Question

The present value of K due in 2 years is 678.80. If the force of discount (same as force of interest ) is cut by a factor of 1/4, that present value would be 766.30. What would the present value be if the rate of discount d is cut by a factor of 1/5?

Answer 1

When initial discount rate be d%, then

PV = K x P/F(d%, 2)

678.8 = K x (1 + d)^-2 ............[1]

When discount rate = d - (1/4)d = (3/4)d = 0.75d,

766.3 = K x (1 + 0.75d)^-2 ............[2]

So,

(2) / (1) yields:

[(1 + 0.75d)^-2] / [(1 + d)^-2] = 1.1289

[(1 + d) / (1 + 0.75d)]² = 1.1289

Taking square root,

(1 + d) / (1 + 0.75d) = 1.0625

1 + d = 1.0625 + 0.7969d

0.2031d = 0.0625

d = 0.3077

d = 30.77%

Using (1),

678.8 = K x (1.3077)^-2

K = 1160.80

When new discount rate = d - (1/5)d = (4/5) x d = 0.8 x 0.3077 = 0.2462,

Present value = 1160.80 x (1.2462)^-2 = 1160.80 x 0.6440 = $747.50