10 000 (1.124)¹⁰
                                    =   $32 185.71
17.  Parts (a) to (d) of this question are answered using Equation 3.6, which is:
        
        
        
        
       Part (e) is answered using Equation 3.11: